Higher-Order optimality conditions for set-valued optimization

2007-2008 > Academic research: refereed > Publication in refereed journa

Extended Well-Posedness of Quasiconvex Vector Optimization Problems.

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems

Minty variational principle for set-valued variational inequalities

It is well known that a solution of a Minty scalar variational inequality of differential type is a solution of the related optimization problem, under lower semicontinuity assumption. This relation is known as "Minty variational principle".In the vector case, the Minty variational principle has been investigated by F. Giannessi [15] and subsequently by X. M. Yang, X. Q. Yang, K. L. Teo [22]. For a differentiable objective function f: R n \u2192 R m it holds only for pseudoconvex functions. In this paper we extend the Minty variational principle to set-valued variational inequalities with respect to an arbitrary ordering cone and non smooth objective function. As a special case of our result we get that of [22

Some remarks on the Minty vector variational principle

AbstractIn scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93–99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193–201]. In these papers, in the particular case of a differentiable objective function f taking values in Rm and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions

Extended well-posedness of vector optimization problems: the convex case

In this paper we investigate a notion of extended well-posedness in vector optimization. Appropriate asymptotically minimizing sequences, when both the objective function and the feasible region are subject to perturbation are introduced. We show that convex problems, i.e. problems in which both the objective function and the perturbations are C−convex, are extended wellposed. Further, we characterize the proposed well-posedness notion both in terms of linear and nonlinear scalarization

Robust optimization: Sensitivity to uncertainty in scalar and vector cases, with applications

The question we address is how robust solutions react to changes in the uncertainty set. We prove the location of robust solutions with respect to the magnitude of a possible decrease in uncertainty, namely when the uncertainty set shrinks, and convergence of the sequence of robust solutions. In decision making, uncertainty may arise from incomplete information about people\u2019s (stakeholders, voters, opinion leaders, etc.) perception about a specific issue. Whether the decision maker (DM) has to look for the approval of a board or pass an act, they might need to define the strategy that displeases the minority. In such a problem, the feasible region is likely to unchanged, while uncertainty affects the objective function. Hence the paper studies only this framework.The question we address is how robust solutions react to changes in the uncertainty set. We prove the location of robust solutions with respect to the magnitude of a possible decrease in uncertainty, namely when the uncertainty set shrinks, and convergence of the sequence of robust solutions. In decision making, uncertainty may arise from incomplete information about people's (stakeholders, voters, opinion leaders, etc.) perception about a specific issue. Whether the decision maker (DM) has to look for the approval of a board or pass an act, they might need to define the strategy that displeases the minority. In such a problem, the feasible region is likely to unchanged, while uncertainty affects the objective function. Hence the paper studies only this framework

Convexity and global well-posedness in set-optimization

Well-posedness for vector optimization problems has been extensively studied. More recently, some attempts to extend thee results to set-valued optimization have been proposed, mainly applying some scalarization. In this paper we propose a new definition of global well-posedness for set-optimization problems. Using an embedding technique proposed by Kuroiwa and Nuriya (2006), we prove well-posedness property of a class of generalized convex set-valued maps