Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods

Abstract

We present effective linear programming based computational techniques for solving nonconvex quadratic programs with box constraints (BoxQP). We first observe that known cutting planes obtained from the Boolean Quadric Polytope (BQP) are computationally effective at reducing the optimality gap of BoxQP. We next show that the Chvátal–Gomory closure of the BQP is given by the odd-cycle inequalities even when the underlying graph is not complete. By using these cutting planes in a spatial branch-and-cut framework, together with a common integrality-based preprocessing technique and a particular convex quadratic relaxation, we develop a solver that can effectively solve a well-known family of test instances. Our linear programming based solver is competitive with SDP-based state of the art solvers on small instances and sparse instances. Most of our computational techniques have been implemented in the recent version of CPLEX and have led to significant performance improvements on nonconvex quadratic programs with linear constraints.

Appendix

1.1 Transforming valid inequalities

We next describe how to transform valid inequalities for ${ \textsc {bqp}}$ to be valid inequalities for the set $\mathbf {conv}(\mathcal {B}(\ell ,u)) =: { \textsc {bqp}}(\ell ,u)$ so that they can be applied to box-constrained quadratic programs with arbitrary bounds. We explain the transformations in the context of finding a valid inequality violated by a point $(\hat{x},\hat{X})$ with variable bounds $\ell _i \le x_i \le u_i \ \forall i \in N$. The first step is to transform the variables via the equations

$$\begin{aligned} x^\prime _i&= \frac{x_i - \ell _i}{u_i-\ell _i} \qquad \forall i \in N \end{aligned}$$

(20)

$$\begin{aligned} X^\prime _{ij}&= \frac{X_{ij} - \ell _i x_j - \ell _j x_i + \ell _i \ell _j}{(u_i - \ell _i)(u_j - \ell _j)} \qquad \forall \{i,j\} \in E. \end{aligned}$$

(21)

The transformation (21) is obtained by scaling each variable in the product

$$\begin{aligned} x_i x_j \rightarrow \left( \frac{x_i - \ell _i}{u_i - \ell _i}\right) \left( \frac{x_j - \ell _j}{u_j - \ell _j}\right) , \end{aligned}$$

and replacing $x_i x_j$ with $X_{ij}$. The point $(x,X) \in { \textsc {bqp}}(\ell ,u)$ if and only if the transformed point $(x^\prime ,X^\prime ) \in { \textsc {bqp}}$. The point $(x^\prime ,X^\prime )$ is given to CPLEX’s standard $0$-$\frac{1}{2}$ cut generator, which attempts to separate $(x^\prime ,X^\prime )$ for ${ \textsc {bqp}}$. If successful, a “canonical,” inequality of the form

$$\begin{aligned} \sum _{i \in N} \alpha _i x_i^\prime + \sum _{\{i,j\}\in E} B_{ij} X_{ij}^\prime \le \gamma \end{aligned}$$

(22)

is returned. We undo the transformation by substituting (20) and (21) into (22) to get the following equivalent inequality for ${ \textsc {bqp}}(\ell ,u)$:

$$\begin{aligned}&\sum _{i \in N} \left( \frac{\alpha _i}{u_i-\ell _i}\right) x_i + \sum _{\{i,j\}\in E} \frac{B_{ij} X_{ij} - B_{ij} \ell _j x_i - B_{ij} \ell _i x_j}{(u_i-\ell _i)(u_j-\ell _j)} \le \gamma + \sum _{i \in N} \frac{\alpha _i \ell _i}{(u_i - \ell _i)} \nonumber \\&\qquad - \sum _{\{i,j\}\in E} \frac{B_{ij} \ell _i \ell _j}{(u_i - \ell _i)(u_j - \ell _j)}. \end{aligned}$$

(23)

To implement inequality (23), the coefficient of the $x_i$ variable in (22) is multiplied by $(u_i-\ell _i)^{-1}$ and the quantity $\sum _{j : \{i,j\}\in E} \ B_{ij} \ell _j /(u_i-\ell _i)(u_j-\ell _j)$ is subtracted from the coefficient. For each $X_{ij}$ variable, we scale the coefficient by coefficients of the $X_{ij}$ variables by $(u_i - \ell _i)(u_j - \ell _j)$. We also add the quantity

$$\begin{aligned} \sum _{i \in N} \frac{\alpha _i \ell _i}{u_i-\ell _i} - \sum _{\{i,j\}\in E} \frac{B_{ij} \ell _i \ell _j}{(u_i - \ell _i)(u_j - \ell _j)} \end{aligned}$$

to the right-hand-side of the returned inequality. When computing the coefficient of each variable, if the difference in bounds for the variable is smaller than a pre-specified tolerance, ($u_i - \ell _i < \epsilon$), we take $u_i = \ell _i + 1$ when calculating the transformations (20) (21) and (23). The resulting inequality is clearly valid, since it is valid for a relaxation of the nodal subproblem being considered. Further, numerical difficulties arising from possibly large coefficient values in (23) are avoided.

When these transformations are applied to the defining inequalities of the Boolean Quadric Polytope, one obtains the well-known general version of the McCormick inequalities:

1.2 bqp Bounds

Table 8 contains the raw numbers that were aggregated to make Tables 1 and 2. The value in the column $z_{BoxQP\ }$ is the optimal solution value, $z_{{ \textsc {bqp}}}^U$ is the upper bound on $z_{BoxQP\ }$ obtained by the bqprestriction, and $z_{{ \textsc {bqp}}}^L$ is the lower bound on $z_{BoxQP\ }$ obtained from the bqprelaxation. The $z_{{ \textsc {bqp}}}^L$(root) column contains the final root relaxation value from CPLEX when solving the integer program associated with $z_{{ \textsc {bqp}}}^L$, and $z_{\Delta ^+}$ is the bound at the root that could be obtained if exact separation of the odd cycle inequalities was done. The column $z_{\mathcal {M}}$ contains the lower bound obtained from optimizing over the McCormick relaxation $\mathcal {M}$.

Table 8 Bounds for BQP relaxations

1.3 Algorithmic improvements

Table 9 contains the raw numbers that were aggregated to make Table 3. The value in the column $z_{BoxQP\ }$ is the optimal solution value, $z_{\mathcal {M}^2}+$ $0$-$\frac{1}{2}$ is the bound obtained from the strengthened McCormick relaxation with CPLEX $0$-$\frac{1}{2}$-cuts added, $z_{\mathcal {M}}+$ $0$-$\frac{1}{2}$the bound from the standard McCormick relaxation with CPLEX $0$-$\frac{1}{2}$-cuts added. The columns $z_{\mathcal {M}^2}$ and $z_{\mathcal {M}}$ show the lower bounds obtained from the strengthened and traditional McCormick relaxation, respectively.

Table 9 Comparing bounds for $\mathcal {M}$ and $\mathcal {M}^2$ relaxations

Table 10 contains raw numbers on the computational performance of algorithmic techniques described in Sects. 4.2 and 4.3.

Table 10 Computational performance of integer strengthening and local cut improvement

1.4 SDP relaxation bounds

This appendix contains computational time and relaxation bound information for various SDP-based approaches for solving BoxQP , as described in Sect. 5.1. For the results of the two methods SBL and Dong, the reader is invited to refer directly to the papers [21, 31] (Table 11).

Table 11 Bound and solution time for the various SDP based relaxations

1.5 Solver comparison

This appendix contains the raw performance numbers comparing different solvers for BoxQP . The experiment is described in Sect. 5.2 (Table 12).

Table 12 CPU time and final optimality gap of each solver on all instances

1.6 CPLEX performance

This appendix contains the raw performance numbers for the computational experiment comparing the performance of CPLEX v12.6.3 with and without the BQP-based cutting planes described in Sect. 5.3 (Table 13).

Table 13 Results of complete optimization without and with cuts