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Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

Jiawang Nie, Jane J. Ye, Suhan Zhong

TL;DR

It is shown that bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker conditions with a sparse type of Lagrange multipliers.

Abstract

This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.

Partial Lagrange Multiplier Expressions and Disjunctive Decompositions for Bilevel Optimization

TL;DR

It is shown that bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker conditions with a sparse type of Lagrange multipliers.

Abstract

This paper studies bilevel polynomial optimization in which lower-level constraint functions depend linearly on lower-level variables. We show that such bilevel program can be reformulated as a disjunctive program by using Karush-Kuhn-Tucker (KKT) conditions with a sparse type of Lagrange multipliers. This kind of Lagrange multipliers can be conveniently represented by polynomials, for which we call partial Lagrange multiplier expressions (PLMEs). By doing this, each branch problem of the disjunctive program can be solved efficiently by polynomial optimization techniques. Solving each branch problem either returns infeasibility or gives a candidate local or global optimizer for the original bilevel optimization. We give necessary and sufficient conditions for these candidates to be global optimizers, and sufficient conditions for the local optimality. Numerical experiments are also presented to show the efficiency of the method.
Paper Structure (21 sections, 11 theorems, 122 equations, 3 figures, 5 tables, 3 algorithms)

This paper contains 21 sections, 11 theorems, 122 equations, 3 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Lagrange multiplier expression approach
  3. Disjunctive decomposition
  4. PLMEs and the KKT system
  5. Existence of LMEs
  6. Partial Lagrange multiplier expressions
  7. Decomposition of the KKT system
  8. The Disjunctive Decomposition
  9. Disjunctive KKT relaxation
  10. Local optimality of BPOPs
  11. The PLME-Disjunctive Algorithm
  12. Solving $(P_J^*)$
  13. Convergence property
  14. Feasible extensions for linear constraints
  15. Linear feasible extensions
  16. ...and 6 more sections

Key Result

Proposition 2.3

For every $J\in \mathcal{P}$, let $\lambda_J(x,y)$ be as in eq:PLMEgen. Then In particular, if $\hbox{rank}\, A = p$, then

Figures (3)

  • Figure 1: The PLME decomposition for the KKT set of Example \ref{['ex:dcpvisual']}
  • Figure 2: For Example \ref{['ex:locmin']}, the solid line is the feasible set $\mathcal{F}$, the dotted curves are contour level curves of $F$, the "$\Box$" denotes the global minimizer and the "$\star$" denotes the local minimizer.
  • Figure 3: The diagram for relationships between branch problems

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • proof
  • Example 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 22 more