Sensitivity Analysis in Convex Optimization through the Circatangent Derivative

Abstract

The main goal of this paper is to analyse the sensitivity of a vector convex optimization problem according to variations in the right-hand side. We measure the quantitative behavior of a certain set of Pareto optimal points characterized to become minimum when the objective function is composed with a positive function. Its behavior is analysed quantitatively by using the circatangent derivative for set-valued maps. Particularly, it is shown that the sensitivity is closely related to a Lagrange multiplier solution of a dual program.

A natural extension of the classical envelope theorem in vector differential programming

“…Nonetheless, [22,Example 3.1] shows that Property R is not sufficient to assure the tangential regularity ofΣ even when Σ is tangentially regular. To guarantee tangential regularity ofΣ, the set-valued map Σ must also to verify an additional property of regularity called S. Here we remember it.…”

“…Finally, [22,Theorem 3.1] shows that if a set-valued map Σ is tangentially regular and satisfies Properties R and S at (p 0 , G 0 ), thenΣ is also tangentially regular at (p 0 , G 0 (p 0 )) and…”

“…for every q ∈ P. [22,Example 3.2] shows that tangential regularity of Σ andΣ do not imply Σ to enjoy Property S nor (1) be satisfied.…”

“…Let us prove the theorem now when Ψ is tangentially regular at (p 0 , G xp 0 ). Since Ψ satisfies property B at (p 0 , G xp 0 ), by using Theorem 3.1 of [22], we have just to prove that Ψ satisfies property S at (p 0 ,…”

“…In this work, by applying a selection in the efficient set, two versions of the envelope theorem for differentiable and convex programs were stated. In the paper the authors used the so-called T -optimal solutions, concept successfully utilized in many other works of sensitivity analysis [13][14][15][16][17][18][19][20][21][22]. These solutions are characterized to become minimum when the objective function is composed with a positive function, T , and under weak requirements are dense in the efficient set.…”