Heuristic Bundle Upper Bound Based Polyhedral Bundle Method for Semidefinite Programming

TL;DR

This work develops a modified polyhedral bundle approach for semidefinite programming by introducing a heuristic upper bound on the number of bundles tied to the primal solution rank and a rank-prediction mechanism when the rank is unknown. The method replaces the SDP semidefinite constraint with a polyhedral lower model, solving a sequence of small quadratic programs augmented with a regularization term to ensure stability. Theoretical convergence is established for the modified scheme, and extensive numerical tests on random sparse SDPs and Max-Cut relaxations show improved efficiency for low-condition problems, with guidance on parameter tuning for harder instances. Overall, the approach offers a scalable alternative to interior-point methods in certain regimes, highlighting a trade-off between bundle count, rank estimation, and subproblem cost that governs practical performance.

Abstract

Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the polyhedral bundle method to standard SDPs. The basic idea of this method is to approximate semidefinite constraints using linear constraints, and thereby transform the SDP problem into a series of quadratic programming subproblems. However, the number of linear constraints often increases continuously during the iteration process, leading to a significant reduction in the solution efficiency. Therefore, based on the idea of limiting the upper bound on the number of bundles, we heuristically select the upper bound through numerical experiments according to the rank of the primal optimal solution and propose a modified subproblem. In this way, under the premise of ensuring the approximation ability of the lower approximation model, we minimize the number of bundles as much as possible to improve the solution efficiency of the subproblems. The algorithm performs well in both Max-Cut problems and random SDP problems. In particular, for random sparse SDP problems with low condition numbers, under the condition of a relative accuracy of \(10^{-4}\), it shows a significant improvement compared with algorithms such as interior-point methods.

Figures (4)

• Figure 2.1: Relationship between the properties of the solution to Subproblem (\ref{['eq:sub6']}) and Parameter $\xi$, where the problem parameters are set as follows: $t_{k} = 1,y_{k} = [0.1; 1],b = [0.1;0.1],B = [0,1;1,1],a = [0;0].$ The blue lines represent the constraints, the light blue area denotes the feasible region, and the green curve is the contour curve corresponding to the objective function and the optimal solution (marked by the green circle).
• Figure 5.1: The suboptimality of random instances with $n=500, m=500$, and $r^{\star}=10$.
• Figure 5.2: The suboptimality of Max-Cut instances G1.
• Figure 5.3: The suboptimality of Max-Cut instances G1.

Theorems & Definitions (5)

• Lemma 1.1
• Theorem 2.1
• Lemma 4.1
• Theorem 4.1
• Lemma 4.2