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Optimality conditions for bilevel programs via Moreau envelope reformulation

Kuang Bai, Jane Ye, Shangzhi Zeng

Abstract

For bilevel programs with a convex lower level program, the classical approach replaces the lower level program with its Karush-Kuhn-Tucker condition and solve the resulting mathematical program with complementarity constraint (MPCC). It is known that when the set of lower level multipliers is not unique, MPCC may not be equivalent to the original bilevel problem, and many MPCC-tailored constraint qualifications do not hold. In this paper, we study bilevel programs where the lower level is generalized convex. Applying the equivalent reformulation via Moreau envelope, we derive new directional optimality conditions. Even in the nondirectional case, the new optimality condition is stronger than the strong stationarity for the corresponding MPCC.

Optimality conditions for bilevel programs via Moreau envelope reformulation

Abstract

For bilevel programs with a convex lower level program, the classical approach replaces the lower level program with its Karush-Kuhn-Tucker condition and solve the resulting mathematical program with complementarity constraint (MPCC). It is known that when the set of lower level multipliers is not unique, MPCC may not be equivalent to the original bilevel problem, and many MPCC-tailored constraint qualifications do not hold. In this paper, we study bilevel programs where the lower level is generalized convex. Applying the equivalent reformulation via Moreau envelope, we derive new directional optimality conditions. Even in the nondirectional case, the new optimality condition is stronger than the strong stationarity for the corresponding MPCC.
Paper Structure (11 sections, 19 theorems, 122 equations)

This paper contains 11 sections, 19 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and preliminary results
  3. Directional KKT conditions
  4. Directional sensitivity analysis of the Moreau envelope function
  5. Sensitivity analysis under weak convexity
  6. Sensitivity analysis without weak convexity
  7. Sensitivity analysis under FOSCMS
  8. Sensitivity analysis under RCRCQ+RS
  9. Necessary optimality conditions for problem (VP)$_\gamma$
  10. Conclusion
  11. Acknowledgments

Key Result

Proposition 2.1

Let $\varphi:\mathbb{R}^n\rightarrow \overline{\mathbb{R}}$ and $\varphi (\bar{x})$ be finite. Suppose $\varphi$ is Lipschitz continuous and directionally differentiable at $\bar{x}$ in direction $u$. Then

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 2.1: Tangent Cone and Normal Cone
  • Definition 2.2: Directional Normal Cone
  • Definition 2.3
  • Definition 2.4: Graphical and Directional Derivatives
  • Definition 2.5: Subdifferentials
  • Definition 2.6: Directional Subdifferentials
  • Proposition 2.1
  • Definition 2.7: Directional Clarke Subdifferential
  • Proposition 2.2
  • ...and 28 more