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Sparse convex relaxations in polynomial optimization

Gennadiy Averkov, Benjamin Peters, Sebastian Sager

TL;DR

The concept of monomial patterns is introduced, which helps to understand existing approaches from different schools of thought, to develop novel relaxation schemes, and to derive a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature.

Abstract

We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering. Historically, different approaches on how to solve nonconvex polynomial optimization problems based on convex relaxations have been developed in different scientific communities. Here, we introduce the concept of monomial patterns. A pattern determines what monomials are to be linked by convex constraints in a convex relaxation of a polynomial optimization problem. This concept helps to understand existing approaches from different schools of thought, to develop novel relaxation schemes, and to derive a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature. We unify different approaches to polynomial optimization including polyhedral approximations, dense semidefinite relaxations, SONC, SAGE, and TSSOS in a self-contained exposition. We also carry out computational experiments to demonstrate the practical advantages of a flexible usage of pattern-based sparse relaxations of polynomial optimization problems.

Sparse convex relaxations in polynomial optimization

TL;DR

The concept of monomial patterns is introduced, which helps to understand existing approaches from different schools of thought, to develop novel relaxation schemes, and to derive a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature.

Abstract

We present a novel, general, and unifying point of view on sparse approaches to polynomial optimization. Solving polynomial optimization problems to global optimality is a ubiquitous challenge in many areas of science and engineering. Historically, different approaches on how to solve nonconvex polynomial optimization problems based on convex relaxations have been developed in different scientific communities. Here, we introduce the concept of monomial patterns. A pattern determines what monomials are to be linked by convex constraints in a convex relaxation of a polynomial optimization problem. This concept helps to understand existing approaches from different schools of thought, to develop novel relaxation schemes, and to derive a flexible duality theory, which can be specialized to many concrete situations that have been considered in the literature. We unify different approaches to polynomial optimization including polyhedral approximations, dense semidefinite relaxations, SONC, SAGE, and TSSOS in a self-contained exposition. We also carry out computational experiments to demonstrate the practical advantages of a flexible usage of pattern-based sparse relaxations of polynomial optimization problems.
Paper Structure (46 sections, 28 theorems, 96 equations, 7 figures)

This paper contains 46 sections, 28 theorems, 96 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The problem class polynomial optimization
  3. Polynomials
  4. Optimization
  5. How to globally solve polynomial optimization problems
  6. Convexification
  7. Monomial Lifting
  8. Formal convexification
  9. Formal sparsification
  10. Modeling choices and optimization paradigms
  11. Duality
  12. Contributions
  13. Structure of the paper
  14. Modeling principles for convexification
  15. Semidefinite modeling principles
  16. ...and 31 more sections

Key Result

Theorem 2.3

\newlabelputinar:thm0 Let $X = \{ x \in \mathbb{R}^n \colon g_1(x) \ge 0,\ldots, g_s(x) \ge 0\}$ be a compact semialgebraic set contained in the ball $\{ x \colon x_1^2 + \cdots + x_n^2 \le \rho\}$ of some radius $\rho \in \mathbb{R}_{>0}$. Let $g_0 (x):=1$ and $g_{s+1}(x) := \rho - (x_1^2 + \cdot

Figures (7)

  • Figure 1: Elementary steps towards a moment relaxation of a one-dimensional cubic polynomial. First row, left: the curve $(x^\alpha)_{\alpha \in A}$ from \ref{['eq_curve']} for $A=\{1,2,3\}$ and $X=[0,1]$, indicating the feasible values for the lifted, monomial variables $v$. Right: the moment body $\mathcal{M}_A(X) = \operatorname{conv} \{ (x^\alpha)_{\alpha \in A} \colon x \in X\}$, see \ref{['eq_momentbody']}. Second row, left to right: projections $\mathcal{M}_{P_i}(X)$ of $\mathcal{M}_A(X)$ in blue for $P_i\in\mathcal{F} := \{\{1,2\}, \,\{1,3\}, \,\{2,3\}\}$. Third row, left to right: liftings of $\mathcal{M}_{P_i}(X)$ into $\mathbb{R}^3$ for all $P_i\in\mathcal{F}$. Last row: the (intersected) feasible region $(v_\alpha)_{\alpha \in P_i} \in \mathcal{M}_{P_i}(X) \ \text{for} \ i \in [3]$ of problem \ref{['P-RLXa']}.
  • Figure 1: Visualization of involved variables $v_\alpha$ in examplary patterns. The title of each subplot refers to the corresponding set $A$ from Example \ref{['rmk:pattern-plot']}. Exponents $\alpha \in A \subset \mathbb{N}^2$ are depicted as red squares. Pattern-specific auxiliary exponents $\alpha \in \cup_i P_i$ are depicted as blue dots. Curves connect all exponents $\alpha \in P_i$. As a rule of thumb, the largest cardinality $| P_i |$ has a strong influence on the overall runtime to solve \ref{['P-RLXa']}. First row: multilinear patterns with $P = \{0,1\}^2$, Section \ref{['sec_polyhedralbodies']} Second row: expression tree, Section \ref{['sec_expression']}, bound-factor product, Section \ref{['sec_polyhedralbodies']}, and moment relaxation, Section \ref{['sec_sdp']} Third row: different truncated submonoids, Section \ref{['deriving:from:mon:subs']} Fourth row: different shifted chains, Section \ref{['sec_shifting']} Fifth row: Families $F^1$, $F^2$, $F^3$ combining multilinear and shifted chains patterns
  • Figure 1: Results for dense exponent sets $A=\mathbb{N}^n_d$. Shown is the median and standard variation for 20 sample coefficient vectors $f$ of the evaluation criterium \ref{['eq_criterium']} for the methods specified in Section \ref{['sec_setup']}. Note that a value close to $0$ indicates a good relaxation. The average computational time in seconds is provided below each method in the bottom row. Note that only instances are selected for which BARON needs the full time limit on either minimization or maximization problem.
  • Figure 2: As in Figure \ref{['boxplot:dense']}, but for sparse exponent sets $S(n,d)$.
  • Figure 3: As in Figure \ref{['boxplot:dense']}, but for sparse exponent sets $S(n,d)$ with $n > d$.
  • ...and 2 more figures

Theorems & Definitions (91)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3: Putinar
  • Theorem 2.4: Dual Putinar
  • Proof 1
  • Remark 2.5
  • Corollary 2.6
  • Proof 2
  • Remark 2.7
  • Remark 2.8
  • ...and 81 more