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Directional first order approach for a class of bilevel programs

Kuang Bai, Wei Yao, Jane J. Ye, Jin Zhang

Abstract

In this paper, we study a class of bilevel optimization program (BP), where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order approach requires the convexity of the lower level program while reformulations involving the value function results in difficult optimization problems. In this paper we propose a directional first order approach which does not require the convexity of the lower level program. First, under some reasonable assumptions, we show that the lower level program can be equivalently characterized by its first order condition over a directional neighborhood. Next, for the resulting single level optimization problem, under common constraint qualifications, we establish directional necessary optimality conditions. Finally, an example of BP with nonconvex lower level program is given, where we demonstrate the failure of the classical first order approach and derive necessary optimality conditions using its directional counterpart.

Directional first order approach for a class of bilevel programs

Abstract

In this paper, we study a class of bilevel optimization program (BP), where the feasible set of the lower level program is independent of the upper level variable. For bilevel programs it is known that the first order approach requires the convexity of the lower level program while reformulations involving the value function results in difficult optimization problems. In this paper we propose a directional first order approach which does not require the convexity of the lower level program. First, under some reasonable assumptions, we show that the lower level program can be equivalently characterized by its first order condition over a directional neighborhood. Next, for the resulting single level optimization problem, under common constraint qualifications, we establish directional necessary optimality conditions. Finally, an example of BP with nonconvex lower level program is given, where we demonstrate the failure of the classical first order approach and derive necessary optimality conditions using its directional counterpart.
Paper Structure (11 sections, 17 theorems, 90 equations, 1 figure)

This paper contains 11 sections, 17 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Application in the principal-agent problem
  3. Existing approaches for deriving optimality conditions
  4. Our new approach
  5. Organization
  6. Preliminaries
  7. Local equivalent characterization of the lower level program constraint
  8. Sufficient conditions for single-valued localization of the first order stationary map
  9. Sufficient conditions for directional inner semi-continuity
  10. First order necessary optimality conditions for (BP)
  11. Conclusion

Key Result

Proposition 2.1

Consider a set constraint system $\phi(z)\in\Omega$, where $\phi(z):\mathbb R^p\rightarrow\mathbb R^s$ is continuously differentiable and $\Omega\subseteq\mathbb R^s$ is locally closed. Let $\phi (\bar{z}) \in \Omega$ and $d\in \mathbb{R}^p$.

Figures (1)

  • Figure 1: Mirrlees example

Theorems & Definitions (34)

  • Definition 2.1: Tangent Cone and Normal Cone
  • Definition 2.2: Directional Normal Cone
  • Definition 2.3
  • Definition 2.4: Directional Neighborhood
  • Definition 2.5: Directional Metric Subregularity
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.6: Directional Inner Semi-continuity
  • Remark 2.1
  • Theorem 3.1
  • ...and 24 more