Lagrangian Duality for Mixed-Integer Semidefinite Programming: Theory and Algorithms

TL;DR

This paper develops a Lagrangian duality framework for mixed-integer semidefinite programming, providing stronger bounds than traditional relaxations and proposing algorithms to compute these bounds effectively.

Contribution

It introduces the Lagrangian duality theory for MISDP, derives dual bounds, and proposes algorithms to compute hierarchical bounds that outperform existing relaxations.

Findings

Lagrangian dual bounds dominate continuous relaxations.

Hierarchical bounds are significantly stronger for max-k-cut.

Algorithms effectively compute dual bounds for MISDP.

Abstract

This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous relaxation of the MISDP problem. We present a hierarchy of Lagrangian dual bounds by exploiting the theory of integer positive semidefinite matrices and propose three algorithms for obtaining those bounds. Our algorithms are variants of well-known algorithms for minimizing non-differentiable convex functions. The numerical results on the max-$k$-cut problem show that the Lagrangian dual bounds are substantially stronger than the semidefinite programming bound obtained by relaxing integrality, already for lower levels in the hierarchy.