Differential stability of convex optimization problems with possibly empty solution sets

TL;DR

This paper investigates the differential stability of convex optimization problems with possibly empty solution sets, providing exact formulas for the $ ext{epsilon}$-subdifferential of the optimal value function using a sum rule, and extending prior work.

Contribution

It introduces a new approach to analyze the differential stability of convex problems with empty solutions, expanding the theoretical understanding and computational tools.

Findings

Derived exact formulas for $ ext{epsilon}$-subdifferentials.

Extended stability analysis to problems with empty solution sets.

Provided illustrative examples demonstrating the formulas.

Abstract

As a complement to two recent papers by An and Yen [An, D.T.V., Yen, N.D.: Differential stability of convex optimization problems under inclusion constraints. Appl. Anal., 94, 108--128 (2015)], and by An and Yao [An, D.T.V., Yao, J.-C.: Further results on differential stability of convex optimization problems. J. Optim. Theory Appl., 170, 28--42 (2016)] on subdifferentials of the optimal value function of infinite-dimensional convex optimization problems, this paper studies the differential stability of convex optimization problems, where the solution set may be empty. By using a suitable sum rule for $\varepsilon$-subdifferentials, we obtain exact formulas for computing the $\varepsilon$-subdifferential of the optimal value function. Several illustrative examples are also given.