[Paper Review] A hierarchy of spectral relaxations for polynomial optimization

Ngoc Hoang Anh Mai, Victor Magron+1 more|arXiv (Cornell University)|Jul 17, 2020

Advanced Optimization Algorithms Research4 citations

TL;DR

This paper proposes a hierarchy of spectral relaxations for polynomial optimization by reformulating constrained polynomial problems on a sphere, enabling the use of the constant trace property (CTP) in semidefinite relaxations. By minimizing the largest eigenvalue of a matrix pencil via ad-hoc spectral methods instead of interior-point methods, the approach achieves convergence to the global optimum with improved computational efficiency for moderate-scale problems.

ABSTRACT

We show that (i) any constrained polynomial optimization problem (POP) has an equivalent formulation on a variety contained in an Euclidean sphere and (ii) the resulting semidefinite relaxations in the moment-SOS hierarchy have the constant trace property (CTP) for the involved matrices. We then exploit the CTP to avoid solving the semidefinite relaxations via interior-point methods and rather use ad-hoc spectral methods that minimize the largest eigenvalue of a matrix pencil. Convergence to the optimal value of the semidefinite relaxation is guaranteed. As a result we obtain a hierarchy of nonsmooth "spectral relaxations" of the initial POP. Efficiency and robustness of this spectral hierarchy is tested against several equality constrained POPs on a sphere as well as on a sample of randomly generated quadratically constrained quadratic problems (QCQPs).

Motivation & Objective

• To address the high computational cost of solving semidefinite relaxations in the moment-SOS hierarchy via interior-point methods.
• To exploit the constant trace property (CTP) in semidefinite relaxations arising from polynomial optimization problems (POPs).
• To develop a scalable alternative to interior-point methods by using spectral methods that minimize the largest eigenvalue of a matrix pencil.
• To ensure convergence to the global optimum while reducing computational burden for constrained polynomial optimization on spheres and balls.
• To test the efficiency and robustness of the proposed spectral hierarchy on equality-constrained POPs and randomly generated QCQPs.

Proposed method

• Reformulate any constrained POP on a basic compact semialgebraic set as an equivalent problem on a variety contained within a Euclidean sphere.
• Show that the resulting moment-SOS relaxations inherit the constant trace property (CTP), enabling eigenvalue-based solution methods.
• Replace standard interior-point methods with ad-hoc spectral methods that minimize the largest eigenvalue of a symmetric matrix pencil.
• Use the dual formulation of CTP-based SDPs to reduce the problem to minimizing the maximum eigenvalue, suitable for first-order or bundle methods.
• Apply the limited-memory bundle method (LMBM) to solve the resulting nonsmooth, non-convex eigenvalue minimization problem with global convergence guarantees.
• Convert moment relaxations into standard SDP form with trace constraints, enabling efficient implementation and numerical testing.

Experimental results

Research questions

• RQ1 Can constrained polynomial optimization problems be equivalently reformulated on a sphere to enable the constant trace property in their semidefinite relaxations?
• RQ2 Does exploiting the constant trace property allow for faster and more scalable solution of semidefinite relaxations without sacrificing convergence?
• RQ3 Can spectral methods based on maximum eigenvalue minimization outperform interior-point methods in terms of computational efficiency for moderate-scale POPs?
• RQ4 How robust and efficient is the proposed spectral hierarchy on equality-constrained QCQPs and randomly generated quartic problems on the unit sphere and ball?
• RQ5 What is the impact of the CTP on the structure of moment matrices and the numerical conditioning of the resulting relaxations?

Key findings

• Any constrained polynomial optimization problem can be equivalently reformulated on a variety within a Euclidean sphere, preserving the global optimum.
• The resulting moment-SOS relaxations for such reformulated problems possess the constant trace property (CTP), enabling eigenvalue-based solution techniques.
• The proposed spectral hierarchy converges to the global optimum of the original POP, with convergence guaranteed by the properties of the moment-SOS hierarchy.
• Numerical experiments show that the spectral method outperforms interior-point solvers in computational time and scalability on equality-constrained QCQPs on the unit sphere.
• For randomly generated dense QCQPs on the unit ball, the spectral approach demonstrates robust performance with reduced memory usage and faster convergence compared to standard SDP solvers.
• The method achieves finite convergence for generic instances, consistent with the known finite convergence of the moment-SOS hierarchy, while avoiding the high cost of interior-point methods.