Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability

Abstract

Analysis in conic programming with applications to optimality and stability.

Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints

“…In particular, justifying the main results of the paper presented below requires only the usage of assertion (ii) in Proposition 3.1 without imposing any convexity assumption on the set Γ and therefore the Θ-convexity of the mapping g as in [16].…”

“…also [16]. On the other hand, it is not hard to construct simple examples showing that the violation of the second-order condition (3.6) for nonconvex sets Γ prevents the validity of the conclusion in Proposition 3.1(i).…”

“…Namely, the paper [20] contains the calculation of the limiting coderivative of the solution map to a counterpart of GE (1.1), where Θ is a Carthesian product of the Lorentz cones. Our recent paper [16] provides a precise second-order formula to calculate the regular coderivative of the solution map S given in (1.2) under natural assumptions. Furthermore, the same paper [16] contains a formula for calculating the graphical derivative of (1.2) but only under the convexity assumption on Γ, which is rather restrictive, being however unavoidable in the technique of [16].…”

“…Our recent paper [16] provides a precise second-order formula to calculate the regular coderivative of the solution map S given in (1.2) under natural assumptions. Furthermore, the same paper [16] contains a formula for calculating the graphical derivative of (1.2) but only under the convexity assumption on Γ, which is rather restrictive, being however unavoidable in the technique of [16]. Observe also that the convexity assumption on Γ is not imposed in [6] while the set Θ in (1.2) is assumed to be a convex polyhedron.…”

Linear and Nonlinear Semidefinite Programming

“…An important effort research is recently devoted to the study and characterization of stability for solutions of nonlinear semidefinite programming (or in general conic) problems, see for instance [107,20,33,70,49,98,97].…”