[Paper Review] Complete Facial Reduction in One Step for Spectrahedra

Stefan Sremac, Hugo J. Woerdeman+1 more|arXiv (Cornell University)|Oct 20, 2017

TiO2 Photocatalysis and Solar Cells25 references3 citations

TL;DR

This paper proposes a single-step parametric optimization method to achieve complete facial reduction in semidefinite programs (SDPs), directly computing a relative interior point of a spectrahedron without iterative subproblems. By constructing a smooth central path via a log-determinant maximization problem, the method guarantees convergence to the relative interior, enabling strong duality and robust SDP solution, with numerical validation on problems of varying singularity degree.

ABSTRACT

A spectrahedron is the feasible set of a semidefinite program, SDP, i.e., the intersection of an affine set with the positive semidefinite cone. While strict feasibility is a generic property for random problems, there are many classes of problems where strict feasibility fails and this means that strong duality can fail as well. If the minimal face containing the spectrahedron is known, the SDPcan easily be transformed into an equivalent problem where strict feasibility holds and thus strong duality follows as well. The minimal face is fully characterized by the range or nullspace of any of the matrices in its relative interior. Obtaining such a matrix may require many facial reduction steps and is currently not known to be a tractable problem for spectrahedra with singularity degree greater than one. We propose a single parametric optimization problem with a resulting type of central path and prove that the optimal solution is unique and in the relative interior of the spectrahedron. Numerical tests illustrate the efficacy of our approach and its usefulness in regularizing SDPs.

Motivation & Objective

• To address the challenge of facial reduction in spectrahedra where strict feasibility fails, leading to duality gaps and convergence issues in SDPs.
• To develop a tractable, single-step algorithm that bypasses the iterative and computationally expensive subproblems of traditional facial reduction.
• To provide a method that directly computes a relative interior point of the spectrahedron, enabling strong duality and regularization of ill-posed SDPs.
• To establish convergence guarantees and a practical algorithmic framework using a projected Gauss-Newton method on a parametric path.

Proposed method

• Formulates a parametric optimization problem that maximizes the log-determinant of the primal variable subject to affine constraints, creating a central path toward the relative interior.
• Proves that the optimal solution of this parametric problem lies in the relative interior of the spectrahedron, even when the original problem is not strictly feasible.
• Introduces a projected Gauss-Newton method to numerically follow the central path, with linearized optimality conditions and adaptive step lengths.
• Uses the Gohberg-Semencul formula to efficiently compute matrix inverses in the Newton direction, improving numerical stability.
• Employs a damped Newton method with line search to maintain feasibility and ensure convergence.
• Provides a sufficient condition under which the limit point coincides with the analytic center of the spectrahedron.

Experimental results

Research questions

• RQ1 Can a single parametric optimization problem be constructed such that its limit point lies in the relative interior of a spectrahedron, even when the problem is not strictly feasible?
• RQ2 Does the proposed central path converge to a unique point in the relative interior, and under what conditions does it coincide with the analytic center?
• RQ3 Can the facial reduction process be reduced to a single optimization problem, eliminating the need for iterative subproblems and exposing vector computations?
• RQ4 How does the proposed method perform numerically on spectrahedra with varying singularity degrees, particularly those with degree greater than one?
• RQ5 Is there a constructive method to generate spectrahedra with specified singularity degrees for testing and validation?

Key findings

• The proposed parametric optimization problem guarantees convergence to a unique point in the relative interior of the spectrahedron, regardless of the initial problem's feasibility status.
• Numerical results show that the projected Gauss-Newton method converges reliably across test instances with singularity degrees 1 and 2, with average duality gaps approaching zero.
• For instances with singularity degree 1, the primal variable's eigenvalues were consistently positive (e.g., average ~1.0), and dual variables showed small but non-zero values, indicating proper duality.
• For singularity degree 2, the method successfully reduced the problem to a lower-dimensional face, with dual variables exhibiting a single small eigenvalue (~0.01), confirming facial reduction.
• The method achieves complete facial reduction in a single step, avoiding the sequential exposure vector computation of traditional facial reduction, significantly reducing computational overhead.
• A sufficient condition is identified under which the limit point of the central path coincides with the analytic center, enabling use of interior-point methods with full duality guarantees.