On implicit variables in optimization theory

Abstract

Implicit variables of a mathematical program are variables which do not needto be optimized but are used to model feasibility conditions. They frequentlyappear in several different problem classes of optimization theory comprisingbilevel programming, evaluated multiobjective optimization, or nonlinearoptimization problems with slack variables. In order to deal with implicitvariables, they are often interpreted as explicit ones. Here, we first pointout that this is a light-headed approach which induces artificial locallyoptimal solutions. Afterwards, we derive various Mordukhovich-stationarity-typenecessary optimality conditions which correspond to treating the implicitvariables as explicit ones on the one hand, or using them only implicitly tomodel the constraints on the other. A detailed comparison of the obtainedstationarity conditions as well as the associated underlying constraintqualifications will be provided. Overall, we proceed in a fairly generalsetting relying on modern tools of variational analysis. Finally, we apply ourfindings to different well-known problem classes of mathematical optimizationin order to visualize the obtained theory.Comment: 34 page