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1.
Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function. 相似文献
2.
Johannes Jahn Truong Xuan Duc Ha 《Journal of Optimization Theory and Applications》2011,148(2):209-236
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. 相似文献
3.
Positivity - In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set... 相似文献
4.
We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero. 相似文献
5.
David Bergman Andre A. Cire Willem-Jan van Hoeve Tallys Yunes 《Journal of Heuristics》2014,20(2):211-234
In this paper we introduce a new method for generating heuristic solutions to binary optimization problems. We develop a technique based on binary decision diagrams. We use these structures to provide an under-approximation to the set of feasible solutions. We show that the proposed algorithm delivers comparable solutions to a state-of-the-art general-purpose optimization solver on randomly generated set covering and set packing problems. 相似文献
6.
In this paper, we propose two kinds of robustness concepts by virtue of the scalarization techniques (Benson’s method and elastic constraint method) in multiobjective optimization, which can be characterized as special cases of a general non-linear scalarizing approach. Moreover, we introduce both constrained and unconstrained multiobjective optimization problems and discuss their relations to scalar robust optimization problems. Particularly, optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem, and these solutions are efficient under uniqueness assumptions. Two examples are employed to illustrate those results. Finally, the connections between robustness concepts and risk measures in investment decision problems are also revealed. 相似文献
7.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2003,2(1):13-29
The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients. 相似文献
8.
《European Journal of Operational Research》2005,162(3):584-601
Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems). 相似文献
9.
A Nonlinear Scalarization Function and Generalized Quasi-vector Equilibrium Problems 总被引:1,自引:0,他引:1
Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization
problems can be found by solving scalar optimization problems. In this paper we introduce a nonlinear scalarization function
for a variable domination structure. Several important properties, such as subadditiveness and continuity, of this nonlinear
scalarization function are established. This nonlinear scalarization function is applied to study the existence of solutions
for generalized quasi-vector equilibrium problems.
This paper is dedicated to Professor Franco Giannessi for his 68th birthday 相似文献
10.
分别在有pre-order的无线性结构的集合和拓扑空间中,给出了有效点的存在性。作为应用,讨论了向量优化问题中解的存在性。最后给出了紧、弱紧、锥紧、锥半紧、上序紧、下序紧、上序半紧、准上序半紧和准下序半紧等之间的关系。 相似文献
11.
This paper establishes a bridge between set optimization problems and vector Ky Fan inequality problems. We introduce a general model, called the bifunction-set optimization problem, that provides a unifying framework for the above-mentioned problems. An existence result in our model is obtained, with the help of KKM–Fan’s lemma. As applications, we derive some new or sharper existence results for set optimization problems and generalized vector Ky Fan inequalities with efficient solutions. 相似文献
12.
In this paper we introduce several concepts of approximate solutions of set-valued optimization problems with vector and set optimization. We prove existence results and necessary and sufficient conditions by using limit sets. 相似文献
13.
Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant
interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic
optimization models involving second order stochastic dominance constraints can be solved by linear programming. However,
problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated
feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained
optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions
by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive
relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure.
We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable
fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio
optimization problem.
This research was supported by the NSF awards DMS-0603728 and DMI-0354678. 相似文献
14.
Vyacheslav V. Chistyakov Boris I. Goldengorin Panos M. Pardalos 《Journal of Global Optimization》2012,53(3):475-495
The currently adopted notion of a tolerance in combinatorial optimization is defined referring to an arbitrarily chosen optimal solution, i.e., locally. In this paper we introduce global tolerances with respect to the set of all optimal solutions, and show that the assumption of nonembededdness of the set of feasible solutions in the provided relations between the extremal values of upper and lower global tolerances can be relaxed. The equality between globally and locally defined tolerances provides a new criterion for the multiplicity (uniqueness) of the set of optimal solutions to the problem under consideration. 相似文献
15.
In this paper, we show how a nonlinear scalarization functional can be used in order to characterize several well-known set order relations and which thus plays a key role in set optimization. By means of this functional, we derive characterizations for minimal elements of set-valued optimization problems using a set approach. Our methods do not rely on any convexity assumptions on the considered sets. Furthermore, we develop a derivative-free descent method for set optimization problems without convexity assumptions to verify the usefulness of our results. 相似文献
16.
In this paper, we introduce a Minty type vector variational inequality, a Stampacchia type vector variational inequality, and the weak forms of them, which are all defined by means of subdifferentials on Hadamard manifolds. We also study the equivalent relations between the vector variational inequalities and nonsmooth convex vector optimization problems. By using the equivalent relations and an analogous to KKM lemma, we give some existence theorems for weakly efficient solutions of convex vector optimization problems under relaxed compact assumptions. 相似文献
17.
In this paper, we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization
problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points
with respect to the order relation. In the case where the lower-level problem of the bilevel optimization problem is convex
and continuously differentiable in the lower-level variables, this order relation is equivalent to a second, more tractable
order relation.
Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the
order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before.
We comment also on the practical and computational implications of our approach. 相似文献
18.
In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends
the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector
equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions
that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence
results for both systems of vector variational-like inequalities and vector optimization problems with variable preference. 相似文献
19.
Efficiency and Henig Efficiency for Vector Equilibrium Problems 总被引:6,自引:0,他引:6
We introduce the concept of Henig efficiency for vector equilibrium problems, and extend scalarization results from vector optimization problems to vector equilibrium problems. Using these scalarization results, we discuss the existence of the efficient solutions and the connectedness of the set of Henig efficient solutions to the vector-valued Hartman–Stampacchia variational inequality. 相似文献
20.
This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles. 相似文献