On implicit variables in optimization theory

TL;DR

This paper investigates the role of implicit variables in optimization problems, highlighting issues with their explicit treatment and deriving new stationarity conditions using variational analysis.

Contribution

It introduces novel stationarity conditions for implicit variables, contrasting explicit and implicit handling, and applies these to various optimization problem classes.

Findings

Explicit treatment of implicit variables can lead to artificial local optima.

Derived new necessary optimality conditions using Mordukhovich-stationarity.

Applied theory to classical optimization problems to illustrate results.

Abstract

Implicit variables of a mathematical program are variables which do not need to be optimized but are used to model feasibility conditions. They frequently appear in several different problem classes of optimization theory comprising bilevel programming, evaluated multiobjective optimization, or nonlinear optimization problems with slack variables. In order to deal with implicit variables, they are often interpreted as explicit ones. Here, we first point out that this is a light-headed approach which induces artificial locally optimal solutions. Afterwards, we derive various Mordukhovich-stationarity-type necessary optimality conditions which correspond to treating the implicit variables as explicit ones on the one hand, or using them only implicitly to model the constraints on the other. A detailed comparison of the obtained stationarity conditions as well as the associated underlying constraint qualifications will be provided. Overall, we proceed in a fairly general setting relying on modern tools of variational analysis. Finally, we apply our findings to different well-known problem classes of mathematical optimization in order to visualize the obtained theory.