[Paper Review] On Positive Duality Gaps in Semidefinite Programming

Gábor Pataki|arXiv (Cornell University)|Dec 31, 2018

Advanced Optimization Algorithms Research33 references3 citations

TL;DR

This paper provides a complete characterization of two-variable semidefinite programs (SDPs) with positive duality gaps using elementary row operations and Gaussian elimination, transforming them into a standard form where the gap is immediately apparent. It further shows that the same structural mechanism underlies positive gaps in higher-dimensional SDPs and identifies the maximum possible singularity degree for such gaps.

ABSTRACT

We present a novel analysis of semidefinite programs (SDPs) with positive duality gaps, i.e. different optimal values in the primal and dual problems. These SDPs are extremely pathological, often unsolvable, and also serve as models of more general pathological convex programs. However, despite their allure, they are not well understood even when they have just two variables. We first completely characterize two variable SDPs with positive gaps; in particular, we transform them into a standard form that makes the positive gap trivial to recognize. The transformation is very simple, as it mostly uses elementary row operations coming from Gaussian elimination. We next show that the two variable case sheds light on larger SDPs with positive gaps: we present SDPs in any dimension in which the positive gap is caused by the same structure as in the two variable case. We analyze a fundamental parameter, the {\em singularity degree} of the duals of our SDPs, and show that it is the largest that can result in a positive gap. We finally generate a library of difficult SDPs with positive gaps (some of these SDPs have only two variables) and present a computational study.

Motivation & Objective

• To fully characterize semidefinite programs (SDPs) with positive duality gaps when there are only two variables.
• To develop a simple transformation using elementary row operations (akin to Gaussian elimination) that renders the positive duality gap trivial to detect.
• To show that the two-variable structure explains positive gaps in higher-dimensional SDPs by identifying a common underlying mechanism.
• To analyze the singularity degree of duals in such SDPs and prove it reaches the theoretical maximum for positive gaps.
• To generate a library of challenging SDPs with positive gaps and perform a computational study to validate the findings.

Proposed method

• Transforms two-variable SDPs into a canonical form using only elementary row operations and matrix congruence transformations.
• Employs facial reduction and the concept of strict facial reduction sequences to analyze the structure of the primal and dual cones.
• Uses the singularity degree of the dual problem as a key invariant to classify pathological SDPs and determine the maximum possible degree for positive gaps.
• Applies a regularized facial reduction sequence to simplify the SDP structure while preserving the duality gap.
• Derives certificates of positive duality gaps via linear combinations of constraint matrices and dual feasibility conditions.
• Constructs a library of difficult SDPs with positive gaps, including minimal two-variable instances, for computational validation.

Experimental results

Research questions

• RQ1 What structural properties characterize two-variable SDPs that exhibit a positive duality gap?
• RQ2 Can such SDPs be transformed into a standard form where the positive gap is immediately recognizable through elementary operations?
• RQ3 Does the mechanism causing positive gaps in two-variable SDPs also explain gaps in higher-dimensional SDPs?
• RQ4 What is the maximum possible singularity degree of the dual problem in SDPs with a positive duality gap?
• RQ5 Can a systematic library of challenging SDPs with positive gaps be constructed for computational testing and validation?

Key findings

• All two-variable SDPs with positive duality gaps can be transformed via elementary row operations into a canonical form where the gap is trivially identifiable.
• The positive duality gap in such SDPs arises from a specific structural configuration in the constraint matrices that is preserved under the transformation.
• The singularity degree of the dual problem in these SDPs reaches the theoretical maximum possible value for a positive gap, indicating a fundamental limit in the structure of pathological SDPs.
• The same structural mechanism responsible for positive gaps in two-variable SDPs underlies gaps in higher-dimensional SDPs, providing a unifying explanation.
• A computational study confirms that the constructed library of SDPs with positive gaps includes instances that are challenging for standard SDP solvers, such as Mosek, which may incorrectly report primal infeasibility.
• The paper establishes that facial reduction and extended duals can remedy such pathologies, but only in theory—practical solvers still fail on these instances due to the inherent ill-conditioning and gap structure.