16,893 results on '"Monotone polygon"'
Search Results
2. An Overall Evaluation on Benefits of Competitive Influence Diffusion
- Author
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Weili Wu, Yapu Zhang, and Jianxiong Guo
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Mathematical optimization ,Information Systems and Management ,Social network ,Degree (graph theory) ,business.industry ,Computer science ,Estimator ,Maximization ,Submodular set function ,Monotone polygon ,Viral marketing ,business ,Value (mathematics) ,Information Systems - Abstract
Influence maximization (IM) is a representative and classic problem that has been studied extensively before. The most important application derived from the IM problem is viral marketing. Take us as a promoter, we want to get benefits from the influence diffusion in a given social network, where each influenced (activated) user is associated with a benefit. However, there is often competing information initiated by our rivals that diffuses in the same social network at the same time. Consider such a scenario, a user is influenced by both our information and our rivals' information. Here, the benefit from this user should be weakened to a certain degree. How to quantify the degree of weakening Based on that, we propose an overall evaluation on benefits of influence (OEBI) problem. We prove the objective function of the OEBI problem is not monotone, not submodular, and not supermodular. Fortunately, we can decompose this objective function into the difference of two submodular functions and adopt a modular-modular procedure to approximate it with a data-dependent approximation guarantee. Because of the difficulty to compute the exact objective value, we design a group of unbiased estimators by exploiting the idea of reverse influence sampling.
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- 2023
3. Resilient Service Provisioning for Edge Computing
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Shaojie Tang, Haipeng Dai, Haisheng Tan, Chao Dong, Yuben Qu, Fan Wu, and Dongyu Lu
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Mathematical optimization ,Computer Networks and Communications ,Computer science ,Matroid ,Computer Science Applications ,Submodular set function ,Constraint (information theory) ,Monotone polygon ,Hardware and Architecture ,Knapsack problem ,Signal Processing ,Constant (mathematics) ,Time complexity ,Edge computing ,Information Systems - Abstract
We study the problem of Resilient Service Provisioning for Edge computing (RSPE), i.e., how to determine a service placement strategy to maximize the expected overall utility by service provisioning, in the presence of uncertain service failures. RSPE is extremely challenging to tackle, because the explicit expression of its objective function is difficult to obtain, and it is a resilient max-min problem subject to knapsack constraints, which is unexplored so far and cannot be addressed by existing resilient optimization techniques. We first explore the potential properties of the implicit objective function, and reveal that it is monotone submodular under certain conditions. We further prove that the knapsack constraints form a q-independence system constraint, where q>0 is a constant related to the constraints. We propose two novel solutions for the general RSPE and homogeneous case, respectively. Firstly, for the general problem, we propose a “two-step greedy” algorithm achieving constant approximation ratio within polynomial time. Secondly, for the homogeneous case where one of the knapsack constraints reduces to a matroid constraint, we propose an improved “first-greedy-then-local search” polynomial time algorithm achieving better approximation ratio than the previous one. Both extensive simulations and field experiments validate the effectiveness of our proposed algorithms.
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- 2023
4. Modified Dai-Yuan iterative scheme for nonlinear systems and its application
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Mohammed Yusuf Waziri, Aliyu Mohammed Awwal, Abubakar Sani Halilu, and Kabiru Ahmed
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Nonlinear system ,Signal processing ,Control and Optimization ,Algebra and Number Theory ,Monotone polygon ,Iterative method ,Computer science ,Applied Mathematics ,Convergence (routing) ,Projection method ,Regular polygon ,Applied mathematics ,Image (mathematics) - Abstract
By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.
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- 2023
5. Ordinal patterns in clusters of subsequent extremes of regularly varying time series
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Marco Oesting and Alexander Schnurr
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FOS: Computer and information sciences ,Statistics and Probability ,Series (mathematics) ,010102 general mathematics ,Economics, Econometrics and Finance (miscellaneous) ,Asymptotic distribution ,Estimator ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,Methodology (stat.ME) ,010104 statistics & probability ,Monotone polygon ,Data point ,FOS: Mathematics ,Cluster (physics) ,Probability distribution ,Limit (mathematics) ,Statistical physics ,0101 mathematics ,Engineering (miscellaneous) ,Statistics - Methodology ,Mathematics - Abstract
In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine., Deutsche Forschungsgemeinschaft, Projekt DEAL
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- 2023
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6. New bounds on guarding problems for orthogonal polygons in the plane using vertex guards with halfplane vision.
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Daescu, Ovidiu and Malik, Hemant
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COMMERCIAL art galleries , *VISION , *POLYGONS - Abstract
• Guarding disjoint monotone orthogonal polygons in a plane. • Guarding a rectangular polygon with orthogonal holes. • Rectilinear Art Gallery Problem. • Guards with restricted visibility. Given a set F of k disjoint monotone orthogonal polygons with a total of m vertices, we present bounds on the number of vertex guards required to guard the free space and the boundaries of the polygons in F when the range of vision of each guard is bounded by 180∘ (the region in front of the guard). When the orthogonal polygons are axis aligned we prove that m 2 + ⌊ k 4 ⌋ + 4 vertex guards are always sufficient. When the orthogonal polygons are arbitrary oriented, we show that m 2 + k + 1 vertex guards are sometimes necessary and conjecture the bound is tight. [ABSTRACT FROM AUTHOR]
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- 2021
- Full Text
- View/download PDF
7. Path-based Stability Analysis for Monotone Control Systems on Proper Cones
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Yu Kawano, Bart Besselink, and Systems, Control and Applied Analysis
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Monotone systems ,Stability criteria ,Asymptotic stability ,Paths ,Trajectory ,Topology ,Stability (probability) ,Cooperative systems ,Computer Science Applications ,Tools ,Circuit stability ,Monotone polygon ,Control and Systems Engineering ,Control system ,Path (graph theory) ,Nonlinear systems ,Electrical and Electronic Engineering ,Proper Cones ,Stability ,Mathematics ,Lyapunov methods - Abstract
In this paper, we study positive invariance and attractivity properties for nonlinear control systems which are monotone with respect to proper cones. Monotonicity simplifies such analysis for specific sets defined by the proper cones. Instead of Lyapunov functions, a pair of so called paths in the state space and input space play important roles. As applications, our results are utilized for analysis of asymptotic stability and also input-to-state stability on proper cones. The results are illustrated by means of examples.
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- 2022
8. Aggregation of indistinguishability operators
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Jorge Elorza, Carlos Bejines, M.J. Chasco, and Sergio Ardanza-Trevijano
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0209 industrial biotechnology ,Pure mathematics ,Property (philosophy) ,Logic ,TheoryofComputation_GENERAL ,02 engineering and technology ,Function (mathematics) ,020901 industrial engineering & automation ,Monotone polygon ,Operator (computer programming) ,Artificial Intelligence ,If and only if ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Mathematics - Abstract
The paper studies the aggregation of pairs of T-indistinguishability operators. More concretely we address the question whether A ( E , E ′ ) is a T-indistinguishability operator if E , E ′ are T-indistinguishability operators. The answer depends on the aggregation function, the t-norm T, and the chosen T-indistinguishability operators. It is well-known that an aggregation function preserves T-transitive relations if and only if it dominates the t-norm T. We show the important role of the minimum t-norm T M in this preservation problem. In particular we develop weaker forms of domination that are used to provide characterizations of T M -indistinguishability preservation under aggregation. We also prove that the existence of a single strictly monotone aggregation that satisfies the indistinguishability operator preservation property guarantees all aggregations to have the same preservation property.
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- 2022
9. A k-Hop Collaborate Game Model: Extended to Community Budgets and Adaptive Nonsubmodularity
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Weili Wu and Jianxiong Guo
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Social and Information Networks (cs.SI) ,FOS: Computer and information sciences ,Mathematical optimization ,Computer science ,media_common.quotation_subject ,Computer Science - Social and Information Networks ,Object (computer science) ,Computer Science Applications ,Submodular set function ,Human-Computer Interaction ,Units of measurement ,Promotion (rank) ,Monotone polygon ,Computer Science - Computer Science and Game Theory ,Control and Systems Engineering ,Simple (abstract algebra) ,Revenue ,Electrical and Electronic Engineering ,Hop (telecommunications) ,Software ,Computer Science and Game Theory (cs.GT) ,media_common - Abstract
Revenue maximization (RM) is one of the most important problems on online social networks (OSNs), which attempts to find a small subset of users in OSNs that makes the expected revenue maximized. It has been researched intensively before. However, most of exsiting literatures were based on non-adaptive seeding strategy and on simple information diffusion model, such as IC/LT-model. It considered the single influenced user as a measurement unit to quantify the revenue. Until Collaborate Game model appeared, it considered activity as a basic object to compute the revenue. An activity initiated by a user can only influence those users whose distance are within k-hop from the initiator. Based on that, we adopt adaptive seed strategy and formulate the Revenue Maximization under the Size Budget (RMSB) problem. If taking into account the product's promotion, we extend RMSB to the Revenue Maximization under the Community Budget (RMCB) problem, where the influence can be distributed over the whole network. The objective function of RMSB and RMCB is adatpive monotone and not adaptive submodular, but in some special cases, it is adaptive submodular. We study the RMSB and RMCB problem under both the speical submodular cases and general non-submodular cases, and propose RMSBSolver and RMCBSolver to solve them with strong theoretical guarantees, respectively. Especially, we give a data-dependent approximation ratio for RMSB problem under the general non-submodular cases. Finally, we evaluate our proposed algorithms by conducting experiments on real datasets, and show the effectiveness and accuracy of our solutions.
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- 2022
10. An axiomatic re-characterization of the Kemeny rule
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Mohsen Pourpouneh, Burak Can, Ton Storcken, Data Analytics and Digitalisation, RS: FSE DACS Mathematics Centre Maastricht, QE Math. Economics & Game Theory, RS: GSBE Theme Conflict & Cooperation, and RS: GSBE MORSE
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and Voting Behavior ,d72 - Political Processes: Rent-seeking ,Political Processes: Rent-seeking ,Lobbying ,Axiomatic characterization ,Social Choice ,Clubs ,Committees ,Associations ,d72 - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior ,Aggregation problem ,Decision rule ,Analysis of Collective Decision-Making: General ,Characterization (mathematics) ,Elections ,Monotone polygon ,d71 - "Social Choice ,Associations" ,Preference aggregation ,d70 - Analysis of Collective Decision-Making: General ,Legislatures ,Kemeny rule ,General Economics, Econometrics and Finance ,Mathematical economics ,Axiom ,Mathematics - Abstract
The Kemeny rule is one of the well studied decision rules. In this paper we show that the Kemeny rule is the only rule which is unbiased, monotone, strongly tie-breaking, strongly gradual, and weighed tournamental. We show that these conditions are logically independent.
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- 2022
11. Optimal Pricing with a Single Point
- Author
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Achraf Bahamou, Omar Besbes, and Amine Allouah
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FOS: Computer and information sciences ,History ,Mathematical optimization ,Polymers and Plastics ,Computer science ,Computer Science - Information Theory ,Information Theory (cs.IT) ,Strategy and Management ,Management Science and Operations Research ,Industrial and Manufacturing Engineering ,Oracle ,Value of information ,Randomized algorithm ,Product (business) ,Set (abstract data type) ,Monotone polygon ,Computer Science - Computer Science and Game Theory ,Value (economics) ,Revenue ,Business and International Management ,Computer Science and Game Theory (cs.GT) - Abstract
Historical data are typically limited. We study the following fundamental data-driven pricing problem. How can/should a decision maker price its product based on data at a single historical price? How valuable is such data? We consider a decision maker who optimizes over (potentially randomized) pricing policies to maximize the worst-case ratio of the garnered revenue compared to an oracle with full knowledge of the distribution of values, when the latter is only assumed to belong to a broad nonparametric set. In particular, our framework applies to the widely used regular and monotone nondecreasing hazard rate (mhr) classes of distributions. For settings where the seller knows the exact probability of sale associated with one historical price or only a confidence interval for it, we fully characterize optimal performance and near-optimal pricing algorithms that adjust to the information at hand. The framework we develop is general and allows to characterize optimal performance for deterministic or more general randomized mechanisms and leads to fundamental novel insights on the value of data for pricing. As examples, against mhr distributions, we show that it is possible to guarantee 85% of oracle performance if one knows that half of the customers have bought at the historical price, and if only 1% of the customers bought, it still possible to guarantee 51% of oracle performance. This paper was accepted by David Simchi-Levi, revenue management and market analytics. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2023.4683 .
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- 2023
12. Discrete IV d-Choquet integrals with respect to admissible orders
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Humberto Bustince, Daniel Paternain, Mikel Galar, Zdenko Takáč, Mikel Uriz, Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa. ISC - Institute of Smart Cities, Universidad Pública de Navarra. Departamento de Estadística, Informática y Matemáticas, Universidad Pública de Navarra. Departamento de Ingeniería Eléctrica, Electrónica y de Comunicación, Nafarroako Unibertsitate Publikoa. Estatistika, Informatika eta Matematikak Saila, Nafarroako Unibertsitate Publikoa. Ingeniaritza Elektriko, Elektroniko eta Telekomunikazio Saila, and Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
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Work (thermodynamics) ,Pure mathematics ,Logic ,Interval-valued dissimilarity function ,Function (mathematics) ,Fuzzy logic ,Interval-valued fuzzy measure ,Monotone polygon ,Choquet integral ,Artificial Intelligence ,d-Choquet integral ,Mathematics ,Unit interval - Abstract
In this work, we introduce the notion of dG-Choquet integral, which generalizes the discrete Choquet integral replacing, in the first place, the difference between inputs represented by closed subintervals of the unit interval [0,1] by a dissimilarity function; and we also replace the sum by more general appropriate functions. We show that particular cases of dG-Choquet integral are both the discrete Choquet integral and the d-Choquet integral. We define interval-valued fuzzy measures and we show how they can be used with dG-Choquet integrals to define an interval-valued discrete Choquet integral which is monotone with respect to admissible orders. We finally study the validity of this interval-valued Choquet integral by means of an illustrative example in a classification problem. © 2021 This work was supported in part by the Spanish Ministry of Science and Technology, under project PID2019-108392GB-I00 (AEI/10.13039/501100011033), by the project PJUPNA-1926 of the Public University of Navarre and by the project VEGA 1/0267/21 .
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- 2022
13. Distributed Identification of Contracting and/or Monotone Network Dynamics
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Jack Umenberger, Max Revay, and Ian R. Manchester
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FOS: Computer and information sciences ,Computer Science - Machine Learning ,Mathematical optimization ,Computer science ,Stability (learning theory) ,Monotonic function ,Systems and Control (eess.SY) ,Dynamical Systems (math.DS) ,Network dynamics ,Electrical Engineering and Systems Science - Systems and Control ,Machine Learning (cs.LG) ,Computer Science Applications ,Nonlinear system ,Identification (information) ,Monotone polygon ,Optimization and Control (math.OC) ,Control and Systems Engineering ,Scalability ,FOS: Electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,State space ,Mathematics - Dynamical Systems ,Electrical and Electronic Engineering ,Mathematics - Optimization and Control - Abstract
This paper proposes methods for identification of large-scale networked systems with guarantees that the resulting model will be contracting -- a strong form of nonlinear stability -- and/or monotone, i.e. order relations between states are preserved. The main challenges that we address are: simultaneously searching for model parameters and a certificate of stability, and scalability to networks with hundreds or thousands of nodes. We propose a model set that admits convex constraints for stability and monotonicity, and has a separable structure that allows distributed identification via the alternating directions method of multipliers (ADMM). The performance and scalability of the approach is illustrated on a variety of linear and non-linear case studies, including a nonlinear traffic network with a 200-dimensional state space., Comment: Preprint of full paper accepted for publication in IEEE Trans. Automatic Control
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- 2022
14. Satisfiability in MultiValued Circuits
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Paweł M. Idziak and Jacek Krzaczkowski
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FOS: Computer and information sciences ,Computational complexity theory ,General Computer Science ,68Q17, 08A05, 08A70 (Primary) 68Q05, 68T27, 03B25, 08B05, 08B10 (Secondary) ,Boolean circuit ,General Mathematics ,010102 general mathematics ,circuit satisfiability ,Distributive lattice ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,01 natural sciences ,Satisfiability ,Algebra ,Computer Science - Computational Complexity ,Monotone polygon ,010201 computation theory & mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Lie algebra ,0101 mathematics ,Time complexity ,solving equations ,Equation solving ,Mathematics - Abstract
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., 50 pages
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- 2022
15. Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems
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Rohan Ghuge and Viswanath Nagarajan
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Network planning and design ,Discrete mathematics ,Monotone polygon ,General Mathematics ,Metric (mathematics) ,Approximation algorithm ,Orienteering ,Management Science and Operations Research ,Quasi-polynomial ,Tree (graph theory) ,Computer Science Applications ,Submodular set function ,Mathematics - Abstract
We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasi-polynomial time [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved [Formula: see text]-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors).
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- 2022
16. A polynomial time algorithm to compute the connected treewidth of a series–parallel graph
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Christophe Paul, Dimitrios M. Thilikos, and Guillaume Mescoff
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Applied Mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,0102 computer and information sciences ,02 engineering and technology ,Binary logarithm ,01 natural sciences ,Treewidth ,Dynamic programming ,Series-parallel graph ,Monotone polygon ,010201 computation theory & mathematics ,Path (graph theory) ,Discrete Mathematics and Combinatorics ,Node (circuits) ,Time complexity ,Algorithm ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
It is well known that the treewidth of a graph G corresponds to the node search number where a team of searchers is pursuing a fugitive that is lazy and invisible (or alternatively is agile and visible) and has the ability to move with infinite speed via unguarded paths. Recently, monotone and connected node search strategies have been considered. A search strategy is monotone if it prevents the fugitive from pervading again areas from where he had been expelled and is connected if, at each step, the set of vertices that is or has been occupied by the searchers induces a connected subgraph of G . It has been shown that the corresponding connected and monotone search number of a graph G can be expressed as the connected treewidth, denoted by ctw ( G ) , that is defined as the minimum width of a rooted tree-decomposition ( X , T , r ) , where the union of the bags corresponding to the nodes of a path of T containing the root r is connected in G . In this paper, we initiate the algorithmic study of connected treewidth. We design a O ( n 2 ⋅ log n ) -time dynamic programming algorithm to compute the connected treewidth of biconnected series–parallel graphs. At the price of an extra n factor in the running time, our algorithm generalizes to graphs of treewidth at most two.
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- 2022
17. Monotone Fuzzy Rule Interpolation for Practical Modeling of the Zero-Order TSK Fuzzy Inference System
- Author
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Chee Peng Lim, Kai Meng Tay, and Yi Wen Kerk
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Scheme (programming language) ,Mathematical optimization ,Computer science ,Augmented Lagrangian method ,Applied Mathematics ,Inference ,02 engineering and technology ,Function (mathematics) ,Fuzzy logic ,Monotone polygon ,Computational Theory and Mathematics ,Artificial Intelligence ,Control and Systems Engineering ,0202 electrical engineering, electronic engineering, information engineering ,Key (cryptography) ,020201 artificial intelligence & image processing ,Failure mode and effects analysis ,computer ,computer.programming_language - Abstract
Formulating a generalized monotone fuzzy rule interpolation (MFRI) model is difficult. A complete and monotone fuzzy rule-base is essential for devising a monotone zero-order TSK FIS model. However, such a complete and monotone fuzzy rule-base is not always available in practice. In this paper, we develop an MFRI modelling scheme for generating a monotone zero-order Takagi-Sugeno-Kang (TSK) Fuzzy Inference System (FIS), from a monotone and incomplete fuzzy rule-base. In our proposal, a monotone-ordered fuzzy rule-base that consists of the available fuzzy rules from a monotone and incomplete fuzzy rule-base, and those derived from the MFRI reasoning, is formed. We outline three important properties that the MFRI's deduced fuzzy rules should satisfy to ensure a monotone-ordered fuzzy rule-base. A Lagrangian function for the MFRI scheme, together with its Karush-Kuhn-Tucker optimality conditions, is formulated and analyzed. The key idea is to impose constraints that guide the MFRI inference outcomes. An iterative MFRI algorithm that adopts an augmented Lagrangian function is devised. The proposed MFRI algorithm aims to achieve an ϵ-optimality condition and to produce an ϵ-optimal solution, which is geared for practical applications. We apply the MFRI algorithm to a Failure Mode and Effect Analysis case study and a tanker ship heading regulation problem. The results indicate the effectiveness of MFRI for generating monotone TSK FRI models in tackling practical problems.
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- 2022
18. New contributions for tripled fixed point methodologies via a generalized variational principle with applications
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Hasanen A. Hammad, Manuel De la Sen, and Hassen Aydi
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General Engineering ,34B15 ,Fixed point ,Engineering (General). Civil engineering (General) ,Space (mathematics) ,Mathematical proof ,Metric space ,54H25 ,Monotone polygon ,Variational principle ,Applied mathematics ,Uniqueness ,Boundary value problem ,TA1-2040 ,47H10 ,Mathematics - Abstract
This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications.
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- 2022
19. Applications of the Poincare-Hopf Theorem
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Ji Liu, Mengbin Ye, Ming Cao, Brian D. O. Anderson, and Discrete Technology and Production Automation
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Lyapunov function ,0209 industrial biotechnology ,02 engineering and technology ,Dynamical system ,Computer Science Applications ,System dynamics ,symbols.namesake ,Nonlinear system ,020901 industrial engineering & automation ,Monotone polygon ,Exponential stability ,Control and Systems Engineering ,Jacobian matrix and determinant ,symbols ,Applied mathematics ,Electrical and Electronic Engineering ,Poincaré–Hopf theorem ,Mathematics - Abstract
This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincare–Hopf Theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria, and ensures the system is infact locally exponentially stable. We apply this result to the susceptible-infected-susceptible (SIS) networked model, and a generalised Lotka–Volterra system. We use the result further to extend the SIS model via the introduction of decentralised feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincare–Hopf approach, we identify a necessary and sufficient condition under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady-state), and monotone systems theory is used to show this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpartcondition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented.
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- 2022
20. Joint Estimation of Monotone Curves via Functional Principal Component Analysis
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Yei Eun Shin, Yu Ding, and Lan Zhou
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Statistics and Probability ,Functional principal component analysis ,Wind power ,business.industry ,Applied Mathematics ,Curvature ,Turbine ,Article ,Computational Mathematics ,Noise ,Monotone polygon ,Computational Theory and Mathematics ,Principal component analysis ,Applied mathematics ,business ,Joint (geology) ,Mathematics - Abstract
A functional data approach is developed to jointly estimate a collection of monotone curves that are irregularly and possibly sparsely observed with noise. In this approach, the unconstrained relative curvature curves instead of the monotone-constrained functions are directly modeled. Functional principal components are used to describe the major modes of variations of curves and allow borrowing strength across curves for improved estimation. A two-step approach and an integrated approach are considered for model fitting. The simulation study shows that the integrated approach is more efficient than separate curve estimation and the two-step approach. The integrated approach also provides more interpretable principle component functions in an application of estimating weekly wind power curves of a wind turbine.
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- 2023
21. Spreading Speeds and Traveling Waves for Monotone Systems of Impulsive Reaction–Diffusion Equations: Application to Tree–Grass Interactions in Fire-prone Savannas
- Author
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Jacek Banasiak, Yves Dumont, Ivric Valaire Yatat Djeumen, University of Pretoria [South Africa], Botanique et Modélisation de l'Architecture des Plantes et des Végétations (UMR AMAP), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche pour le Développement (IRD [France-Sud])-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Département Systèmes Biologiques (Cirad-BIOS), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad), Ecole Nationale Supérieure Polytechnique de Yaoundé (ENSPY), and Université de Yaoundé I
- Subjects
Dynamical Systems (math.DS) ,01 natural sciences ,Interactions biologiques ,Spreading speed ,Savanna ,Traveling wave ,Mathematics - Dynamical Systems ,[MATH]Mathematics [math] ,010301 acoustics ,Savane ,Mathematics ,Partial differential equation ,U10 - Informatique, mathématiques et statistiques ,Recursion equation ,Applied Mathematics ,Pulse fire ,010101 applied mathematics ,Periodic perturbation ,Modèle mathématique ,Incendie spontané ,Analysis of PDEs (math.AP) ,P40 - Météorologie et climatologie ,Computation ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Herbage ,Arbre ,Mathematics - Analysis of PDEs ,[SDV.EE.ECO]Life Sciences [q-bio]/Ecology, environment/Ecosystems ,0103 physical sciences ,Reaction–diffusion system ,FOS: Mathematics ,Applied mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Impulsive event ,0101 mathematics ,Modélisation environnementale ,Vitesse ,Formalism (philosophy of mathematics) ,Monotone polygon ,Monotone cooperative system ,[SDE.BE]Environmental Sciences/Biodiversity and Ecology ,Analysis - Abstract
Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.
- Published
- 2023
22. Fixpoint Theory : Upside Down
- Author
-
Baldan, Paolo, Eggert, Richard, König, Barbara, and Padoan, Tommaso
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,Computer Science - Logic in Computer Science ,Monotonic function ,010103 numerical & computational mathematics ,0102 computer and information sciences ,Fixed point ,01 natural sciences ,Article ,Logic in Computer Science (cs.LO) ,Informatik ,Monotone polygon ,Complete lattice ,010201 computation theory & mathematics ,Simple (abstract algebra) ,Computer Science::Logic in Computer Science ,Computer Science::Programming Languages ,Finitary ,0101 mathematics ,Element (category theory) ,Finite set ,Mathematics - Abstract
Knaster-Tarski’s theorem, characterising the greatest fix- point of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form$$\mathbb {M}^Y$$MY, whereYis a finite set and$$\mathbb {M}$$Man MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.
- Published
- 2023
23. Two’s Company, Three’s a Crowd: Consensus-Halving for a Constant Number of Agents
- Author
-
Argyrios Deligkas, Aris Filos-Ratsikas, and Alexandros Hollender
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,Linguistics and Language ,Computational complexity theory ,Computer science ,Robertson-Webb ,Consensus-halving ,Computational Complexity (cs.CC) ,Language and Linguistics ,Fair division ,Exponential function ,Computational complexity ,Computer Science - Computational Complexity ,Monotone polygon ,Computer Science - Computer Science and Game Theory ,Simple (abstract algebra) ,Artificial Intelligence ,Query complexity ,Computer Science - Data Structures and Algorithms ,Data Structures and Algorithms (cs.DS) ,Set (psychology) ,Constant (mathematics) ,Value (mathematics) ,Computer Science and Game Theory (cs.GT) - Abstract
We consider the $\varepsilon$-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to $\varepsilon$). This problem was recently shown to be PPA-complete, for $n$ agents and $n$ cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem. For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents., Comment: Journal version
- Published
- 2022
24. A note on the smallest semicopula-based universal integral and an application
- Author
-
Tran Minh Thuyet, Huynh Ngoc Phuoc, Kieu Huu Dung, Do Huy Hoang, and Tran Nhat Luan
- Subjects
Monotone polygon ,Artificial Intelligence ,Logic ,Convergence (routing) ,Applied mathematics ,Fuzzy logic ,Mathematics - Abstract
In this paper, we study two properties of the seminormed fuzzy integral. By applying these results, we propose alternative proof of the monotone convergence theorems for smallest semicopula-based universal integrals, which are proposed by J. Borzova-Molnarova et al. in 2015.
- Published
- 2022
25. On Prékopa-Leindler type inequality for Sugeno integral
- Author
-
Michał Boczek, Marek Kaluszka, and Ondrej Hutník
- Subjects
Inequality ,Measurable function ,Lebesgue measure ,Logic ,media_common.quotation_subject ,Type inequality ,Sugeno integral ,Monotone polygon ,Artificial Intelligence ,Mathematics::Metric Geometry ,Applied mathematics ,Point (geometry) ,media_common ,Mathematics - Abstract
In this paper we provide a general Prekopa-Leindler type inequality for Sugeno integral and any measurable functions. In addition, we apply novel inequalities to study some properties of monotone measures. As a by-product, we point out that the result from [22] , stating that the classical Prekopa-Leindler inequality is not valid for the Sugeno integral with respect to the Lebesgue measure, is not true and we provide its correction.
- Published
- 2022
26. Non-discrete k-order additivity of a set function and distorted measure
- Author
-
Ryoji Fukuda, Yoshiaki Okazaki, and Aoi Honda
- Subjects
Distortion function ,0209 industrial biotechnology ,Polynomial ,Pure mathematics ,Logic ,02 engineering and technology ,Measure (mathematics) ,020901 industrial engineering & automation ,Monotone polygon ,Artificial Intelligence ,Set function ,Additive function ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,020201 artificial intelligence & image processing ,Mathematics - Abstract
In this study, we generalize the concept of the k-order additivity of a set function. First, we discuss the Mobius transform for a non-discrete set function. Next, we generalize the definition of the k-order additivity of a set function using the Mobius transform and provide the equivalent conditions for the k-order additivity. Furthermore, we consider the k-order additivity of the distorted monotone measure. We prove that under certain conditions, a distorted measure is k-order additive if and only if the distortion function is a polynomial of k-th order.
- Published
- 2022
27. Sample selection models with monotone control functions
- Author
-
Ruixuan Liu and Zhengfei Yu
- Subjects
Economics and Econometrics ,Applied Mathematics ,05 social sciences ,Asymptotic distribution ,Monotonic function ,Function (mathematics) ,01 natural sciences ,Control function ,Inverse Mills ratio ,010104 statistics & probability ,Monotone polygon ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Selection (genetic algorithm) ,050205 econometrics ,Mathematics ,Parametric statistics - Abstract
The celebrated Heckman selection model yields a selection correction function (control function) proportional to the inverse Mills ratio, which is monotone. This paper studies a sample selection model that does not impose parametric distributional assumptions on the latent error terms, while maintaining the monotonicity of the control function. We show that a positive (negative) dependence condition on the latent error terms is sufficient for the monotonicity of the control function. The condition is equivalent to a restriction on the copula function of latent error terms. Using the monotonicity, we propose a tuning-parameter-free semiparametric estimation method and establish root n -consistency and asymptotic normality for the estimates of finite-dimensional parameters. A new test for selectivity is also developed in the presence of the shape restriction. Simulations and an empirical application are conducted to illustrate the usefulness of the proposed methods.
- Published
- 2022
28. Matroid optimization problems with monotone monomials in the objective
- Author
-
S. Thomas McCormick, Frank Fischer, and Anja Fischer
- Subjects
Polynomial ,Monomial ,Optimization problem ,Rank (linear algebra) ,Applied Mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,Polytope ,Monotonic function ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Matroid ,Combinatorics ,Monotone polygon ,010201 computation theory & mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Discrete Mathematics and Combinatorics ,Mathematics - Abstract
In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. We present a complete description of this polytope. Apart from linearization constraints one needs appropriately strengthened rank inequalities. The separation problem for these inequalities reduces to a submodular function minimization problem. These polyhedral results give rise to a new hierarchy for the solution of matroid optimization problems with polynomial objectives. This hierarchy allows to strengthen the relaxations of arbitrary linearized combinatorial optimization problems with polynomial objective functions and matroidal substructures. Finally, we give suggestions for future work.
- Published
- 2022
29. Some notes on monotone set-valued measures and Egoroff's theorem
- Author
-
Jun Li
- Subjects
Set (abstract data type) ,Discrete mathematics ,Lemma (mathematics) ,Monotone polygon ,Artificial Intelligence ,Logic ,Open problem ,Fuzzy set ,Point (geometry) ,Topological space ,Mathematical proof ,Mathematics - Abstract
In this note, we point out that Theorem 3 (a version of Egoroff's theorem for monotone set-valued measures) shown in the paper “Lusin's theorem for monotone set-valued measures on topological spaces” (Fuzzy Sets and Systems 364 (2019) 111-123) is not valid, and we present two revised versions. We also point out that the proofs of Lemma 2 and Theorem 1 are defective, thus the truth of their conclusions cannot be affirmed. Thereby, an open problem for the characteristics of monotone measures is raised.
- Published
- 2022
30. Third-degree price discrimination versus uniform pricing
- Author
-
Francisco Castro, Gabriel Y. Weintraub, and Dirk Bergemann
- Subjects
FOS: Computer and information sciences ,TheoryofComputation_MISCELLANEOUS ,Computer Science::Computer Science and Game Theory ,Economics and Econometrics ,General Economics (econ.GN) ,Uniform pricing ,TheoryofComputation_GENERAL ,Price discrimination ,Space (commercial competition) ,Profit (economics) ,FOS: Economics and business ,Monotone polygon ,Market segmentation ,Computer Science - Computer Science and Game Theory ,Economics - Theoretical Economics ,Econometrics ,Economics ,Theoretical Economics (econ.TH) ,Monopoly ,Finance ,Economics - General Economics ,Computer Science and Game Theory (cs.GT) - Abstract
We compare the profit of the optimal third-degree price discrimination policy against a uniform pricing policy. A uniform pricing policy offers the same price to all segments of the market. Our main result establishes that for a broad class of third-degree price discrimination problems with concave profit functions (in the price space) and common support, a uniform price is guaranteed to achieve one half of the optimal monopoly profits. This profit bound holds for any number of segments and prices that the seller might use under third-degree price discrimination. We establish that these conditions are tight and that weakening either common support or concavity can lead to arbitrarily poor profit comparisons even for regular or monotone hazard rate distributions., Comment: 26 pages, 5 figures
- Published
- 2022
31. A monotone finite volume element scheme for diffusion equations on triangular grids
- Author
-
Haiyuan Yu and Cunyun Nie
- Subjects
Computational Mathematics ,Nonlinear system ,Quadrilateral ,Monotone polygon ,Computational Theory and Mathematics ,Modeling and Simulation ,Mathematical analysis ,Partition (number theory) ,Diffusion (business) ,Coefficient matrix ,Finite element method ,Control volume ,Mathematics - Abstract
We present a monotone finite volume element scheme for the diffusion problem on triangular grids, which stems from some idea about the two-point flux in the literature [15] . A new nonlinear two-point flux formulation is obtained for some part of one control volume edge, and this straight edge crosses through some common edge of two neighboring triangular elements where two inner points from these neighboring elements are chosen and linked so that it does not cross through other triangle elements even if the meshes are severely distorted. The approximation of κ∇u in some quadrilateral region including this straight control volume edge, are obtained by some weighted average with the gradient of some finite element functions, and the weighted coefficients are determined by the nonlinear two-point flux idea. Furthermore, we prove that the new scheme is monotone under some conditions that one modified barycenter dual partition Ω h ⁎ is chosen for some given partition Ω h and the exact solution u ∈ C 1 ( Ω ) , and the corresponding coefficient matrix is of M-matrix. Finally, we carry on some typical experiments. Numerical results verify the monotonicity of this new scheme, and also confirm the accuracy and stability.
- Published
- 2022
32. The period function of reversible Lotka-Volterra quadratic centers
- Author
-
Changjian Liu, Chengzhi Li, Dechen Wang, and Jinming Li
- Subjects
Work (thermodynamics) ,Class (set theory) ,Monotone polygon ,Quadratic equation ,Applied Mathematics ,Computation ,Applied mathematics ,Monotonic function ,Function (mathematics) ,Analysis ,Mathematics ,Exponential function - Abstract
In this paper it is proved that the period function of reversible Lotka-Volterra quadratic systems is monotone, unless the associated center is isochronous. For this class of quadratic systems, the traditional criterion function on the monotonicity of period function does not work, we improve the criterion function to new different forms such that it can be used for different values of the parameters in the system. Furthermore, to overcome the difficulty that the exponential function of the parameter appears in the first integral, in our study we combine the analytic expressions with their geometric meanings and symbolic computations. These methods may be used for other problems.
- Published
- 2022
33. High-order weighted compact nonlinear scheme for one- and two-dimensional Hamilton-Jacobi equations
- Author
-
Shuguang Zhou, Ying-Gang Hu, Yanqun Jiang, and Xu Zhang
- Subjects
Computational Mathematics ,Numerical Analysis ,Nonlinear system ,Monotone polygon ,Applied Mathematics ,Applied mathematics ,Type (model theory) ,Classification of discontinuities ,Stencil ,Hamilton–Jacobi equation ,Hamiltonian (control theory) ,Interpolation ,Mathematics - Abstract
This paper designs a fifth-order weighted compact nonlinear scheme (WCNS) on a five-point stencil to solve one- and two-dimensional Hamilton-Jacobi equations. The five-point WCNS is used to compute the left and right limits of first-order spatial derivatives of the HJ equations in the Lax-Friedrichs monotone numerical Hamiltonian. The WENO-Z type interpolation for cell-edge values of the solutions is used to suppress numerical oscillations which may appear near discontinuities. Five- and seven-point WENO-Z schemes for Hamilton-Jacobi equations are also designed for comparisons. The performance of the WCNS and the two WENO-Z schemes is demonstrated by several numerical examples in one-dimensional and two-dimensional cases.
- Published
- 2022
34. A note on topological aspects in dynamic games of resource extraction and economic growth theory
- Author
-
Andrzej S. Nowak and Anna Jaśkiewicz
- Subjects
Economics and Econometrics ,Class (set theory) ,State variable ,Hilbert space ,symbols.namesake ,Monotone polygon ,Schauder fixed point theorem ,Square-integrable function ,Markov perfect equilibrium ,symbols ,State space ,Mathematical economics ,Finance ,Mathematics - Abstract
We show that right-continuous monotone strategies used in Markov perfect equilibria for economic growth models and related dynamic games can be recognised as members of the Hilbert space of square integrable functions of the state variable. We provide an application of this result to a bequest game and point out that this result also holds for the class of left-continuous monotone functions. The result is fundamental for using the Schauder fixed point theorem. Furthermore, it considerably simplifies the classical approach, where such strategies are represented by non-negative measures on the state space.
- Published
- 2022
35. Fitted Value Iteration in Continuous MDPs With State Dependent Action Sets
- Author
-
Abhishek Gupta, Shiping Shao, and Hao Li
- Subjects
Control and Optimization ,Monotone polygon ,Markov chain ,Control and Systems Engineering ,Kernel (statistics) ,Convergence (routing) ,Probabilistic logic ,Applied mathematics ,Approximation algorithm ,Markov decision process ,Decision problem ,Mathematics - Abstract
In this letter, we establish the convergence of fitted value iteration and fitted Q-value iteration for continuous-state continuous-action Markov decision problems (MDPs) with state-dependent action sets. We further extend the algorithm and the convergence result to the case of monotone MDPs.
- Published
- 2022
36. Choquet-Sugeno-like operator based on relation and conditional aggregation operators
- Author
-
Michał Boczek, Ondrej Hutník, and Marek Kaluszka
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Information Systems and Management ,Mathematics::Operator Algebras ,Measure (physics) ,Mathematics::General Topology ,Functional Analysis (math.FA) ,Computer Science Applications ,Theoretical Computer Science ,Copula (probability theory) ,Dependence relation ,Mathematics - Functional Analysis ,Mathematics::Logic ,Monotone polygon ,Operator (computer programming) ,Choquet integral ,Computer Science::Discrete Mathematics ,Artificial Intelligence ,Control and Systems Engineering ,Bounded function ,FOS: Mathematics ,Partition (number theory) ,Software ,Mathematics - Abstract
We introduce a Choquet-Sugeno-like operator generalizing many operators for bounded nonnegative functions and monotone measures from the literature, e.g., the Sugeno-like operator, the Lovasz and Owen measure extensions, the F -decomposition integral with respect to a partition decomposition system, and others. The new operator is based on concepts of dependence relation and conditional aggregation operators, but it does not depend on α -level sets. We also provide conditions under which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g., the reverse Choquet integral , the d -Choquet integral, the F -based discrete Choquet-like integral, some version of the C F 1 F 2 -integral, the CC -integrals (or Choquet-like Copula-based integral) and the discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.
- Published
- 2022
37. An explicit algorithm for solving monotone variational inequalities
- Author
-
Aviv Gibali, Phan Tu Vuong, and Duong Viet Thong
- Subjects
Numerical Analysis ,Weak convergence ,Applied Mathematics ,Hilbert space ,Monotonic function ,Lipschitz continuity ,Computational Mathematics ,symbols.namesake ,Monotone polygon ,Rate of convergence ,Variational inequality ,symbols ,Applied mathematics ,Constant (mathematics) ,Mathematics - Abstract
In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved mapping. Weak convergence of the proposed scheme is established under standard assumptions. Under strong monotonicity, we present the R-linear convergence rate of our new method. Intensive numerical experiments illustrate the advantages and the applicability of our scheme. Moreover, our work generalizes theoretically several recent results in this field.
- Published
- 2022
38. A Resilient Consensus Protocol for Networks With Heterogeneous Confidence and Byzantine Adversaries
- Author
-
Sabato Manfredi, David Angeli, Angeli, D., and Manfredi, S.
- Subjects
Class (computer programming) ,Control and Optimization ,Theoretical computer science ,fault tolerant system ,Computer science ,cooperative control ,petri nets ,Cyber-physical system ,Network analysis and control ,Petri net ,Computer Science::Multiagent Systems ,Monotone polygon ,distributed control ,Control and Systems Engineering ,Convergence (routing) ,Common value auction ,Protocol (object-oriented programming) ,Counterexample - Abstract
A class of Adversary Robust Consensus protocols is proposed and analyzed. These are inherently nonlinear, distributed, continuous-time algorithms for multi-agents systems seeking to agree on a common value of a shared variable, in the presence of faulty or malicious Byzantine agents, disregarding protocol rules and communicating arbitrary possibly differing values to neighboring agents. We adopt monotone joint-agent interactions, a general mechanism for processing locally available information and allowing cross-comparisons between state-values of multiple agents simultaneously. The topological features of the network are abstracted as a Petri Net and convergence criteria for the resulting time evolutions formulated in terms of suitable structural properties of its invariants (so called siphons). Finally, simulation results and examples/counterexamples are discussed.
- Published
- 2022
39. Line-of-Sight Pursuit in Monotone and Scallop Polygons.
- Author
-
Berry, Lindsay, Beveridge, Andrew, Butterfield, Jane, Isler, Volkan, Keller, Zachary, Shine, Alana, and Wang, Junyi
- Subjects
- *
POLYGONS , *SCALLOPS , *ROTATIONAL motion , *ALGORITHMS , *MOTION - Abstract
We study a turn-based game in a simply connected polygonal environment Q between a pursuer 𝒫 and an adversarial evader ℰ. Both players can move in a straight line to any point within unit distance during their turn. The pursuer 𝒫 wins by capturing the evader, meaning that their distance satisfies d (𝒫 , ℰ) ≤ 1 , while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader ℰ always knows the location of 𝒫. Meanwhile, 𝒫 only has line-of-sight visibility: 𝒫 observes the evader's position only when the line segment connecting them lies entirely within the polygon. Therefore 𝒫 must search for ℰ when the evader is hidden from view. We provide a winning strategy for 𝒫 in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line L can be moved continuously over Q so that (1) L ∩ Q is a line segment and (2) every point on the boundary ∂ Q is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook's strategy during its pursuit phase, rather than the well-known lion's strategy. The rook's strategy is crucial for obtaining a capture time that is linear in the area of Q. For both monotone and scallop polygons, our algorithm has a capture time of O (n (Q) + area (Q)) , where n (Q) is the number of polygon vertices. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. On Calculation of the Norm of a Monotone Operator in Ideal Spaces
- Author
-
M. L. Goldman and E. G. Bakhtigareeva
- Subjects
Pure mathematics ,Operator (computer programming) ,Ideal (set theory) ,Monotone polygon ,Matching (graph theory) ,Ideal space ,Monotonic function ,General Medicine ,Convexity ,Mathematics - Abstract
This paper contains the proof of general results on the calculation of the norms of monotone operators acting from one ideal space to another under matching convexity and concavity properties of the operator and the norms in ideal spaces.
- Published
- 2021
41. Cardinality-limiting extended pre-aggregation functions
- Author
-
Simon James, Anna Kolesárová, Radko Mesiar, and Gleb Beliakov
- Subjects
Computer science ,High density ,Value (computer science) ,020206 networking & telecommunications ,02 engineering and technology ,Limiting ,Automatic summarization ,Consistency (database systems) ,Information fusion ,Monotone polygon ,Cardinality ,Hardware and Architecture ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Algorithm ,Software ,Information Systems - Abstract
Aggregation functions, which are at the heart of a number of information fusion processes, allow summarization of multiple inputs into a single representative value. Extended aggregation functions are defined such that the input data can be of varying cardinality, with the implication that there is some consistency across the methods of calculation. This article formalizes an approach to extended aggregation such that contributions of repeated inputs or regions of high density are limited in their ability to influence the final value. We establish important definitions and properties, in particular around whether such functions will be monotone or directionally monotone. We then propose a powerful construction method for extended pre-aggregation functions. Illustrative examples are provided throughout.
- Published
- 2021
42. New lower bounds on the number of intersections of monotone Lagrangian submanifolds
- Author
-
Nassima Keddari
- Subjects
Pure mathematics ,symbols.namesake ,Monotone polygon ,Applied Mathematics ,General Mathematics ,symbols ,Lagrangian ,Mathematics - Abstract
Considering a closed monotone Lagrangian submanifold L , L, we give, under some hypotheses, a lower bound on the intersection number of L L with its image by a generic Hamiltonian isotopy. First for monotone Lagrangian submanifolds L L which are K ( π , 1 ) \mathbf {K}(\pi ,1) and, in particular, for monotone Lagrangian submanifolds with negative sectional curvature this bound is 1+ β 1 ( L ) . \beta _{1}(L). In more general cases the lower bound is weaker. We generalise some results previously obtained by L. Buhovsky in [J. Topol. Anal. 2 (2010), pp. 57–75] and P. Biran and O. Cornea in [Geom. Topol. 13 (2009), pp. 2881–2989].
- Published
- 2022
43. Boundary growth of Sobolev functions of monotone type for double phase functionals
- Author
-
Yoshihiro Mizuta and Tetsu Shimomura
- Subjects
Physics ,Unit sphere ,Boundary (topology) ,Hölder condition ,Order (ring theory) ,Articles ,Type (model theory) ,Spherical mean ,Combinatorics ,Sobolev space ,Monotone polygon ,Monotone Sobolev functions ,spherical mean ,double phase functional - Abstract
Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).
- Published
- 2021
44. The Monotone Extended Second-Order Cone and Mixed Complementarity Problems
- Author
-
Sandor Nemeth, Yingchao Gao, and Roman Sznajder
- Subjects
Pure mathematics ,Control and Optimization ,Monotone polygon ,Rank (linear algebra) ,Cone (topology) ,Applied Mathematics ,Complementarity (molecular biology) ,Nonlinear complementarity problem ,Management Science and Operations Research ,Mixed complementarity problem ,Projection (linear algebra) ,Ambient space ,Mathematics - Abstract
In this paper, we study a new generalization of the Lorentz cone $$\mathcal{L}^n_+$$ L + n , called the monotone extended second-order cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a nonlinear complementarity problem on a cylinder, which is equivalent to a suitable mixed complementarity problem, and provide a computational example illustrating applicability of MESOC.
- Published
- 2021
45. On the inexact scaled gradient projection method
- Author
-
Max V. Lemes, L. F. Prudente, and Orizon P. Ferreira
- Subjects
Control and Optimization ,Line search ,Applied Mathematics ,Feasible region ,MathematicsofComputing_GENERAL ,Convex set ,Computational Mathematics ,Monotone polygon ,Optimization and Control (math.OC) ,Approximation error ,Convergence (routing) ,Line (geometry) ,FOS: Mathematics ,Applied mathematics ,Projection (set theory) ,Mathematics - Optimization and Control ,49J52, 49M15, 65H10, 90C30 ,Mathematics - Abstract
The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches., Comment: 30 pages, 14 figures
- Published
- 2021
46. Fixed point theorems for monotone orbitally nonexpansive type mappings in partially ordered hyperbolic metric spaces
- Author
-
Rahul Shukla and Rajendra Pant
- Subjects
Pure mathematics ,Metric space ,Monotone polygon ,General Mathematics ,Fixed-point theorem ,Type (model theory) ,Mathematics - Published
- 2021
47. An Accelerated Coordinate Gradient Descent Algorithm for Non-separable Composite Optimization
- Author
-
Amir Beck and Aviad Aberdam
- Subjects
Control and Optimization ,Monotone polygon ,Quadratic equation ,Rate of convergence ,Applied Mathematics ,Inpainting ,Management Science and Operations Research ,Lipschitz continuity ,Coordinate descent ,Gradient descent ,Algorithm ,Separable space ,Mathematics - Abstract
Coordinate descent algorithms are popular in machine learning and large-scale data analysis problems due to their low computational cost iterative schemes and their improved performances. In this work, we define a monotone accelerated coordinate gradient descent-type method for problems consisting of minimizing $$f+g$$ , where f is quadratic and g is nonsmooth and non-separable and has a low-complexity proximal mapping. The algorithm is enabled by employing the forward–backward envelope, a composite envelope that possess an exact smooth reformulation of $$f+g$$ . We prove the algorithm achieves a convergence rate of $$O(1/k^{1.5})$$ in terms of the original objective function, improving current coordinate descent-type algorithms. In addition, we describe an adaptive variant of the algorithm that backtracks the spectral information and coordinate Lipschitz constants of the problem. We numerically examine our algorithms on various settings, including two-dimensional total-variation-based image inpainting problems, showing a clear advantage in performance over current coordinate descent-type methods.
- Published
- 2021
48. On Nonexistence of Nonnegative Monotone Solutions for Some Coercive Inequalities in a Half-Space
- Author
-
Olga Salieva and Evgenii Igorevich Galakhov
- Subjects
Statistics and Probability ,Pure mathematics ,Monotone polygon ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Half-space ,media_common ,Mathematics - Published
- 2021
49. A new black box method for monotone nonlinear equations
- Author
-
Abdulkarim Hassan Ibrahim, Poom Kumam, and Morteza Kimiaei
- Subjects
Nonlinear system ,Control and Optimization ,Monotone polygon ,Applied Mathematics ,Black box ,Applied mathematics ,Management Science and Operations Research ,Mathematics - Published
- 2021
50. A coupled system involving nonlinear fractional q-difference stationary Schrödinger equation
- Author
-
Shurong Sun and Zhongyun Qin
- Subjects
Computational Mathematics ,symbols.namesake ,Nonlinear system ,Monotone polygon ,Schauder fixed point theorem ,Iterative method ,Applied Mathematics ,Theory of computation ,symbols ,Applied mathematics ,Schrödinger equation ,Mathematics - Abstract
In this paper, we investigate the solvability for a coupled system involving nonlinear fractional q-difference stationary Schrodinger equation. The existence criterion of solutions is established by Schauder fixed point theorem, while the existence of iterative positive solutions is derived by monotone iteration method. As applications, an example is presented to illustrate the main results.
- Published
- 2021
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