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Degree reduction techniques for polynomial optimization problems

Brais González-Rodríguez, Joe Naoum-Sawaya

TL;DR

Computational results show that reducing the degree of the polynomial to a degree that is higher than two provides computational advantages in certain cases compared to fully quadrifying the problem.

Abstract

This paper presents a new approach to quadrify a polynomial programming problem, i.e. reduce the polynomial program to a quadratic program, before solving it. The proposed approach, QUAD-RLT, exploits the Reformulation-Linearization Technique (RLT) structure to obtain smaller relaxations that can be solved faster and still provide high quality bounds. QUAD-RLT is compared to other quadrification techniques that have been previously discussed in the literature. The paper presents theoretical as well as computational results showing the advantage of QUAD-RLT compared to other quadrification techniques. Furthermore, rather than quadrifying a polynomial program, QUAD-RLT is generalized to reduce the degree of the polynomial to any degree. Computational results show that reducing the degree of the polynomial to a degree that is higher than two provides computational advantages in certain cases compared to fully quadrifying the problem. Finally, QUAD-RLT along with other quadrification/degree reduction schemes are implemented and made available in the freely available software RAPOSa.

Degree reduction techniques for polynomial optimization problems

TL;DR

Computational results show that reducing the degree of the polynomial to a degree that is higher than two provides computational advantages in certain cases compared to fully quadrifying the problem.

Abstract

This paper presents a new approach to quadrify a polynomial programming problem, i.e. reduce the polynomial program to a quadratic program, before solving it. The proposed approach, QUAD-RLT, exploits the Reformulation-Linearization Technique (RLT) structure to obtain smaller relaxations that can be solved faster and still provide high quality bounds. QUAD-RLT is compared to other quadrification techniques that have been previously discussed in the literature. The paper presents theoretical as well as computational results showing the advantage of QUAD-RLT compared to other quadrification techniques. Furthermore, rather than quadrifying a polynomial program, QUAD-RLT is generalized to reduce the degree of the polynomial to any degree. Computational results show that reducing the degree of the polynomial to a degree that is higher than two provides computational advantages in certain cases compared to fully quadrifying the problem. Finally, QUAD-RLT along with other quadrification/degree reduction schemes are implemented and made available in the freely available software RAPOSa.
Paper Structure (11 sections, 5 theorems, 20 equations, 3 tables)

This paper contains 11 sections, 5 theorems, 20 equations, 3 tables.

Table of Contents

  1. Introduction
  2. The Reformulation-Linearization Technique and Quadrification Schemes for Polynomial Problems
  3. The Reformulation-Linearization Technique
  4. Quadrification Schemes
  5. New Dimension Results for Different Quadrification Schemes
  6. A new quadrification scheme
  7. Computational results
  8. Degree reduction
  9. Computational results
  10. Conclusion
  11. Results evaluating tightness and size at the root node

Key Result

Theorem 1

Let $N^{LP}_1$, $N^{LP}_2$, and $N^{LP}_3$ be the number of variables in eq:LPsch1, eq:LPsch2, and eq:LPsch3, respectively. Let $R^{LP}_1$, $R^{LP}_2$, and $R^{LP}_3$ be the number of constraints in eq:LPsch1, eq:LPsch2, and eq:LPsch3, respectively. It holds that $N^{LP}_1\leq N^{LP}_2 \leq N^{LP}_3

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • Example 2
  • Theorem 3
  • proof
  • Example 3
  • Theorem 4
  • ...and 3 more