arXiv subjects

Stefan Sremac

Publications and source records attributed to Stefan Sremac.

5 recordsLinked to original sources

arXiv2026

Finding Maximum Determinant Principal Submatrices via Hadamard Bounds and Projection Methods

An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and machine learning. We study several exact and approximate approaches to this problem. We first derive an upper bound based on Hadamard's inequality, along with a projection scheme based on the Gram--Schmidt process without normalization. This combination yields a highly effective upper bound and leads to an exact branch-and-bound algorithm for moderate-sized instances. For larger scale problems we propose a continuous relaxation that facilitates reliable performance evaluation when the exact method returns only near-optimal solutions. We further prove that the projection scheme strengthens the upper bound derived from this relaxation. Numerical experiments demonstrate the effectiveness of the proposed methods across a broad range of datasets.

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arXiv2025

A Convex Optimization Approach to the Discrete Hanging Chain Problem

In this paper we investigate the discrete version of the classical hanging chain problem. We generalize the problem, by allowing for arbitrary mass and length of each link. We show that the shape of the chain can be obtained by solving a convex optimization problem. Then we use optimality conditions to show that the problem can be further reduced to solving a single non-linear equation, when the links of the chain have symmetric mass and length.

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arXiv2019

Error Bounds and Singularity Degree in Semidefinite Programming

In semidefinite programming a proposed optimal solution may be quite poor in spite of having sufficiently small residual in the optimality conditions. This issue may be framed in terms of the discrepancy between forward error (the unmeasurable `true error') and backward error (the measurable violation of optimality conditions). In his seminal work, Sturm provided an upper bound on forward error in terms of backward error and singularity degree. In this paper we provide a method to bound the maximum rank over all solutions and use this result to obtain a lower bound on forward error for a class of convergent sequences. This lower bound complements the upper bound of Sturm. The results of Sturm imply that semidefinite programs with slow convergence necessarily have large singularity degree. Here we show that large singularity degree is, in some sense, also a sufficient condition for slow convergence for a family of external-type `central' paths. Our results are supported by numerical observations.

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arXiv2018

Maximum determinant positive definite Toeplitz completions

We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite replaced by positive semidefinite, and maximum determinant replaced by maximum rank. These results are used to determine the singularity degree of a family of semidefinite optimization problems.

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arXiv2017

Complete Facial Reduction in One Step for Spectrahedra

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.

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