Lagrange Duality in Set Optimization

Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. “Saddle sets” replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand, and the existence of a saddle set for the Lagrangian on the other hand. Applications to set-valued risk measures are indicated.     

New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set P(Y)\mathcal{P}(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \preccurlyeq\preccurlyeq: minimal elements, semicompactness, completeness, domination property of a subset of Q\mathcal{Q}, and semicontinuity of a set-valued map with values in Q\mathcal{Q} in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) from which one can easily derive similar results for the case, when F takes values on P(Y)\mathcal{P}(Y) equipped with various order relations.     

广播网络的最佳源点集

广播是研究通信网络的某个成员的消息如何尽快地传递给所有其它成员的消息传递问题,有两类常见的通信模式,一类是shouting模式,即在一个单位时间内,一个顶点能够和它的户斤有邻点通信;另一类是whispering模式,即在一个单位时间以内,一个顶点最多只能和它的一个邻点通信,通信网络通常用图来描述,最初贮存消息的网络成员称为源点。     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

Characterizations of the Set Less Order Relation in Nonconvex Set Optimization

Journal of Optimization Theory and Applications - For nonconvex set optimization problems based on the set less order relation, this paper presents characterizations of optimal sets and gives...     

Generalized B-Well-Posedness for Set Optimization Problems

This paper aims at studying the generalized well-posedness in the sense of Bednarczuk for set optimization problems with set-valued maps. Three kinds of B-well-posedness for set optimization problems are introduced. Some relations among the three kinds of B-well-posedness are established. Necessary and sufficient conditions of well-posedness for set optimization problems are obtained.     

About Asymptotic Analysis and Set Optimization

Set-Valued and Variational Analysis - We introduce, in a natural way, a new notion of asymptotic map for a set-valued map. The non-uniqueness of such an asymptotic map leads to consider equivalent...     

集优化问题的广义适定性与解的稳定性

本文研究了集优化问题的适定性与解的稳定性. 首次利用嵌入技术引入了集优化问题的广义适定性概念, 得到了此类适定性的一些判定准则和特征, 并给出其充分条件. 此外, 借助一类广义Gerstewitz 函数, 建立了此类适定性与一类标量优化问题广义适定性之间的等价关系. 最后, 在适当条件下研究了含参集优化问题弱有效解映射的上半连续性和下半连续性.     

Active Constraint Set Invariancy Sensitivity Analysis in Linear Optimization

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant.For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: “what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?”. The differences of the results between the linear and convex quadratic optimization problems are highlighted too.     

Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

Journal of Optimization Theory and Applications - In this paper, we give some properties concerned with weak minimal solutions of nonconvex set optimization problems. We also give some properties...     

On the Stability of the Feasible Set in Linear Optimization

This paper deals with the stability of two families of linear optimization problems, each one formed by the dual problems to the members of the other family. We characterize the problems of these families that are stable in the sense that they remain consistent (inconsistent) under sufficiently small arbitrary perturbations of all the data. This characterization is established in terms of the lower semicontinuity property of the feasible set mapping and the boundedness of the optimal set of the corresponding coupled problem. Other continuity properties of the feasible set mapping are also derived. This stability theory extends some well-known theorems of Williams and Robinson on the stability of ordinary linear programming problems to linear optimization problems with infinitely many variables or constraints.     

On Optimization over the Efficient Set in Linear Multicriteria Programming

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example. The research was partly done while the third author was visiting the Department of Mathematics, University of Trier with the support by the Alexander von Humboldt Foundation. He thanks the university as well as the foundation.     

On Essential Stable Sets of Solutions in Set Optimization Problems

This paper introduces the essential stability for set optimization problems. Some kinds of essential stable sets of weakly minimal and minimal solutions are shown. The graph of minimal solution mappings is not necessarily closed, which is different from weakly minimal solution mappings. The existence of minimum essential sets of minimal solutions is proved.     

积分——水平集求全局最优的遗传算法实现

Chew Soo Hong,Zheng Q uan提出了一个积分——水平集求全局最优的概念性算法及M on te-C ar-lo随机投点的实现途径,并在很多实际问题中得到了很好的应用,但这一实现算法的收敛性是个未解决的问题.利用近年来广泛应用的遗传算法,给出了这一算法的另一种实现途径,并从理论和数值两个方面验证了算法的可行性.     

Stochastic Optimization over a Pareto Set Associated with a Stochastic Multi-Objective Optimization Problem

We deal with the problem of minimizing the expectation of a real valued random function over the weakly Pareto or Pareto set associated with a Stochastic Multi-objective Optimization Problem, whose objectives are expectations of random functions. Assuming that the closed form of these expectations is difficult to obtain, we apply the Sample Average Approximation method in order to approach this problem. We prove that the Hausdorff–Pompeiu distance between the weakly Pareto sets associated with the Sample Average Approximation problem and the true weakly Pareto set converges to zero almost surely as the sample size goes to infinity, assuming that our Stochastic Multi-objective Optimization Problem is strictly convex. Then we show that every cluster point of any sequence of optimal solutions of the Sample Average Approximation problems is almost surely a true optimal solution. To handle also the non-convex case, we assume that the real objective to be minimized over the Pareto set depends on the expectations of the objectives of the Stochastic Optimization Problem, i.e. we optimize over the image space of the Stochastic Optimization Problem. Then, without any convexity hypothesis, we obtain the same type of results for the Pareto sets in the image spaces. Thus we show that the sequence of optimal values of the Sample Average Approximation problems converges almost surely to the true optimal value as the sample size goes to infinity.     

Vectorization hierarchies of some graph quantifiers

We give a sufficient condition for the inexpressibility of the k-th extended vectorization of a generalized quantifier in , the extension of first-order logic by all k-ary quantifiers. The condition is based on a model construction which, given two -equivalent models with certain additional structure, yields a pair of -equivalent models. We also consider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity. Received: 15 October 1996     

Reduced Vertex Set Result for Interval Semidefinite Optimization Problems

In this paper we propose a reduced vertex result for the robust solution of uncertain semidefinite optimization problems subject to interval uncertainty. If the number of decision variables is m and the size of the coefficient matrices in the linear matrix inequality constraints is n×n, a direct vertex approach would require satisfaction of 2 ^{n(m+1)(n+1)/2} vertex constraints: a huge number, even for small values of n and m. The conditions derived here are instead based on the introduction of m slack variables and a subset of vertex coefficient matrices of cardinality 2 ⁿ⁻¹, thus reducing the problem to a practically manageable size, at least for small n. A similar size reduction is also obtained for a class of problems with affinely dependent interval uncertainty. This work is supported by MIUR under the FIRB project “Learning, Randomization and Guaranteed Predictive Inference for Complex Uncertain Systems,” and by CNR RSTL funds.