[Paper Review] Constraint qualifications and optimality conditions in bilevel optimization

Jane J. Ye|arXiv (Cornell University)|Oct 9, 2019

Optimization and Variational Analysis4 citations

TL;DR

This paper develops checkable constraint qualifications and optimality conditions for bilevel optimization problems, focusing on convex and nonconvex lower-level programs. It introduces a relaxed constant positive linear dependence (RCPLD) condition for the combined program and a sharp necessary optimality condition via generalized equations, improving upon traditional mathematical programs with complementarity constraints (MPCC). The key contribution is a verifiable M-stationarity condition based on value function upper estimates.

ABSTRACT

In this paper we study constraint qualifications and optimality conditions for bilevel programming problems. We strive to derive checkable constraint qualifications in terms of problem data and applicable optimality conditions. For the bilevel program with convex lower level program we discuss drawbacks of reformulating a bilevel programming problem by the mathematical program with complementarity constraints and present a new sharp necessary optimality condition for the reformulation by the mathematical program with a generalized equation constraint. For the bilevel program with a nonconvex lower level program we propose a relaxed constant positive linear dependence (RCPLD) condition for the combined program.

Motivation & Objective

• To address the limitations of reformulating bilevel programs as mathematical programs with complementarity constraints (MPCC), especially when the lower-level problem is nonconvex or has multiple multipliers.
• To develop checkable, verifiable constraint qualifications—specifically the relaxed constant positive linear dependence (RCPLD) condition—for bilevel programs with nonconvex lower-level problems.
• To establish a sharp necessary optimality condition for bilevel programs by reformulating them as mathematical programs with generalized equation constraints.
• To provide a new M-stationarity condition based on upper estimates of the value function, which is weaker and more applicable than existing conditions.
• To ensure the optimality conditions are applicable under Lipschitz continuity of the value function and verify their validity via linearized cones and normal cone theory.

Proposed method

• Reformulates the bilevel program as a mathematical program with a generalized equation constraint using the KKT conditions of the lower-level problem.
• Introduces the relaxed constant positive linear dependence (RCPLD) condition for the combined program, defined via the rank of a Jacobian matrix involving gradients of the lower-level Lagrangian.
• Applies the value function approach to derive an M-stationarity condition by upper-estimating the limiting subdifferential of the value function using the set $W(\bar{x})$.
• Uses the MPEC-linearized cone ${\cal L}^{MPEC}$ to define the tangent cone to the feasible region of the reformulated problem.
• Employs variational analysis tools, including Fréchet and Mordukhovich normal cones, to characterize stationarity and constraint qualifications.
• Establishes weak calmness of the problem at a solution point to ensure the existence of a multiplier and validity of the optimality condition.

Experimental results

Research questions

• RQ1 Can a checkable constraint qualification be derived for bilevel programs with nonconvex lower-level problems?
• RQ2 How can the classical MPCC reformulation be improved to avoid non-equivalence in local optimality when multiple multipliers exist?
• RQ3 What is a sharp necessary optimality condition for bilevel programs reformulated as mathematical programs with generalized equation constraints?
• RQ4 Under what conditions is the M-stationarity condition based on an upper estimate of the value function valid?
• RQ5 How does the RCPLD condition relate to the existence of a solution and stationarity in bilevel optimization?

Key findings

• The RCPLD condition is verified when the rank of the matrix $J^{*}$ equals $m+n+r+s-|I_{u}|$, ensuring the validity of the M-stationarity condition based on the value function.
• The M-stationarity condition based on an upper estimate of the value function is weaker than the one based on the exact value function, making it more broadly applicable.
• The value function constraint $f(x,y)-V(x)\leq 0$ is not required in the verification of the optimality condition, simplifying the reformulation.
• When the value function is Lipschitz continuous at $\bar{x}$, the RCPLD condition holds and the M-stationarity condition is necessary for a local solution.
• The MPEC-linearized cone ${\cal L}^{MPEC}$ is used to define the tangent cone to the feasible region of the reformulated problem, enabling the derivation of stationarity conditions.
• Weak calmness of the problem at a solution point, combined with compactness of $W(\bar{x})$, ensures that the solution satisfies the M-stationarity condition based on an upper estimate.