Rank conditions for exactness of semidefinite relaxations in polynomial optimization (2501.06052v1)

Abstract: We consider the Moment-SOS hierarchy in polynomial optimization. We first provide a sufficient condition to solve the truncated K-moment problem associated with a given degree-$2n$ pseudo-moment sequence $\phi$ n and a semi-algebraic set $K \subset \mathbb{R}^d$. Namely, let $2v$ be the maximum degree of the polynomials that describe $K$. If the rank $r$ of its associated moment matrix is less than $nv + 1$, then $\phi^n$ has an atomic representing measure supported on at most $r$ points of $K$. When used at step-$n$ of the Moment-SOS hierarchy, it provides a sufficient condition to guarantee its finite convergence (i.e., the optimal value of the corresponding degree-n semidefinite relaxation of the hierarchy is the global minimum). For Quadratic Constrained Quadratic Problems (QCQPs) one may also recover global minimizers from the optimal pseudo-moment sequence. Our condition is in the spirit of Blekherman's rank condition and while on the one-hand it is more restrictive, on the other hand it applies to constrained POPs as it provides a localization on $K$ for the representing measure.

• The paper presents a novel rank condition that ensures finite convergence of the SDP relaxation to the global optimum in polynomial optimization problems.
• It details a methodology that links moment matrix ranks with the existence of an atomic measure, enabling extraction of global minimizers for constrained and unconstrained POPs.
• The findings offer practical stopping criteria for the Moment-SOS hierarchy and enhance computational efficiency in solving complex, non-convex optimization problems.

Rank Conditions for Exactness of Semidefinite Relaxations in Polynomial Optimization

Polynomial optimization problems (POPs) are central in mathematical optimization, where the objective is to minimize a polynomial function over a feasible region defined by polynomial inequalities. The complexity and non-convexity of these problems have prompted researchers to explore various solution approaches, among which the Moment-SOS hierarchy stands out as a powerful technique that relies on semidefinite programming (SDP) relaxations.

This paper by Jean B. Lasserre focuses on the role of rank conditions in ensuring the exactness of SDP relaxations within the Moment-SOS hierarchy. The primary contribution is the establishment of rank conditions, offering sufficient criteria to guarantee the finite convergence of the hierarchy to global optima in polynomial optimization problems. The paper extends the utility of semidefinite relaxations by providing practical conditions under which they yield exact solutions—converging to the global minimum of the original problem—thus facilitating both theoretical understanding and computational efficiency in solving POPs.

Mathematical Framework

The Moment-SOS hierarchy transforms a polynomial optimization problem into a sequence of SDP relaxations that provide increasingly accurate approximations of the global minimum. For a given degree $2n$, each relaxation in the hierarchy involves constructing a moment matrix $M_n(y)$ associated with a pseudo-moment sequence $y$ and testing for positive semidefiniteness. The convergence to the global minimum is influenced by the rank of the moment matrix.

The paper introduces a pivotal rank condition: if the rank $r$ of the moment matrix $M_n(y)$ satisfies $r \leq n - v + 1$, where $v$ is the maximum degree of the polynomials defining the semi-algebraic feasible set, then the pseudo-moment sequence has an atomic measure supported on at most $r$ points of the feasible set. This result parallels Blekherman's rank condition but importantly extends it to constrained polynomial optimization problems by incorporating a localization condition.

Implications for POPs

1. Exactness of Relaxations: The established rank condition provides a formal criterion under which the SDP relaxation at a given hierarchy step exactly recovers the global minimum of the polynomial optimization problem. This is significant for applications where exact solutions are critical and computational resources are limited, permitting termination of the hierarchy once the condition is met.
2. Recovering Global Minimizers: For quadratic constrained quadratic problems (QCQPs), the conditions enable the extraction of global minimizers from optimal pseudo-moment sequences. This facilitates not only the identification of the optimal value but also the constructing the solution set.
3. Unconstrained POPs: For unconstrained problems, the results corroborate that even when no feasible set is defined, the rank condition can still ensure exactness of the solution provided by the single SDP relaxation associated with the POP.

Theoretical and Practical Advancements

The theoretical advancements presented in this paper bridge a crucial gap by connecting moment matrix ranks with solution exactness, thereby enhancing the applicability of semidefinite relaxations in polynomial optimization. Practically, these results allow researchers to effectively monitor the progress of the Moment-SOS hierarchy and to make informed decisions about stopping criteria, ultimately optimizing the computational process.

Speculations and Future Directions

Future research could explore the refinement of these rank conditions for broader classes of optimization problems, including those involving higher-dimensional convex bodies and non-polynomial constraints. Additionally, advancements in numerical methods for large-scale semidefinite programming could benefit significantly from these theoretical insights, particularly in hybrid optimization frameworks integrating POP and non-polynomial elements.

In conclusion, the paper's contributions not only advance the theoretical foundation of the Moment-SOS methodology for polynomial optimization but also offer practical tools for efficiently solving complex optimization problems using semidefinite programming techniques.