Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving bilevel programs based on lower-level Mond-Weir duality

Yu-Wei Li, Gui-Hua Lin, Xide Zhu

TL;DR

The lower-level Mond-Weir duality is applied to present a new reformulation, called MDP, for bilevel program and it is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense.

Abstract

This paper focuses on developing effective algorithms for solving bilevel program. The most popular approach is to replace the lower-level problem by its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. Furthermore, in order to compare the new MDP approach with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments showed that MDP has evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time.

Solving bilevel programs based on lower-level Mond-Weir duality

TL;DR

The lower-level Mond-Weir duality is applied to present a new reformulation, called MDP, for bilevel program and it is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense.

Abstract

This paper focuses on developing effective algorithms for solving bilevel program. The most popular approach is to replace the lower-level problem by its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. Furthermore, in order to compare the new MDP approach with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments showed that MDP has evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time.
Paper Structure (13 sections, 16 theorems, 99 equations, 17 tables, 1 algorithm)

This paper contains 13 sections, 16 theorems, 99 equations, 17 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constraint qualifications
  4. Mond-Weir duality for constrained optimization problems
  5. New reformulation based on lower-level Mond-Weir duality
  6. MDP may satisfy the MFCQ at its feasible points
  7. Relations among MPCC, WDP, and MDP
  8. Extension of MDP
  9. Numerical experiments
  10. Test examples
  11. Relaxation schemes for MDP and eMDP
  12. Comparison of MPCC, WDP, MDP, and eMDP
  13. Conclusions

Key Result

Theorem 2.1

(weak Mond-Weir duality) Suppose that $f$ is pseudoconvex and $u^{T} g + v^{T} h$ is quasiconvex over $Y\cup \mathcal{Z}(u,v)$ for any $u\in \mathbb{R}^p_+$ and $v\in \mathbb{R}^q$. Then, the optimal value of the primal problem is no less than its Mond-Weir dual problem, i.e.,

Theorems & Definitions (27)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Example 3.1
  • ...and 17 more