Abstract
This paper focuses on formulas for the ε-subdifferential of the optimal value function of scalar and vector convex optimization problems. These formulas can be applied when the set of solutions of the problem is empty. In the scalar case, both unconstrained problems and problems with an inclusion constraint are considered. For the last ones, limiting results are derived, in such a way that no qualification conditions are required. The main mathematical tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the vector case, unconstrained problems are studied and exact formulas are derived by linear scalarizations. These results are based on a concept of infimal set, the notion of cone proper set and an ε-subdifferential for convex vector functions due to Taa.
Highlights
Studying differential stability of optimization problems usually means to study differentiability properties of the optimal value function in parametric mathematical programming
Ioffe and Tihomirov [21], HiriartUrruty and Lemarechal [18, 19], Phelps [30], Zalinescu [37] and Borwein and Vanderwerff [9] presented a beautiful theory about convex sets and convex functions in finite and infinite dimensional spaces with many significant applications in mathematical programming, classical variational calculus, and optimal control theory
Proof Notice that μc coincides with the optimal value function μ given by φ + δgphG and Mηc coincides in domG with the approximate solution map Mη defined by φ + δgph G
Summary
Studying differential stability of optimization problems usually means to study differentiability properties of the optimal value function in parametric mathematical programming. This paper concerns with the study of differential stability properties to scalar and vector convex programming problems. This paper concerns with differential stability of vector optimization problems that would not satisfy the domination property (in particular with empty solution set). Differential stability properties in convex scalar and vector optimization constraint is considered and a limiting formula for ε-subdifferential of the optimal value function is obtained.
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