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

TL;DR

This paper introduces disjunctive cuts based on second-order cones for solving a specific class of integer bilevel nonlinear programs, improving solution efficiency through a branch-and-cut and cutting-plane methods.

Contribution

It develops novel second-order cone-based disjunctive cuts and strategies for their separation, enhancing solution algorithms for bilevel programs with cone constraints.

Findings

Proposed methods outperform a state-of-the-art solver on diverse instances.

Enhanced algorithms significantly improve computational performance.

Effective handling of binary and integer variables with multiple linking constraints.

Abstract

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose 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 an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of 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 binary instances.