The object of this paper is to perform an analysis of the sensitivity for convex vector programs with inequality constraints by examining the quantitative behavior of a certain set of optima according to changes of right-hand side parameters included in the program. The results in the paper prove that the sensitivity of the program depends on the solution of a dual program and its sensitivity.
The problem of analyzing the sensitivity in vector programming has drawn the attention of many authors from the works by Kuhn and Tucker in 1951. It is known that one of the difficulties that appears treating this problem lies in the fact that, in vector programming, the set of the optimal values most of times is not a singleton. Thus whereas in the scalar case of scalar programming the optimal value reached is a minimum value, and therefore unique, in the case of vector programming the optimal ones are multiple values. This implies that, in general, it turns out to be more complicated to analyze sensitivity in vector optimization programs than in scalar optimization programs, since in the vectorial case the analysis of the sensitivity may necessitate to study a set-valued map (the set-valued map that assigns to each value of some parameter the set of optimal values reached by its associated program) whilst in the scalar case the analysis of sensitivity with respect to a parameter consists of the study of a function (the function that assigns to each value of the parameter the minimum value reached by its associated program).
One of the techniques used in sensitivity analysis is to reduce the problem by choosing a particular point in the efficient set. This is the case if we are interested in the best alternative which minimizes a specific scalar utility function as in [1], where the authors reduce to an optimization problem with scalar objective by minimizing the distance between some fixed desirable point and the efficient set, or as in [2], where the scalarization is done by the weighted sum approach, etc.
When dealing with a subset or the whole set of efficient points, there are several procedures. One is to assume the existence of an adequate selection of particular efficient points as in [3] where the authors study sensitivity taking a selection of the balance points introduced by E. Galperin and further developed in [4]. In [5], [6], [7], [8] and [9] the authors consider the so-called TT-optimal solutions and also assume the existence of a Fréchet differentiable selection. However, there are several approaches which deal with sets of efficient points and focus on the behavior of some set-valued perturbation maps (e.g., [10], [11] and [12], the two survey papers [13] and [14], and the references therein).
Continuing the line of inquiry of [7], a sensitivity analysis is performed in [15] for differential vector programs with equality constraints with respect to the right-hand side, proving that sensitivity of the problem depends not only on a suitable Lagrange multiplier but also on the derivative of a set-valued function of Lagrange multipliers.
Following a similar procedure to [15], the present paper performs an analysis of sensitivity for convex vector programs with inequality constraints, analyzing the quantitative behavior of certain set of optimal points (that are dense in the efficient line) according to changes of the right-hand side parameters included in the program.
The paper is organized as follows. Section 2 introduces notation, basic concepts, and some results that will be used throughout the paper. Section 3 is devoted to study two important properties (Theorem 4 and Theorem 5) of the set-valued function being the solution of a dual program that constitutes the base for the analysis of sensitivity developed in the next section. In Section 4, Theorem 6 proves that the sensitivity of the program depends on the set-valued function studied in Section 3 and the sensitivity of this set-valued function. A particularization of Theorem 6 for p-homogeneous programs is also presented in Theorem 7, because of its special usefulness since linear programs are 1-homogeneous.
For a vector convex optimization problem in which the feasible set depends on a parameter vector, we have considered the behavior of the set-valued map which associates to each parameter value the set of TT-optimal values. The concept of contingent derivatives of set-valued maps has been used because it depends on the point in the graph of the set-valued function, fixed when studying the sensitivity. By using the dual program introduced in [5], a set-valued dual function has been defined, and we have found a set-valued version of the relationship in [5] between the sensitivity of the program and the dual solution. It has been stated that the contingent derivative of the TT-optimal set-valued function depends on the dual solution plus the projection on Ker TT of its derivative. In Theorem 7, the above results are simplified when applied to pp-homogeneous programs and then to linear programs.
