Estimates of generalized Hessians for optimal value functions in mathematical programming

TL;DR

This paper develops second-order generalized Hessian estimates for the optimal value function in mathematical programming, facilitating advanced stability analysis and robust algorithm design.

Contribution

It introduces novel second-order generalized Hessian estimates for the optimal value function, extending beyond first-order analysis in nonsmooth optimization.

Findings

Derived estimates of the generalized Hessian for the optimal value function.

Utilized optimal solution and Lagrange multiplier mappings for derivative estimates.

Facilitates development of second-order optimization algorithms.

Abstract

The optimal value function is one of the basic objects in the field of mathematical optimization, as it allows the evaluation of the variations in the cost/revenue generated while minimizing/maximizing a given function under some constraints. In the context of stability/sensitivity analysis, a large number of publications have been dedicated to the study of continuity and differentiability properties of the optimal value function. The differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivatives and subdifferentials, given that the function is typically nonsmooth. With the progress made in the last two to three decades in major subfields of optimization such as robust, minmax, semi-infinite and bilevel optimization, and their connection to the optimal value function, there is a crucial need for a second order analysis of the generalized differentiability properties of this function. This type of analysis will enable the development of robust solution algorithms, such as the Newton method. The main goal of this paper is to provide results in this direction. In fact, we derive estimates of the generalized Hessian for the optimal value function. Our results are based on two handy tools from parametric optimization, namely the optimal solution and Lagrange multiplier mappings, for which completely detailed estimates of their generalized derivatives are either well-known or can easily be obtained.