Using a conic bundle method to accelerate both phases of a quadratic convex reformulation

TL;DR

The paper introduces MIQCR-CB, an improved algorithm for solving mixed-integer quadratic programs by accelerating the convex reformulation phase using a conic bundle method, resulting in faster overall solution times.

Contribution

It develops a subgradient algorithm within a Lagrangian duality framework to efficiently solve large-scale semidefinite problems, reducing reformulation size and computation time.

Findings

Significant speed-up in the convex reformulation phase.

Smaller reformulated problems lead to faster second-phase solutions.

Extensive computational results demonstrate improved efficiency.

Abstract

We present algorithm MIQCR-CB that is an advancement of method MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem $(SDP)$, and, in the second phase, the equivalent formulation is solved by a standard solver. As the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers, already for medium sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving $(SDP)$ that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm.