## Invited Session Wed.3.H 2051

#### Wednesday, 15:15 - 16:45 h, Room: H 2051

**Cluster 24: Variational analysis** [...]

### Some stability aspects in optimization theory

**Chair: Abderrahim Hantoute and Rafael Correa**

**Wednesday, 15:15 - 15:40 h, Room: H 2051, Talk 1**

**Abderrahim Hantoute**

On convex relaxation of optimization problems

**Coauthor: Rafael Correa**

**Abstract:**

We relate a given optimization problem * ∈ f*_{X}{f} to its lsc convex relaxation * ∈ f*_{X}{(cl-co)(f)}; here *(cl-co)(f)* is the lsc convex hull of *f*. We establish a complete characterization of the solutions set of the relaxed problem by means exclusively of "some kind'' of the solution of the initial problem. Consequently, under some natural conditions, of coercivity type, this analysis yields both existence and characterization of the solution of the initial problem. Our main tools rely on the subdifferetial analysis of the so-called Legendre-Fenchel function.

**Wednesday, 15:45 - 16:10 h, Room: H 2051, Talk 2**

**C. H. Jeffrey Pang**

First order analysis of set-valued maps and differential inclusions

**Abstract:**

The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Its analysis requires the use of set-valued maps. For a set-valued map, the tangential derivative and coderivatives separately characterize a first order sensitivity analysis property, or more precisely, a pseudo strict differentiability property. The characterization using tangential derivatives requires fewer assumptions. In finite dimensions, the coderivative characterization establishes a bijective relationship between the convexified limiting coderivatives and the pseudo strict derivatives. This result can be used to estimate the convexified limiting coderivatives of limits of set-valued maps. We apply these results to the study of differential inclusions by calculating the tangential derivatives and coderivatives of the reachable map, which leads to the subdifferential and subderivative dependence of the value function in terms of the initial conditions. These results in turn furthers our understanding of the Euler-Lagrange and transversality conditions in differential inclusions.

**Wednesday, 16:15 - 16:40 h, Room: H 2051, Talk 3**

**Vladimir Shikhman**

Implicit vs. inverse function theorem in nonsmooth analysis

**Abstract:**

We study the application of implicit and inverse function theorems to systems of complementarity equations. The goal is to characterize the so-called topological stability of those systems. Here, stability refers to homeomorphy invariance of the solution set under small perturbations of the defining functions. We discuss the gap between the nonsmooth versions of implicit and inverse function theorems in the complementarity setting. Namely, for successfully applying the nonsmooth implicit function theorem one needs to perform first a linear coordinate transformation. We illustrate how this fact becomes crucial for the nonsmooth analysis.