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Currently, most traffic control methods at intersections rely on the control of signal lights. However, most signal lights operate in the traditional fixed timing mode, which cannot adjust the timing based on the time-varying traffic flow. To solve the problem, this paper constructs a signal timing control model to optimize road capacity, delay time and the number of stops at the intersections, under the following constraints: cycle time, effective green light time and the maximum number of vehicles in each direction of intersection. To solve the model, the standard dragonfly algorithm (DA) was improved by a hybrid mutation operator, which ensures the diversity of solution set. The proposed model and algorithm were compared with the Webster model through simulations in an actual scene and on a virtual platform. The comparison fully proves the advantages of our model and algorithm.

In order to solve the constrained multi-objective problem more effectively, the constrained multi-objective water evaporation optimization algorithm based on decomposition with e-constraint handling technology (MOEA/D-WEO-eCHT) is proposed in this paper. MOEA/D-WEO-eCHT uses the MOEA/D framework to deal with multi-objective, e-constraint handling technology to handle constraints and WEO algorithm to perform the main body evolution operation. Firstly, an external memory population in which the abandoned excellent parent can be kept is established for each subproblem, which make the abandoned parent be fully utilized by participating in the subsequent evolution. Secondly, some strategies of WEO, such as calculation and normalization method of individual fitness and calculation method of step size S, are improved to make WEO suitable for solving constrained multi-objective problems and helpful to obtain the Pareto solution set with better convergence and distribution. Thirdly, the e(t) calculation method and retention mechanism of individuals are improved to reasonably coordinate feasible and infeasible individuals in different evolution stages. To evaluate the performance of the proposed algorithm, the CTP and CF benchmark instances and an engineering optimization problem are studied. The experimental results illustrate that MOEA/D-WEO-eCHT is more effective than other related algorithms on these benchmark instances, in terms of convergence and distribution of the obtained solution sets.

In the past few years, Global Navigation Satellite Systems (GNSS) based attitude determination has been widely used thanks to its high accuracy, low cost, and real-time performance. This paper presents a novel 3-D GNSS attitude determination method based on Riemannian optimization techniques. The paper first exploits the antenna geometry and baseline lengths to reformulate the 3-D GNSS attitude determination problem as an optimization over a non-convex set. Since the solution set is a manifold, in this manuscript we formulate the problem as an optimization over a Riemannian manifold. The study of the geometry of the manifold allows the design of efficient first and second order Riemannian algorithms to solve the 3-D GNSS attitude determination problem. Despite the non-convexity of the problem, the proposed algorithms are guaranteed to globally converge to a critical point of the optimization problem. To assess the performance of the proposed framework, numerical simulations are provided for the most challenging attitude determination cases: the unaided, single-epoch, and single-frequency scenarios. Numerical results reveal that the proposed algorithms largely outperform state-of-the-art methods for various system configurations with lower complexity than generic non-convex solvers, e.g., interior point methods.

For a class of quasi-variational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to {monotone} perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed., Comment: 29 pages

In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.

This paper proposes a framework for generation of collision-free reference trajectories in a cooperative multi-agent setting. The approach is hierarchical: a high-level controller schedules groups of cooperating agents, for which trajectories are then determined by a lower-level trajectory planner. Admissible behaviors of a cooperative group are encoded by so-called maneuvers, which are modeled by hybrid automata. This allows to plan trajectories by solving hybrid optimal control problems. Controllable sets characterize the solution sets of these problems and are used for quick assessment of feasibility prior to planning. This enables the high-level controller to quickly assess the feasibility of different maneuver options and cooperative groups. Special emphasis is put on safety and feasibility of the framework. The efficacy of the approach is demonstrated by simulation of a highway entry scenario for autonomous vehicles.

This article presents a fuzzy system-based modeling approach to estimate the weld bead geometry (WBG) from the welding process variables (WPVs) and to achieve a specific weld bead shape. The bacterial memetic algorithm (BMA) is applied to solve these problems in two different roles, as a supervised trainer, and as an optimizer. As a supervised trainer, the BMA is applied to tune two different WBG models. The bead geometry properties (BGP) model follows a traditional approach providing the WBG properties as outputs. The direct profile measurement (DPM) model describes the bead profiles points by a non-linear function realized in the form of fuzzy rules. As an optimizer, the BMA utilizes the developed fuzzy systems to find the solution sets of WPVs to acquire the desired WBG. The best performance is achieved by applying six rules in the BGP model and eleven rules in the DPM model. The results indicate that the normalized root means square error for the validation data set lies in the range of 0.40 - 1.56% for the BGP model and 4.49 - 7.52% for the DPM model. The comparative analysis suggests that the BGP model estimates the BWG in a superior manner when several WPVs are altered. The developed fuzzy systems provide a tool for interpreting the effects of the WPVs. The developed optimizer provides multiple valid set of WPVs to produce the desired WBG, thus supporting the selection of those process variables in applications.

Set optimization is an indispensable part of theory and method of optimization, and has been received wide attentions due to its extensive applications in group decision and group game problems. This paper focus on the continuity of the strict (weak) minimal solution set mapping of parametric set-valued vector optimization problems with the lower set less order relation. We firstly introduce a concept of strict lower level mapping of parametric set-valued vector optimization problems. Moreover, the upper and lower semicontinuity of the strict lower level mapping are obtained under some suitable conditions. Lastly, the sufficient condition for the continuity of the strict minimal solution set mappings of parametric set optimization problems are established by a new proof method, which is different from that in [ 27 , 28 ].

Many decision making situations are characterized by a hierarchical structure where a lower-level (follower) optimization problem appears as a constraint of the upper-level (leader) one. Such kind of situations is usually modeled as a BLOP (Bi-Level Optimization Problem). The resolution of the latter usually has a heavy computational cost because the evaluation of a single upper-level solution requires finding its corresponding (near) optimal lower-level one. When several objectives are optimized in each level, the BLOP becomes a multi-objective task and more computationally costly as the optimum corresponds to a whole non-dominated solution set, called the PF (Pareto Front). Despite the considerable number of recent works in multi-objective evolutionary bi-level optimization, the number of methods that could be applied to the combinatorial (discrete) case is much reduced. Motivated by this observation, we propose in this paper an Indicator-Based version of our recently proposed Co-Evolutionary Migration-Based Algorithm (CEMBA), that we name IB-CEMBA, to solve combinatorial multi-objective BLOPs. The indicator-based search choice is justified by two arguments. On the one hand, it allows selecting the solution having the maximal marginal contribution in terms of the performance indicator from the lower-level PF. On the other hand, it encourages both convergence and diversity at the upper-level. The comparative experimental study reveals the outperformance of IB-CEMBA on a multi-objective bi-level production-distribution problem. From the effectiveness viewpoint, the upper-level hyper-volume values and inverted generational distance ones vary in the intervals [0.8500, 0.9710] and [0.0072, 0.2420], respectively. From the efficiency viewpoint, IB-CEMBA has a good reduction rate of the Number of Function Evaluations (NFEs), lying in the interval [30.13%, 54.09%]. To further show the versatility of our algorithm, we have developed a case study in machine learning, and more specifically we have addressed the bi-level multi-objective feature construction problem.

The typical stand-alone microgrid (MG) composed of wind turbine (WT), photovoltaic (PV), diesel generator (DE), and battery energy storage (BES) is taken as the research object. Firstly, a multi-objective optimal capacity configuration model considering economic efficiency, reliability, and environmental protection is established. Secondly, in view of the complex characteristics of the optimization model, such as strong nonlinearity and multi-constrained conditions, combining the enumeration that can find the complete real Pareto solution set with the intelligent algorithm that has the characteristics of fast convergence, a search algorithm referencing adjacent points based on SPEA2 (SARAP) is proposed. The algorithm obtains the contour of the real Pareto optimal solution set by SPEA2, and then repeatedly extracts the adjacent points that satisfy certain conditions from the solution set solved by SPEA2. A small search space based on the adjacent points is constructed, and an omnidirectional search in this space to achieve the complete real Pareto optimal solution set is performed. The algorithm performance analysis of both computational complexity and convergence shows that the operation speed of SARAP is approximately five times higher than that of the enumeration, and the obtained result is close to the complete real Pareto optimal solution set. Finally, the optimization calculation and typical daily production simulation are carried out according to the resource and load characteristics of a region, and the results further prove the rationality and validity of the proposed algorithm.

A discrete particle swarm optimization algorithm with adaptive inertia weight (DPSO-AIW) is proposed to solve the multiobjective Flexible Job-shop Scheduling Problem. The algorithm uses a two-layer coding structure to encode the chromosomes, namely operation sequence (OS) and machine assignment (MA). The initial population combined random selection of OS and the global selection based on operation (GSO) of MA. In order to obtain the Pareto optimal solution, non-dominated fronts are obtained by rapid non-dominated sorting. In the evolution process, the discrete particle swarm optimization algorithm is used to directly solve the values of the next generation chromosomes in the discrete domain, and the population diversity is enhanced by adaptively adjusting the variation of the inertia weight $\omega $ , and the Pareto optimal solution obtained in the process is stored in the Pareto optimal solution set (POS). Finally, numerical simulation based on two sets of international standard instances and comparisons with some existing algorithms are carried out. The comparative results demonstrate the effectiveness and practicability of the proposed DPSO-AIW in solving the multiobjective Flexible Job-shop Scheduling Problem.

This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group., 11 pages

Dominance robustness (DR) has been proposed for assessing the ability of the Pareto-optimal solutions to remain to be nondominated when the decision variables are subject to noise. There are two main challenges in search for dominance robust optimal solutions in dominance robust multiobjective optimization (MOP), namely, accurate estimation of the DR measure and a good balance between convergence and DR in the presence of uncertainty. In this article, a novel robust MOP evolutionary algorithm based on decision variable assortment (DVA) is proposed to tackle these challenges. To be specific, an indicator, termed as dominance robust indicator, is proposed to measure the DR based on the dominance level and dominance relationship of the sampled points. Then, the decision variables are divided into low DR-related variables and high DR-related variables based on the DVA strategy. Finally, low and high DR related variables are optimized separately to obtain the dominance robust optimal solutions. In addition, performance indicators to quantify the performance of dominance robust optimal solution set obtained by robust MOP algorithm are proposed. Experimental results have demonstrated that the proposed algorithm is competitive in search for dominance robust optimal solutions.

Мақолада интервал алгоритмлар учун ҳисоблаш дастурларини яратиш муаммолари тадқиқ қилинган. Бунда интервал ҳисоблашларни ташкил қилишнинг ўзига хос жиҳатлари, соҳада мавжуд дастурий маҳсулотларнинг имкониятлари асосан иккита йўналишда, яъни интервал ҳисоблашлар учун яратилган махсус дастурлаш тиллари ва мавжуд дастурлаш муҳитлари учун интервал кутубхоналар яратиш соҳасида ўрганилди. Шунингдек, чизиқли интервал тенгламалар системаларининг мумкин бўлган ечимлар тўпламини 3D ҳамда 2D визуаллаштириш дастурий таъминоти имкониятлари аниқ мисолларда кўрсатиб берилди., The article examines the problems of creating computational programs for interval algorithms. At the same time, the features of the organization of interval calculations, the possibilities of existing software products created in the field of interval analysis were studied, mainly in two directions: special programming languages developed for interval calculations and interval libraries for existing software environments. Also, the possibilities of software products for 3D and 2D visualization of tolerance sets of solutions of interval linear systems of equations in specific examples are shown.

In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.

For targeted motion control tasks of manipulators, it is frequently necessary to make use of full levels of joint actuation to guarantee successful motion planning and path tracking. Such way of motion planning and control may keep the joint actuation in a non-sparse manner during motion control process. In order to improve sparsity of joint actuation for manipulator systems, a novel motion planning scheme which can optimally and sparsely adopt joint actuation is proposed in this paper. The proposed motion planning strategy is formulated as a constrained L1 norm optimization problem, and an equivalent enhanced optimization solution dealing with bounded joint velocity is proposed as well. A new primal dual neural network with a new solution set division is further proposed and applied to solve such bounded optimization which can sparsely adopt joint actuation for motion control. Simulation and experiment results demonstrate the efficiency, accuracy and superiority of the proposed method for optimally and sparsely adopting joint actuation. The average sparsity (i.e., -||˙θ||p where θ denotes the joint angle) of the joint motion of the manipulator can be increased by 39.22% and 51.30% for path tracking tasks in X-Y and X-Z planes respectively, indicating that the sparsity of joint actuation can be enhanced.

This paper presents a Markov sampling-based framework, called Asymptotic Bayesian Optimization, for solving a class of constrained design optimization problems. The optimization problem is converted into a unified two-phase sample generation problem which is solved by an effective Markov chain Monte Carlo simulation scheme. First, an exploration phase generates designs distributed over the feasible design space. Based on this information, an exploitation phase obtains a set of designs lying in the vicinity of the optimal solution set. The proposed formulation can handle continuous, discrete, or mixed discrete-continuous design variables. Appropriate adaptive proposal distributions for the continuous and discrete design variables are suggested. The set of optimal solutions provides valuable sensitivity information of the different quantities involved in the problem with respect to the design variables. Representative examples including an analytical problem involving nonlinear benchmark functions, a classical engineering design problem, and a performance-based design optimization problem of a structural system under stochastic excitation are presented to show the effectiveness and potentiality of the proposed optimization scheme. Validation calculations show that the scheme is a flexible, efficient and competitive choice for solving a wide range of classical and complex engineering design problems.

We investigate the complexity of satisfiability problems parameterized by the size of the solution set compared to all solution candidates. Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time $$O^*(\varepsilon ^{-0.617})$$ where $$\varepsilon $$ is the fraction of assignments that are satisfying. This improves significantly over the trivial sampling bound of expected $$\Theta ^*(\varepsilon ^{-1})$$, and on all previous algorithms whenever $$\varepsilon = \Omega (0.708^n)$$, where n is the number of variables. We also consider algorithms for 3-SAT with an $$\varepsilon $$ fraction of satisfying assignments, and prove that we can output a satisfying assignment in $$O^*(\varepsilon ^{-0.936})$$ randomized time, and sample uniformly a satisfying assignment in time $$O^*(\varepsilon ^{-0.908}1.021^n)$$. In the end we also present sampling results in the cases of 1-IN-3-SAT, monotone bounded degree 3-SAT and planar k-SAT.

In this paper, we obtain the Painleve–Kuratowski upper convergence and the Painleve–Kuratowski lower convergence of the approximate solution sets for set optimization problems with the continuity and convexity of objective mappings. Moreover, we discuss the extended well-posedness and the weak extended well-posedness for set optimization problems under some mild conditions. We also give some examples to illustrate our main results.

Metaheuristic algorithms have popularly been used to solve a wide range of complex engineering optimization problems. In order to solve the problems with two or more objectives multi-objective algorithms (MOAs) are used. Application of MOAs to solve multiple conflicting objectives yields a Pareto-optimal solution set. In this paper, we propose a multi-objective decomposition-based moth flame optimization (MOMFO/D) algorithm, that decomposes the objectives into multiple single objectives which are optimized simultaneously. The algorithm is used to solve the relay node placement problem, that is modeled as a bi-objective problem with the goal of minimization of average intra-cluster distance and average hop-count to improve the network lifetime. The Pareto-optimal fronts obtained through the simulations are evaluated using three distinct quality indicators namely the Inverted Generational Distance, Spacing Metric and Maximum Spread in order to evaluate the performance. The obtained results considered over a number of runs are compared with other existing optimizers in the literature such as multi-objective non-dominated sorted moth flame optimizer, and multi-objective evolutionary algorithm based on decomposition. The results demonstrate the superiority in the performance of the proposed algorithm over others. The statistical analysis of the experimental work has been carried out by conducting Friedman’s and Quade test.

We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constrai...

Traditional Belief-Rule-Based (BRB) ensemble learning methods integrate all of the trained sub-BRB systems to obtain better results than a single belief-rule-based system. However, as the number of BRB systems participating in ensemble learning increases, a large amount of redundant sub-BRB systems are generated because of the diminishing difference between subsystems. This drastically decreases the prediction speed and increases the storage requirements for BRB systems. In order to solve these problems, this paper proposes BRBCS-PAES: a selective ensemble learning approach for BRB Classification Systems (BRBCS) based on Pareto-Archived Evolutionary Strategy (PAES) multi-objective optimization. This system employs the improved Bagging algorithm to train the base classifier. For the purpose of increasing the degree of difference in the integration of the base classifier, the training set is constructed by the repeated sampling of data. In the base classifier selection stage, the trained base classifier is binary coded, and the number of base classifiers participating in integration and generalization error of the base classifier is used as the objective function for multi-objective optimization. Finally, the elite retention strategy and the adaptive mesh algorithm are adopted to produce the PAES optimal solution set. Three experimental studies on classification problems are performed to verify the effectiveness of the proposed method. The comparison results demonstrate that the proposed method can effectively reduce the number of base classifiers participating in the integration and improve the accuracy of BRBCS.

We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. In this framework, we first consider the minimax Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. Recasting this approximation problem as a problem of tropical pseudo-quadratic programming we then write out a closed-form solution to it. This solution might be either a unique score vector (up to a positive factor) or a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques of maximizing and minimizing the Hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores, and thus are well representative of the entire solution set., Comment: 37 pages (added a new example and a discussion). arXiv admin note: text overlap with arXiv:1801.10524

In vicariance biogeography, widespread or sympatric taxa can be dealt with under assumptions 0, 1, and 2. Data from cladogenetic relationships among taxa of a monophyletic group and their distribution over areas are assumed, in the order 0 → 1 → 2, to represent decreasing information about vicariance events. A less strict assumption carries a larger solution set, i.e., the number of possible area cladograms increases with the decrease in strictness of the assumption applied. We formulate two requirements for obtaining valid general area cladograms from data of several monophyletic groups of taxa. First, the assumptions, and with them the sets of area cladograms derived under these assumptions, should be inclusive. Second, sets of single group area cladograms should be compared for different monophyletic groups under a single assumption. When these two requirements are met, area cladograms become consistent with respect to the processes (vicariance, extinction, and dispersal) that are a priori assumed. The explanatory power increases for any particular monophyletic group of taxa when the set of valid general area cladograms contains a subset of area cladograms derived under a more strict assumption. We discuss examples from literature of how violation of these two requirements affects the results.

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

Decisions on how to manage future flood risks are frequently informed by both sophisticated and computationally expensive models. This complexity often limits the representation of uncertainties and the consideration of strategies. Here we use an intermediate complexity model framework that enables us to analyze a richer set of strategies, a wider range of objectives, and greater levels of uncertainty than are typically considered by more sophisticated and computationally expensive models. We find that allowing for more combinations of risk mitigation strategies can help expand the solution set, help explain synergies and trade‐offs, and point to strategies that can improve outcomes.

We present a method to determine local gravity fields for the Moon using Gravity Recovery and Interior Laboratory (GRAIL) data. We express gravity as gridded gravity anomalies on a sphere, and we estimate adjustments to a background global start model expressed in spherical harmonics. We processed GRAIL Ka‐band range‐rate data with a short‐arc approach, using only data over the area of interest. We determine our gravity solutions using neighbor smoothing constraints. We divided the entire Moon into 12 regions and 2 polar caps, with a resolution of 0.15°×0.15° (which is equivalent to degree and order 1199 in spherical harmonics), and determined the optimal smoothing parameter for each area by comparing localized correlations between gravity and topography for each solution set. Our selected areas share nodes with surrounding areas and they are overlapping. To mitigate boundary effects, we patch the solutions together by symmetrically omitting the boundary parts of overlapping solutions. Our new solution has been iterated, and it has improved correlations with topography when compared to a fully iterated global model. Our method requires fewer resources, and can easily handle regionally varying resolution or constraints. The smooth model describes small‐scale features clearly, and can be used in local studies of the structure of the lunar crust., Key Points We determine 14 local gravity field solutions covering the entire Moon using GRAIL intersatellite range‐rate dataWe patch the local solutions into a global map, which shows no seams, minimizing boundary effectsOur solution has improved correlations between gravity and topography compared to standard global solutions, with less computational effort

The purpose of this paper is to present a new version of the Bihari inequality with singular kernel and give a simple proof of the fractional Gronwall lemma. Our new ideas rest on the use of Young’s and Holder’s inequalities to simplify the complex inequalities. Based on this new type of Bihari inequality we can relax many results of fractional differential equations and inclusions and stochastic differential equations. Also, the obtained inequalities can be used to analyze a specific class of fractional differential equations, both linear and nonlinear. Using the Caputo fractional derivative, the study of an initial valued problem for a fractional differential equation provides some topological proprieties for the solution set, and shows it is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser’s theorem on the solution structure of the ordinary differential equation and relax some results about the fractional differential equation. Also, we establish existence results for Caputo fractional stochastic differential equations. Finally, we study the existence of solution for fractional differential inclusion in Banach lattice.

We investigate the probabilistic feasibility of randomized solutions to two distinct classes of uncertain multi-agent optimization programs. We first assume that only the constraints of the program are affected by uncertainty, while the cost function is arbitrary. Leveraging recent developments on a posteriori analysis within the scenario approach, we provide probabilistic guarantees for all feasible solutions of the program under study. This is particularly useful in cases where the numerical implementation of the solution-seeking algorithm prevents the exact quantification of the optimal solution. Furthermore, this result provides guarantees for the entire solution set of optimization programs with uncertain convex constraints and (possibly) non-convex cost function. We then focus on optimization programs with deterministic constraints, where the cost function depends on uncertainty and admits an aggregate representation of the agents’ decisions. By exploiting the structure of the program under study and leveraging the so called support rank notion, we provide agent-independent robustness certificates for the optimal solution, i.e., the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of each agent’s decision space. This substantially reduces the amount of samples required to achieve a certain level of probabilistic robustness for a larger number of agents. All robustness certificates provided in this paper are distribution-free and can be used alongside any optimization algorithm. Our theoretical results are accompanied by a numerical case study of a charging control problem for a fleet of electric vehicles.

We study the problem of how to compute the boolean abstraction of the solution set of 1 a linear equation system over the positive reals. We call a linear equation system φ exact for the 2 boolean abstraction if the abstract interpretation of φ over the structure of booleans is equal to the 3 boolean abstraction of the solution set of φ over the positive reals. Abstract interpretation over 4 the booleans is thus complete for the boolean abstraction when restricted to exact linear equation 5 systems, while it is not complete more generally. We present a new rewriting algorithm that makes 6 linear equation systems exact for the boolean abstraction while preserving the solutions over the 7 positive reals. The rewriting algorithm is based on the elementary modes of the linear equation 8 system. The computation of the elementary modes may require exponential time in the worst 9 case, but is often feasible in practice with freely available tools. For exact linear equation systems 10 we can compute the boolean abstraction by finite domain constraint programming. This yields a 11 solution of the initial problem that is often feasible in practice. Our exact rewriting algorithm has 12 two further applications. First, it can be used to compute the sign abstraction of linear equation 13 systems over the reals, as needed for analysing function programs with linear arithmetics. And 14 second it can be applied to compute the difference abstraction of a linear equation system as used 15 in change prediction algorithms for flux networks in systems biology.

Selecting a subset of the best solutions among large-scale problems is an important area of research. When the alternative solutions are stochastic in nature, then it puts more burden on the problem. The objective of this paper is to select a set that is likely to contain the actual best solutions with high probability. If the selected set contains all the best solutions, then the selection is denoted as correct selection. We are interested in maximizing the probability of this selection; P(CS). In many cases, the available computation budget for simulating the solution set in order to maximize P(CS) is limited. Therefore, instead of distributing these computational efforts equally likely among the alternatives, the optimal computing budget allocation (OCBA) procedure came to put more effort on the solutions that have more impact on the selected set. In this paper, we derive formulas of how to distribute the available budget asymptotically to find the approximation of P(CS). We then present a procedure that uses OCBA with the ordinal optimization (OO) in order to select the set of best solutions. The properties and performance of the proposed procedure are illustrated through a numerical example. Overall results indicate that the procedure is able to select a subset of the best systems with high probability of correct selection using small number of simulation samples under different parameter settings.

This paper presents a multi-objective algorithm based on tissue P system (MO TPS for short) for solving the tri-objective vehicles routing problem with time windows (VRPTW). Unlike most of the work where just the accuracy or extensibility of the solution is the core, the proposed algorithm focuses on searching the boundaries of solution sets and ensuring the solutions have better extensibility and uniformly distributed. In MO TPS, the cells of the tissue P system are divided into two groups. The first group, consisting of only one cell, aims at approaching to the Pareto front by the NSGA-II while second group, consisting of six cells, focuses on searching boundaries by the artificial bee colony algorithm with different prioritization rules. The main ideas of the MO TPS are to utilize the evolution of two groups of cells with different functions in the tissue P system for searching the boundaries of solution sets, obtaining solution sets which are uniformly distributed and have better extensibility and approaching to the Pareto front on the premise of preserving the elite boundaries. 56 Solomon benchmarks are utilized to test algorithm performance. Experimental results show that on the premise of ensuring accuracy, the proposed approach outperforms compared algorithms in terms of three metrics.

It is an important task for us to forecast whether it will rain or not. Nowadays, the prediction methods based on neural network have great changes to the final prediction result with the different selection of hyper-parameters. In order to find an appropriate hyperparameter and build a stable neural network model, this paper proposes a GA-CNN-LSTM model which uses genetic algorithm to optimize the parameters of neural network. The model is divided into two parts. One part is CNN-LSTM, which serves as the mapping rule of input and output and plays the role of fitness function. The other part is genetic algorithm (GA), whose function is to find the optimal hyperparameter and its corresponding solution set. Applied to the real Australian weather data to test, the test results show that the accuracy of this model is 1.3527953% higher than that of the single CNN and 0.0934936% higher than that of the unoptimized CNN-LSTM model.

A robust static output feedback (SOF) Stackelberg game with multiple followers in stochastic linear parameter varying (LPV) systems is investigated. The conditions for the existence of a robust SOF Stackelberg strategy set under H∞ constraints are established by the cross-coupled matrix inequality (CCMI). To determine this strategy set, the optimization problems corresponding to the relevant cost bounds are defined, and their solution sets are derived using the Karush-Kuhn-Tucker conditions. The results show that the robust SOF Stackelberg strategy set can be obtained by solving higher-order cross-coupled matrix equations (CCMEs). Because CCMEs are complex and difficult to solve numerically, a heuristic algorithm is developed by combining the CCMEs with the CCMIs. The convergence property is proven using the Krasnoselskii-Mann (KM) iteration algorithm. Finally, two numerical examples are solved to demonstrate the reliability and usefulness of the proposed heuristic algorithm.

In this work, we study a parallel viscosity extragradient method for approximating a common solution of a finite system of pseudomonotone equilibrium problems and common fixed point problem for nonexpansive mappings in Hadamard spaces. We propose an iterative method and prove its strong convergence to an element in the intersection of the solution set of finite system of equilibrium problems and the fixed points set of nonexpansive mappings. Furthermore, we give an example in a Hadamard space which is not an Hilbert space to support the convergence theorem in the paper. This result generalizes and extends recent results in the literature.

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

This paper mainly studies the implementation of flower pollination algorithm in multi-objective optimization problems, flower pollination algorithm mostly uses the way of setting weight to transform multi-objective optimization into single objective optimization, but the setting of weight coefficient mostly depends on the experience of experts, it makes the whole algorithm more subjective, the final performance of the algorithm largely depends on the research level of researchers. This paper proposes a multi-objective flower pollination algorithm based on the traditional flower pollination algorithm, which uses non-dominated sorting to find the elite solution of each generation solution set, and then finds the Pareto optimal solution according to the crowding distance. The algorithm is compared with other three kinds of multi-objective optimization algorithms. The results show that multi objective flower pollination algorithm based on non-dominated sorting(NSFPA) has a good effect on solving multi-objective problems.

With the development of Artificial Intelligence and Big Data, more and more heuristic multi-objective algorithms are applied to the training process of data sets. In this paper, four heuristic multi-objective optimizations which were widely used in the data set training in recent years, are selected for the performance analysis and comparison. Through the comparison and analysis of the performance index for these algorithms on the benchmark problem, the advantages and disadvantages of these strategies in ensuring the convergence of the algorithm and maintaining the diversity of the solution sets are systematically expounded. The simulation results show that these algorithms have their own advantages and disadvantages in solving different specific problems, and the setting of the parameters and the initialization of the solution sets will have a great impact on the performance of the algorithm. Moreover, the different methods have the different abilities in maintaining the convergence and diversity of the solution sets. Although the complex optimization method has a better solution effect, the calculation time cost is higher. In practical application, it is necessary to select the appropriate algorithm flexibly according to the actual problems and conditions.

With the access of DG distributed energy such as single-phase photovoltaic, the inherent three-phase unbalance and negative sequence voltage of distribution network become more prominent. The traditional planning ignores the partial phase operation of distributed PV and the negative sequence voltage in the distribution network, resulting in a slightly poor effect. In this paper, aiming to restrain the influence of negative sequence voltage in distribution network connecting single phase photovoltaic power, an optimal objective function to minimize negative sequence voltage, active power losses on a transmission line and the degree of voltage instability is set up. The Bacterial Colony Chemotaxis Optimization algorithm is used to search the Pareto optimal solution set. The optimal solution is obtained only if the sum of Pareto subset among the optimal solution set. The proposed approach in the paper can effectively solve the multi-objective optimization problem and improve the convergence.

An improved multi-objective particle swarm optimization (MOPSO) variant known as the MOPSO with alternate learning strategies (MOPSOALS) is proposed to overcome the drawbacks of most existing MOPSO variants that can only solve the selected categories of optimization problems with good performance due to the limited directional information brought by search operators. Particularly, both of the current and memory swarm evolution are incorporated into MOPSOALS as the more robust mechanisms in handling different types of problems. Two search operators are introduced in current swarm evolution to determine the particle’s new velocity, while three operators are proposed to fine tune the particle’s personal best position. These five proposed search operators are anticipated to guide all MOPSOALS particles to perform thorough searching in the solution search spaces with various exploration and exploitation strengths by fully utilizing all useful information contained in the non-dominated solution set. The proposed MOPSOALS is reported to have better performance in solving all selected test functions than the five peer algorithms.

When ranking search results for multiple objectives, such as maximizing the relevance and diversity of retrieved documents, competing objectives can induce a space of optimal solutions, each reflecting a different optimal trade-off over objectives. This raises several important questions. Firstly, what Pareto optimal solution set is induced by objective functions? Secondly, what are the best solutions achievable on a given dataset? Finally, how closely do the best dataset solutions approach the true Pareto optimal? We present a clear conceptual framing of these questions, with supporting terminology and visualizations, distinguishing Pareto vs. "dataset-best" solutions and providing strong intuition about how and why different optimization problems lead to different shapes and forms of solutions (regardless of optimization technique). We also provide benchmark problems for verifying the correctness of any Pareto machinery and show how existing multi-objective optimization (MOO) and filter methods can be used to provide accurate and interpretable answers to the above questions. Finally, we show how user-defined constraints imposed on the solution space can be effectively handled. In sum, we provide conceptual, translational, and practical contributions to solving MOO problems in retrieval.

Dominance move (DoM) is a quality indicator that compares two solution sets in a Pareto-optimal sense. The main issue related to DoM is its computational expense. A recent paper proposed a mixed-integer programming (MIP) approach for computing DoM that exhibited a computational complexity that is linear to the number of objectives and polynomial to the number of solutions. Even with this property, considering practical situations, the MIP-DoM calculation on some problems may take many hours. This paper presents an approximation method to deal with the problem using a cluster-based and divide-and-conquer strategy. Some experiments are tested, showing that the cluster based-algorithm is computationally much faster and makes a small percentage error from the original DoM value.