SOCP-based disjunctive cuts for a class of integer nonlinear bilevel programs

TL;DR

This paper introduces a novel disjunctive cut approach based on second-order cones for solving a specific class of bilevel integer programs, improving solution methods for these complex problems.

Contribution

It develops the first disjunctive cuts for discrete bilevel optimization and establishes branch-and-cut and cutting plane algorithms for the problem class.

Findings

Proposed methods outperform state-of-the-art solvers on test instances.

Disjunctive cuts effectively separate infeasible points in bilevel problems.

Algorithms show promising computational results for specific problem variants.

Abstract

We study a class of bilevel integer programs with second-order cone constraints at the upper level and a convex quadratic objective and linear constraints at the lower level. We develop disjunctive cuts to separate bilevel infeasible points using a second-order-cone-based cut-generating procedure. To the best of our knowledge, this is the first time disjunctive cuts are studied in the context of discrete bilevel optimization. Using these disjunctive cuts, we establish a branch-and-cut algorithm for the problem class we study, and a cutting plane method for the problem variant with only binary variables. We present a preliminary computational study on instances with no second-order cone constraints at the upper level and a single linear constraint at the lower level. Our study demonstrates that both our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our test instances, where the non-linearities are linearized in a McCormick fashion.