Simple Bilevel Programming and Extensions, Part-1: Theory

TL;DR

This paper explores the theoretical foundations of simple bilevel programming and SMPEC, establishing optimality conditions and proposing an algorithm, thereby advancing the understanding of these complex hierarchical optimization problems.

Contribution

It provides a comprehensive study of optimality conditions for SBP and SMPEC, including sequential conditions without constraint qualifications, and introduces a schematic algorithm with convergence analysis.

Findings

Optimality conditions for SBP and SMPEC are characterized.

Sequential optimality conditions do not require constraint qualifications.

A schematic algorithm with proven convergence is proposed.

Abstract

In this paper we begin by discussing the simple bilevel programming problem (SBP) and its extension the simple mathematical programming problem under equilibrium constraints (SMPEC). Here we first define both these problems and study their interrelations. Next we study the various types of necessary and sufficient optimality conditions for the (SMPEC) problems; which occur under various reformulations. The optimality conditions for the (SBP) problem are special cases of the results obtained here when the lower level objective is the gradient of a convex function. Among the various optimality conditions presented here are the sequential optimality conditions, which do not need any constraint qualification. We also present here a schematic algorithm for the (SMPEC) problem, where the sequential optimality conditions play a key role in the convergence analysis.