[Paper Review] Global Solutions of Nonconvex Standard Quadratic Programs via Mixed Integer Linear Programming Reformulations

Jacek Gondzio, E. Alper Yıldırım|arXiv (Cornell University)|Oct 4, 2018

Advanced Optimization Algorithms Research29 references19 citations

TL;DR

This paper proposes two mixed integer linear programming (MILP) reformulations for solving nonconvex standard quadratic programs (StQPs) globally. The first uses KKT conditions and big-M constraints with binary variables; the second leverages a tight overestimating function of the quadratic objective. Computational results show both formulations significantly outperform other global solvers, especially on large instances, with orders-of-magnitude speedups and higher optimality rates.

ABSTRACT

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of orders of magnitude.

Motivation & Objective

• To develop efficient global solution methods for nonconvex standard quadratic programs (StQPs), which are NP-hard and arise in diverse applications.
• To reformulate StQPs as mixed integer linear programs (MILPs) to leverage powerful modern MILP solvers like CPLEX and Gurobi.
• To design MILP formulations that exploit the structure of StQPs to improve solution speed and optimality recovery.
• To evaluate the performance of the proposed formulations against state-of-the-art global optimization solvers on benchmark instances.

Proposed method

• First formulation: Reformulates StQP as a linear program with complementarity constraints (LPCC), then linearizes complementarity constraints using binary variables and big-M constraints.
• Second formulation: Replaces the quadratic objective with a piecewise-linear overestimator based on the support of the solution, which is exact at KKT points.
• Derives valid bounds for big-M parameters using lower bounds on the optimal value, exploiting the unit simplex and symmetric matrix structure of StQP.
• Introduces two lower bounds: ℓ₁(Q) (computationally cheap) and ℓ₂(Q) (tighter but more expensive), to improve bounds in the MILP models.
• Enhances formulations with valid inequalities to strengthen the relaxation and improve convergence.
• Employs a branch-and-bound framework within the MILP solvers, with problem-specific tightening strategies for improved performance.

Experimental results

Research questions

• RQ1 Can standard quadratic programs be effectively reformulated as mixed integer linear programs to enable use of high-performance MILP solvers?
• RQ2 How do the proposed MILP formulations compare in performance to general-purpose global solvers like CPLEX-QP and BARON?
• RQ3 What is the impact of using tighter lower bounds (ℓ₂(Q)) versus cheaper bounds (ℓ₁(Q)) on solution speed and optimality?
• RQ4 How effective are valid inequalities in improving the performance of the MILP formulations?
• RQ5 Can the proposed MILP approaches scale effectively to large-scale StQP instances?

Key findings

• On the IS1 instance set, all eight variants of the proposed MILP formulations outperform QP-IP, which in turn outperforms CPLEX-QP and BARON.
• MILP2-L2 achieved the best solution time, followed by MILP1-L2-VI and MILP2-L2-VI, with the latter two solving a higher percentage of instances within the time limit.
• On the larger IS2 set, MILP2-L1 clearly outperformed all other formulations, followed by MILP2-L1-VI, with QP-IP being the next best performer.
• The addition of valid inequalities improved the number of instances solved to optimality for both MILP1 and MILP2 formulations, though performance varied by instance type.
• The relaxed MILP2 formulation achieved the best solution times most frequently, especially when combined with the tighter ℓ₂(Q) bound, though computing ℓ₂(Q) is computationally costly on large instances.
• Overall, the proposed MILP formulations demonstrated orders-of-magnitude improvements in performance over existing global solution approaches, particularly on large-scale instances.