Bilevel Disjunctive Optimization on Affine Manifolds

Abstract

Bilevel optimization is a special kind of optimization where one problem is embedded within another. The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lowerlevel optimization task. These problems involve two kinds of variables: upper-level variables and lower-level variables. Bilevel optimization was first realized in the field of game theory by a German economist von Stackelberg who published a book (1934) that described this hierarchical problem. Now the bilevel optimization problems are commonly found in a number of real-world problems: transportation, economics, decision science, business, engineering, and so on. In this chapter, we provide a general formulation for bilevel disjunctive optimization problem on affine manifolds. These problems contain two levels of optimization tasks where one optimization task is nested within the other. The outer optimization problem is commonly referred to as the leaders (upper level) optimization problem and the inner optimization problem is known as the followers (or lower level) optimization problem. The two levels have their own objectives and constraints. Topics affine convex functions, optimizations with auto-parallel restrictions, affine convexity of posynomial functions, bilevel disjunctive problem and algorithm, models of bilevel disjunctive programming problems, and properties of minimum functions.