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Distributed primal-dual methods have been widely used for solving large-scale constrained optimization problems. The majority of existing results focus on the problems with decoupled constraints. Some recent works have studied the problems subject to separable globally coupled constraints. This paper considers the distributed optimization problems with globally coupled constraints over networks without requiring the separability of the globally coupled constraints. This is made possible by the local estimates of the constraint violations. For solving such a problem, we propose a primal-dual algorithm in the augmented Lagrangian framework, combining the average consensus technique. We first establish a non-ergodic convergence rate of $\mathcal{O}\left(1/k\right)$ in terms of the objective residual for solving a distributed constrained convex optimization problem, where k is the iteration counter. Specifically, the global objective function is the aggregate of the local convex and possibly non-smooth costs, and the coupled constraint is the sum of the local linear equality constraints. The numerical results illustrate the performance of the proposed method.

In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.

We introduce and investigate the coordinated log-$ h $-convexity for interval-valued functions. Also, we prove some new Jensen type inequalities and Hermite-Hadamard type inequalities, which generalize some known results in the literature. Moreover, some examples are given to illustrate our results.

We address the problem of motion planning for aerial drones to allow them to fly through confined spaces and narrow gaps between obstacles. Finding safe and feasible trajectories for steering multiple drones towards some desired positions in tight spaces or guiding them through gaps smaller in width than their diameters is impossible without taking the drones’ orientations into account. To incorporate the orientation into the motion planning problem we employ an ellipsoid model of the drone body, and utilize the separating hyperplane theorem for convex sets to derive constraints that guarantee collision avoidance between the ellipsoid-shaped drone and obstacles modeled as ellipsoids, spheres, or polygons. The resulting set of constraints is seamlessly integrated into the motion planning method based on Bezier curve whose properties are exploited for efficient evaluation of the constraints. The efficacy of the proposed method in generating feasible and collision-free trajectories is demonstrated via different simulations involving single and multiple drones.

This paper studies the distributed convex optimization problem with a common decision variable, a global inequality constraint, and local constraint sets over a time-varying multi-agent network, the objective function of which is a sum of agents' local convex cost functions. To solve such problem, a penalty-based distributed continuous-time subgradient algorithm with time-varying gain is developed for each agent to seek the saddle point of the penalty Lagrangian function. It is shown that an exact primal optimal solution can be obtained with certain assumption on time-varying gain. Moreover, the proposed algorithm adopts fixed-time projection scheme to ensure that for any initial state value, each local state estimate converges to its convex constraint set within fixed time. Finally, numerical examples are provided to show the effectiveness of the theoretical results.

The objective of the authors is to introduce the new class of convex fuzzy-interval-valued functions (convex-FIVFs), which is known as $ p $-convex fuzzy-interval-valued functions ($ p $-convex-FIVFs). Some of the basic properties of the proposed fuzzy-interval-valued functions are also studied. With the help of $ p $-convex FIVFs, we have presented some Hermite-Hadamard type inequalities ($ H-H $ type inequalities), where the integrands are FIVFs. Moreover, we have also proved the Hermite-Hadamard-Fejér type inequality ($ H-H $ Fejér type inequality) for $ p $-convex-FIVFs. To prove the validity of main results, we have provided some useful examples. We have also established some discrete form of Jense's type inequality and Schur's type inequality for $ p $-convex-FIVFs. The outcomes of this paper are generalizations and refinements of different results which are proved in literature. These results and different approaches may open new direction for fuzzy optimization problems, modeling, and interval-valued functions.

It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (H·H) and Hermite-Hadamard Fejér (H·H Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.

In this paper, we consider stochastic approximation algorithms for least-square and logistic regression with no strong-convexity assumption on the convex loss functions. We develop two algorithms with varied step-size motivated by the accelerated gradient algorithm which is initiated for convex stochastic programming. We analyse the developed algorithms that achieve a rate of $ O(1/n^{2}) $ where $ n $ is the number of samples, which is tighter than the best convergence rate $ O(1/n) $ achieved so far on non-strongly-convex stochastic approximation with constant-step-size, for classic supervised learning problems. Our analysis is based on a non-asymptotic analysis of the empirical risk (in expectation) with less assumptions that existing analysis results. It does not require the finite-dimensionality assumption and the Lipschitz condition. We carry out controlled experiments on synthetic and some standard machine learning data sets. Empirical results justify our theoretical analysis and show a faster convergence rate than existing other methods.

We consider facility location problems where n facilities are present in a convex polygon in the rectilinear plane, over which continuous and uniform demand is distributed and within which a convex polygonal barrier is located (removing all demand and preventing all travel within the barrier), and the optimal location for an additional facility is sought. We start with an in-depth analysis of the representation of the bisectors of two facilities affected by the barrier and how it is affected by the position of the additional facility. Following this, a detailed investigation into the changes in the structure of the Voronoi diagram caused by the movement of this additional facility, which governs the form of the objective function for numerous facility location problems, yields a set of linear constraints for a general convex barrier that partitions the market space into a finite number of regions within which the exact solution can be found in polynomial time. This allows us to formulate a polynomial exact algorithm that makes use of a triangular decomposition of the incremental Voronoi diagram and the first order optimality conditions.

Convex optimization is explored in this paper to maximize the security against power analysis attacks while minimizing the power and performance overheads for a cryptographic circuit employs a random dynamic voltage and frequency scaling (RDVFS) technique. To realize this optimization goal, firstly, non-convex functions are used for modeling all the critical parameters related to the RDVFS technique to create a non-convex objective function. Subsequently, the non-convex objective function is appropriately converted into a convex objective function in order to grab the global optimum value. By transforming the acquired convex objective function into the Lagrange dual function and using Karush–Kuhn–Tucker (KKT) conditions to solve this optimization problem, the best RDVFS strategy can be uncovered. As shown in the result, after employing convex optimization for a cryptographic circuit with a RDVFS technique, the corresponding figure-of-merit (FOM) is enhanced by 43.4%. Moreover, in comparison with the conventional machine learning technique, the proposed convex optimization technique improves the overall FOM related with RDVFS technique by 17.6% while reducing 85.3% computational time.

This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

In this present paper, we introduce and explore certain new classes of uniformly convex and starlike functions related to the Liu–Owa integral operator. We explore various properties and characteristics, such as coefficient estimates, rate of growth, distortion result, radii of close-to-convexity, starlikeness, convexity, and Hadamard product. It is important to mention that our results are a generalization of the number of existing results in the literature.

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe–Aumann) mixtures of two acts f and g, [Formula: see text]. We propose a preference condition, monotonicity in optimal mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between monotonicity in optimal mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg paradoxes. We find that monotonicity in optimal mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class. This implies, for example, that maxmin expected utility preferences are consistent with monotonicity in optimal mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between monotonicity in optimal mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which monotonicity in optimal mixtures places upper bounds on the intensity of ambiguity-averse behavior. This paper was accepted by Manel Baucells, decision analysis.

In this paper we present efficient algorithms to compute the radius, diameter, incenter, circumcenter, width and k dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in ℜ d for several types of inputs. We get optimal algorithms when the time complexities are linear. Let the size of input (convex polyhedron P /set S of points/convex polyhedra) be n / N , respectively, size of distance function convex polygon/polyhedron Q be m and size of constraint convex polyhedral region R be r in plane or ℜ d . The radius, incenter and circumcenter of the convex polygon P in ℜ 2 can be optimally computed in linear time, i.e. O ( n + m ) and in time O ( n + m + r ) , if the incenter or circumcenter is additionally constrained in a convex polygon/polyhedron R . In ℜ d , the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O ( N m ) and O ( N m + r ) time respectively, for convex constraint polyhedral region R . We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O ( N m ) -time or O ( N m + r ) -time respectively for convex constraint polyhedral region R in ℜ d . The diameter of a convex polygon in plane can be computed in O ( n + m ) time and the diameter of a convex polyhedron in ℜ d can be computed in O ( n m ) time. The width of a convex polyhedron can be computed in O ( n + m ) time and O ( n 2 m ) time, for ℜ 2 and ℜ d , respectively. The diameter and width of a set of convex polyhedra can be computed in O ( N m ) -time and ( N 2 m ) , respectively, in any dimensions. We also show how the k dimensional minimum enclosing/stabbing cylinder for the convex polyhedron P can be computed in O ( n d − k + 2 m 2 ) in ℜ d and in O ( n d m ( n + m ) ) in ℜ d for k = 1 . The k dimensional minimum enclosing/stabbing cylinder for the set of convex polyhedra S in ℜ d can be computed in O ( N d − k + 2 m 2 ) -time. The diameter and width problems can also be solved for a set of points in ℜ d , either in time O ( n m ) or in time O ( m T ( n ) ) , where T ( n ) is the time complexity of the best convex hull computation algorithm for the given set of points.

Recently chamfer distances have been developed not only on the usual integer grids, but also on some non traditional grids including grids which are not lattices. In this paper the truncated hexagonal grid is considered: its pixels are dodecagons and two shaped (oriented) triangles. Two types of ‘natural’ neighborhood relations are considered on the grid, consequently two weights are used to describe the chamfer distances. Formulae to compute the minimal weights of a connecting path, i.e., the distance of any two pixels, are provided to cases depending on the relative ratio of the weights. Some properties of these distances, including metricity are also analysed. Digital disks based on the weighted distances are also investigated. In some cases, these disks may not be convex, moreover they may contain holes. The conditions of holes are characterised.

We propose a novel class of convex risk measures, based on the concept of the Fr\'echet mean, designed in order to handle uncertainty which arises from multiple information sources regarding the risk factors of interest. The proposed risk measures robustly characterize the exposure of the firm, by filtering out appropriately the partial information available in individual sources into an aggregate model for the risk factors of interest. Importantly, the proposed risks can be expressed in closed analytic forms allowing for interesting qualitative interpretations as well as comparative statics and thus facilitate their use in the everyday risk management process of the insurance firms. The potential use of the proposed risk measures in insurance is illustrated by two concrete applications, capital risk allocation and premia calculation under uncertainty., Comment: 32 pages

In the present work, we consider Zuckerberg's method for geometric convex-hull proofs introduced in [Geometric proofs for convex hull defining formulations, Operations Research Letters 44(5), 625-629 (2016)]. It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg's proof technique, building on ideas from [Extended formulations for convex hulls of some bilinear functions, Discrete Optimization 36, 100569 (2020)]. We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg's proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

The approximate Caratheodory theorem states that given a compact convex set $${\mathcal {C}}\subset {\mathbb {R}}^n$$ and $$p\in [2,+\infty [$$ , each point $$x^*\in {\mathcal {C}}$$ can be approximated to $$\epsilon $$ -accuracy in the $$\ell _p$$ -norm as the convex combination of $${\mathcal {O}}(pD_p^2/\epsilon ^2)$$ vertices of $${\mathcal {C}}$$ , where $$D_p$$ is the diameter of $${\mathcal {C}}$$ in the $$\ell _p$$ -norm. A solution satisfying these properties can be built using probabilistic arguments or by applying mirror descent to the dual problem. We revisit the approximate Caratheodory problem by solving the primal problem via the Frank-Wolfe algorithm, providing a simplified analysis and leading to an efficient practical method. Furthermore, improved cardinality bounds are derived naturally using existing convergence rates of the Frank-Wolfe algorithm in different scenarios, when $$x^*$$ is in the interior of $${\mathcal {C}}$$ , when $$x^*$$ is the convex combination of a subset of vertices with small diameter, or when $${\mathcal {C}}$$ is uniformly convex. We also propose cardinality bounds when $$p\in [1,2[\cup \{+\infty \}$$ via a nonsmooth variant of the algorithm. Lastly, we address the problem of finding sparse approximate projections onto $${\mathcal {C}}$$ in the $$\ell _p$$ -norm, $$p\in [1,+\infty ]$$ .

In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.

We establish the following Helly-type theorems: Let $${\mathcal {K}}$$ be a family of sets in $${\mathbb {R}}^3$$ , and let j be a fixed integer, $$0 \le j \le 3$$ . For every countable subfamily of $${\mathcal {K}}$$ , assume that the corresponding intersection is consistent relative to staircase paths, is staircase convex, and contains a convex set of dimension at least j. Then $$\cap \{K \, : \, K \text { in } {\mathcal {K}} \}$$ has these properties as well. For d a fixed integer, $$d \ge 1$$ , define a function f on $$\{0,1\}$$ by $$f(0)=d+1$$ , $$f(1)=2d$$ . Let $${\mathcal {K}}$$ be a finite family of orthogonal polytopes in $${\mathbb {R}}^d$$ . For j fixed in $$\{ 0,1\}$$ , assume that every f(j) members of $${\mathcal {K}}$$ intersect in a set that is staircase convex and contains a convex subset of dimension at least j. Then $$\cap \{K : K \text { in } {\mathcal {K}} \}$$ has these properties, too. The number f(j) is best possible for $$j=0$$ and for $$j=1$$ . There is no analogous Helly number for $$2 \le j \le d$$ .

In this paper, we classify all isolated, completely isolated, and (left/right) convex subsemigroups of $${\mathcal {C}}_{n}$$ , the semigroup of all order-preserving and decreasing transformations on $$X_{n}=\{1,\ldots ,n\}$$ under its natural order. Moreover, we find the rank of each convex subsemigroup of $${\mathcal {C}}_{n}$$ .

This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

Due to applications in almost every area of mathematics, the theory of convex and nonconvex functions becomes a hot area of research for many mathematicians. In the present research, we generalize the Hermite–Hadamard-type inequalities for p , h -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for p , h -convex functions. Finally, the applications of our main findings are also given.

We propose a new grid unfolding without refinement of a new sub-class of polycubes that we call Manhattan towers with an H-convex base. Such poly-cubes can be seen both as a Manhattan tower and as an orthostack. At the end of the paper we show that an extension of this algorithm to Up-and-Down Orthoterrains is also possible.

In the paper, by using the properties of Schur m-power convex function, we discuss Schur m-power convexity of a new class of symmetric functions where i1, i2, …, ir are non-negative integers, and p ∈ N+. We obtain that is Schur m-power convex for m ≤ 0 and Schur m-power concave for m ≥ p. We also give a counter example to illustrate is neither Schur convex nor Schur concave for p>1. As applications, a Klamkin-Newman type inequality and some analytic inequalities are derived.