Sensitivity analysis of variational inequalities via twice epi-differentiability and proto-differentiability of the proximity operator

TL;DR

This paper develops a sensitivity analysis framework for parameterized variational inequalities in Hilbert spaces, extending epi- and proto-differentiability concepts to handle data perturbations and deriving explicit formulas for solution derivatives.

Contribution

It introduces new notions of twice epi-differentiability and proto-differentiability for parameterized convex functions, linking them via convergent supporting hyperplanes, and applies these to derive explicit derivatives of the proximity operator.

Findings

Derived exact formula for the proto-derivative of the proximity operator.

Proved differentiability of solutions with respect to parameters.

Applied results to convex optimization problems with constraints.

Abstract

In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space. The challenge of the present work is to take into account a perturbation on all the data of the problem. This requires special adjustments in the definitions of the generalized first- and second-order differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly studied by R.T. Rockafellar, of twice epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous convex function and its subdifferential respectively. The link between these two notions is tied to Attouch's theorem and to the new concept, introduced in this paper, of convergent supporting hyperplanes. The previous tools allow us to derive an exact formula of the proto-derivative of the generalized proximity operator associated to a parameterized variational inequality, and deduce the differentiability of the associated solution with respect to the parameter. Furthermore, the derivative is shown to be the solution of a new variational inequality involving semi- and twice epi-derivatives of the data. An application is given to parameterized convex optimization problems involving the sum of two convex functions (one of them being smooth). The case of smooth convex optimization problems with inequality constraints is discussed in details. This approach seems to be new in the literature and open several perspectives towards theoretical and computational issues in nonlinear optimization.