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This thesis is concerned with the phenomenon of implicit variables in optimization theory. Roughly speaking, a variable is called implicit whenever it is used to model the feasible set but does not appear in the objective function. At the first glance, such variables seem to be less relevant for the purpose of optimization.
First, we provide a theoretical study on optimization problems with implicit variables. Therefore, we rely on a model program which covers several interesting problem classes from optimization theory such as bilevel optimization problems, evaluated multiobjective optimization problems, or optimization problems with cardinality constraints. We start our analysis by clarifying that the interpretation of implicit variables as explicit ones induces additional local minimizers. Afterwards, we study three reasonable stationarity systems of Mordukhovich-stationarity-type for the original problem as well as some comparatively weak associated constraint qualifications. The obtained results are applied to the three example classes mentioned above.
Second, we introduce switching- and or-constrained optimization problems. Exploiting the observation that each or-constrained optimization problem can be transferred into a switching-constrained optimization problem with the aid of slack variables, one can interpret or-constrained programs as optimization problems comprising implicit variables. Necessary optimality conditions and constraint qualifications for both problem classes are derived. Furthermore, some approaches for the numerical solution of both problem classes are discussed and results of computational experiments are presented. The shortcomings of implicit variables are highlighted in terms of or-constrained optimization.
Third, we study three different scenarios where optimality conditions and constraint qualifications for challenging optimization problems can be constructed while abstaining from the introduction of implicit variables. We start by deriving a generalized version of the linear independence constraint qualification as well as second-order necessary and sufficient optimality conditions for so-called disjunctive optimization problems, which cover several interesting but inherently irregular problem classes like mathematical programs with complementarity, switching, or-, and cardinality constraints. Afterwards, we exploit several different single-level reformulations of standard bilevel optimization problems in order to find first- and second-order sufficient optimality conditions. Finally, we study sequential stationarity and regularity conditions for nonsmooth mathematical problems with generalized equation constraints with the aid of the popular limiting variational analysis. The investigated model problem covers the one we use for the theoretical analysis of implicit variables.