Minimal conic quadratic reformulations and an optimization model

doi:10.1016/j.orl.2019.09.004

Operations Research Letters

Volume 47, Issue 6, November 2019, Pages 489-493

https://doi.org/10.1016/j.orl.2019.09.004 Get rights and content

Abstract

In this paper, we consider a particular form of inequalities which involves product of multiple variables with rational exponents. These inequalities can equivalently be represented by a number of conic quadratic forms called cone constraints. We propose an integer programming model and a heuristic algorithm to obtain the minimum number of cone constraints which equivalently represent the original inequality. The performance of the proposed algorithm and the computational effect of reformulations are numerically illustrated.

Introduction

In this paper, we consider a particular form of inequalities of the $n$ -block-power- $m$ type as described in Definition 1. These are encountered in a wide range of optimization models in different fields such as engineering, finance, robust optimization and combinatorics. Special cases of them commonly appear in the constraints set of optimization models including convex power functions [1], antenna design, portfolio optimization, truss design and signal filter design (see [8]). In the following sections we will show that an $n$ -block-power- $m$ inequality can equivalently be expressed with a number of second-order (quadratic) cones (see Definition 2), which will be referred as conic reformulation in the rest of the paper. Reformulating the original optimization problem with inequalities of more general forms equivalently with cone constraints facilitates their solutions by user-developed programs or available commercial solvers. For example, when a linear optimization problem involves an inequality of the form given in (1), it may be difficult to solve. However, when the problem is expressed in a form that involves only quadratic cone constraints, then it belongs to the well known class of Second Order Cone Programming (SOCP), for which a vast number of solution approaches exist. Hence, the contribution of our work lies in an intermediate stage of an optimization problem where a general inequality given in the form above is converted to cone inequalities that are handled with more ease. We will also see below that a conic reformulation results in creation of several simpler inequities that equivalently represent the original one, but such a reformulation is not unique. Hence from both practical and theoretical point of view it is desirable to obtain an equivalent representation with minimum number of conic constraints. We will refer to a conic reformulation with minimum possible number of conic constraints as the minimal reformulation. For the reformulation, we provide an iterative power reduction scheme which creates a new inequality at each iteration. We propose an Integer Programming (IP) model and a fast heuristic algorithm to obtain the minimum number of cone constraints which are equivalent to an $n$ -block power- $m$ inequality in its initial form. The performance of the proposed algorithm and the computational effect of reformulations are also illustrated.

The rest of the manuscript is organized as follows. In Section 2, we give preliminary definitions and a brief discussion of the related literature to demonstrate how an $n$ -block-power- $m$ inequality is convertible to cone constraints. In Section 3 we provide our reformulation scheme together with analytical results. Then we propose an IP model and a heuristic algorithm for reformulation. In Section 4 we present the performance of the proposed algorithm in generating minimal reformulations, and a short numerical study to illustrate the computational benefit of minimal reformulations. We conclude the paper with potential applications of our results in Section 5.

Section snippets

Preliminaries and literature review

In this section we provide the basic definitions needed for the reformulation of an $n$ -block-power- $m$ inequality. In line with the usual convention of the optimization models, throughout this article all variables are assumed to be non-negative. Also, ${(.)}^{⊺}$ denotes the transpose of a vector $(.)$ .

We begin by introducing the special form of inequalities mentioned above.

Definition 1

For $m, n \in N$ , an inequality of the form $t_{0}^{2^{m}} \leq t_{1}^{r_{1}} t_{2}^{r_{2}} \dots t_{n}^{r_{n}}$ where $t_{0} \in R$ , the variables $t_{1}, t_{2}, \dots, t_{n} \in R^{+}$ are non-identical and $r_{1} + r_{2} + \dots + r_{n} = 2^{m}$

Problem statement

Conic reformulation of an $n$ -block-power- $m$ inequality rests on a simple fact given in the following lemma.

Lemma 1

The inequality $t_{0}^{2^{m}} \leq \prod_{i = 1}^{n} t_{i}^{r_{i}}$ is equivalent to, $t_{0}^{2^{m}} \leq t_{ı_{1}}^{r_{ı_{1}} - α} t_{ı_{2}}^{r_{ı_{2}} - α} w^{2 α} \prod_{i \neq ı_{1}, ı_{2}}^{} t_{i}^{r_{i}},$ $w^{2} \leq t_{ı_{1}} t_{ı_{2}},$ wherein $t_{ı_{1}}^{α}$ and $t_{ı_{2}}^{α}$ are paired and $0 \leq α \leq min (r_{ı_{1}}, r_{ı_{2}})$ .

The following example illustrates how Lemma 1 is invoked successively to reformulate a 3-block-power-4 inequality.

Example 1

$t_{0}^{8} \leq t_{1}^{2} t_{2}^{3} t_{3}^{3}$ : $\Leftrightarrow t_{0}^{8} \leq t_{2}^{1} t_{3}^{3} w_{1}^{4}, w_{1}^{2} \leq t_{1} t_{2}, (α_{1} = 2); \Leftrightarrow t_{0}^{8} \leq t_{3}^{2} w_{1}^{4} w_{2}^{2}, w_{2}^{2} \leq t_{2} t_{3}, (α_{2} = 1); \Leftrightarrow t_{0}^{8} \leq w_{1}^{4} w_{3}^{4}, w_{3}^{2} \leq t_{3} w_{2}, (α_{3} = 2); \Leftrightarrow t_{0}^{2} \leq w_{1} w_{3}, (α_{4} = 4) .$

The

Numerical study

First we discuss the performance of Algorithm 1 in producing minimal reformulations. To this end, we used 35 instances of $n$ -block-power- $m$ inequalities ( $2 \leq m, n \leq 7$ ), whose exponents were purposefully chosen to be difficult. (For the construction of the test bed, the reader is referred to Appendix.)

The number of cone constraints, required for the conic reformulation of the above-mentioned instances, are illustrated in Fig. 1.

According to our unreported additional tests, Algorithm 1 was able to

Conclusion

In this paper, we considered a special type of inequalities termed as $n$ -block-power- $m$ and proposed a reformulation algorithm to equivalently express it by conic inequalities. The performance of the reformulation is also demonstrated by numerical examples. Recently, a class of problems called $p$ -order conic optimization is introduced by Mosek (although has not been put into practice in its interface yet) and it covers problems with constraints including $p$ -norm. To the best of our knowledge (see

Acknowledgments

The research described in this paper was funded by Iran’s National Elites Foundation, I.R. IRAN; The Scientific and Technological Research Council of Turkey (TUBİTAK) [Grant Number 110M307] which are gratefully acknowledged here. We thank the two anonymous reviewers whose comments helped to improve the original version of the paper.

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Minimal conic quadratic reformulations and an optimization model

Abstract

Introduction

Section snippets

Preliminaries and literature review

Problem statement

Numerical study

Conclusion

Acknowledgments

Oper. Res. Lett.

Linear Algebra Appl.

European J. Oper. Res.

Second-order cone programming

Math. Program.

Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, Vol. 2

On polyhedral approximations of the second-order cone

Math. Oper. Res.