[Paper Review] Polynomial Optimization with Real Varieties

Jiawang Nie|arXiv (Cornell University)|Nov 8, 2012

Polynomial and algebraic computation8 references3 citations

TL;DR

This paper establishes finite convergence of Lasserre’s hierarchy for polynomial optimization over real varieties, proving that finite convergence occurs when the real variety $ V_{\mathbb{R}}(h) $ is finite, even if the complex variety $ V_{\mathbb{C}}(h) $ is infinite. It further shows that finite convergence is independent of polynomial representation of the real variety and extends to a refined preordering-based hierarchy when the feasible set is finite.

ABSTRACT

We consider the optimization problem of minimizing a polynomial f(x) subject to polynomial constraints h(x)=0, g(x)>=0. Lasserre's hierarchy is a sequence of sum of squares relaxations for finding the global minimum. Let K be the feasible set. We prove the following results: i) If the real variety V_R(h) is finite, then Lasserre's hierarchy has finite convergence, no matter the complex variety V_C(h) is finite or not. This solves an open question in Laurent's survey. ii) If K and V_R(h) have the same vanishing ideal, then the finite convergence of Lasserre's hierarchy is independent of the choice of defining polynomials for the real variety V_R(h). iii) When K is finite, a refined version of Lasserre's hierarchy (using the preordering of g) has finite convergence.

Motivation & Objective

• To resolve the open question of whether Lasserre’s hierarchy exhibits finite convergence when the real variety $ V_{\mathbb{R}}(h) $ is finite but the complex variety $ V_{\mathbb{C}}(h) $ is infinite.
• To establish that finite convergence of Lasserre’s hierarchy is independent of the choice of defining polynomials for the real variety $ V_{\mathbb{R}}(h) $, provided $ K $ and $ V_{\mathbb{R}}(h) $ have the same vanishing ideal.
• To develop and analyze a refined hierarchy using the preordering of constraints $ g $, showing finite convergence when the feasible set $ K $ is finite.
• To provide theoretical foundations for the finite convergence of sum-of-squares relaxations in polynomial optimization, particularly in cases with finite real solution sets.

Proposed method

• Uses Lasserre’s hierarchy of sum-of-squares (SOS) relaxations defined via truncated ideals $ \langle h \rangle_{2k} $ and quadratic modules $ Q_k(g) $, with dual semidefinite programs involving moment and localizing matrices.
• Applies Positivstellensatz theorems and flat truncation criteria to analyze finite convergence, particularly leveraging the existence of interpolating polynomials when $ K $ is finite.
• Introduces a refined SOS relaxation using the preordering $ Pr_k(g) $, which includes all products of the constraints $ g_j $, to strengthen the relaxation and ensure finite convergence for finite $ K $.
• Employs a perturbation argument via Lemma 2.1 with parameter $ \epsilon $, constructing a decomposition $ f - (f_{\min} - \epsilon) = \sigma_\epsilon + \phi_\epsilon $, where $ \sigma_\epsilon \in Pr_N(g) $ and $ \phi_\epsilon \in \langle h \rangle_{2N} $, to prove finite convergence.
• Relies on the archimedean condition and Putinar’s Positivstellensatz for asymptotic convergence, and uses the structure of real algebraic varieties to derive finite convergence results.
• Uses interpolation polynomials $ \varphi_i $ vanishing at all but one point of the finite set $ K $, and constructs an SOS polynomial $ a $ to express $ f - f_{\min} $ as a sum of squares on $ K $, enabling finite convergence.

Experimental results

Research questions

• RQ1 Does Lasserre’s hierarchy have finite convergence when $ V_{\mathbb{R}}(h) $ is finite but $ V_{\mathbb{C}}(h) $ is infinite?
• RQ2 Is the finite convergence of Lasserre’s hierarchy independent of the choice of defining polynomials for the real variety $ V_{\mathbb{R}}(h) $, provided $ K $ and $ V_{\mathbb{R}}(h) $ have the same vanishing ideal?
• RQ3 Can a stronger SOS relaxation using the preordering of constraints $ g $ achieve finite convergence when the feasible set $ K $ is finite?
• RQ4 What conditions ensure that the sequence $ \{f_k^{\text{pre}}\} $ of preordering-based relaxations converges finitely to the global minimum?
• RQ5 Can the finite convergence of the dual problem of the preordering-based hierarchy be extended to the primal relaxation sequence?

Key findings

• Lasserre’s hierarchy has finite convergence if $ V_{\mathbb{R}}(h) $ is finite, even when $ V_{\mathbb{C}}(h) $ is infinite, thus resolving an open question in Laurent’s survey.
• Finite convergence of Lasserre’s hierarchy is independent of the choice of defining polynomials for $ V_{\mathbb{R}}(h) $, provided $ K $ and $ V_{\mathbb{R}}(h) $ have the same vanishing ideal.
• When the feasible set $ K $ is finite, the preordering-based hierarchy $ \{f_k^{\text{pre}}\} $ has finite convergence to $ f_{\min} $, even though the standard hierarchy $ \{f_k\} $ may not.
• For finite $ K $, there exists an integer $ N $ such that $ f_k^{\text{pre}} = f_{\min} $ for all $ k \geq N $, as shown via a perturbation argument and decomposition into preordering and ideal components.
• The proof constructs a decomposition $ f - (f_{\min} - \epsilon) = \sigma_\epsilon + \phi_\epsilon $ with $ \sigma_\epsilon \in Pr_N(g) $ and $ \phi_\epsilon \in \langle h \rangle_{2N} $, ensuring finite convergence for $ k \geq N $.
• An example with $ K = \{(0,0)\} $ demonstrates that $ f_k^{\text{pre}} = 0 $ for all $ k \geq 6 $, confirming finite convergence in practice.