Sensitivity Analysis in Set-Valued Optimization and Vector Variational Inequalities
Sensitivity analysis, the quantitative analysis of the perturbation map, is of paramount interest in optimization theory and has applications in several branches of pure and applied mathematics. During the last five decades, substantial progress has been
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Sensitivity Analysis in Set-Valued Optimization and Vector Variational Inequalities
Sensitivity analysis, the quantitative analysis of the perturbation map, is of paramount interest in optimization theory and has applications in several branches of pure and applied mathematics. During the last five decades, substantial progress has been made in sensitivity analysis for optimization problems with scalar objectives. On the other hand, the differentiability issues of the perturbation map for vector optimization problems and set optimization problems are rather involved and they require modern tools from variational analysis. The main difficulty here stems from the fact that the perturbation map for such problems is, in general, set-valued. To fix these ideas, we begin by describing the general framework for such problems. Let X , Y , and Z be normed spaces, and let C Y be a nonempty cone in Y . Let H W X ÃZ and G W X ZÃY be set-valued maps. We consider the following parametrized family of set-valued optimization problems defined by P .x/ D Min.F .x/; C /;
(13.1)
F .x/ WD fy 2 G.x; z/j z 2 H.x/g:
(13.2)
where
Here we interpret X as the parameter space, Z as the decision space, and Y as the objective space, with the cone C inducing a partial ordering on objective values in Y . For each value of the parameter x, the set H.x/ defines a feasible set in Z, whereas the set G.x; z/ gives a set of objectives in the space Y . The set F .x/ comprises the graph of the objective values over the entire feasible region of the optimization problem. In this setting, P .x/ singles out those objective values that cannot be improved relative to the ordering induced by C . Evidently, when G is single-valued, the above model then recovers the setting of parametric vector optimization. If additionally, Y C R and C D RC , then we are in the framework of scalar parametric optimization. The objective of this chapter is to employ derivatives
© Springer-Verlag Berlin Heidelberg 2015 A.A. Khan et al., Set-valued Optimization, Vector Optimization, DOI 10.1007/978-3-642-54265-7__13
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13 Sensitivity Analysis in Set-Valued Optimization and Vector Variational Inequalities
of set-valued maps to estimate how sets F .x/ and P .x/ vary with changes in the parameter x. For this purpose, we will use graphical derivatives as well as the coderivatives of these set-valued maps. For this study, a crucial role is played by the derivatives of F , F C C , P , P C C and their minimal points. In this chapter, we will also study differentiability properties of set-valued gap functions associated to vector variational inequalities and sensitivity analysis for vector variational inequalities. For set-valued optimization and for vector variational inequalities, both the first-order as well as the second-order results are given.
13.1 First Order Sensitivity Analysis in Set-Valued Optimization Many authors have focused on sensitivity analysis using first-order contingent derivatives. For instance, Tanino [564, 565] derived many interesting relations
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