An active-set algorithm for norm constrained quadratic problems

• Published in: Mathematical programming Vol. 193; no. 1; pp. 447 - 483
• Main Authors: Rontsis, Nikitas, Goulart, Paul J., Nakatsukasa, Yuji
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Constraints; Full Length Paper; Linear functions; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Principal components analysis; Quadratic equations; Quadratic programming; Theoretical; Trust-region; Dimensionality reduction; Nonconvex optimization; Sequential quadratic programming; 65F15; 90C90; 90C26; Active-set
• ISSN: 0025-5610; 1436-4646; 1436-4646
• DOI: 10.1007/s10107-021-01617-2

We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of trust-region subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied in the literature, resulting in remarkably efficient algorithms and software. We extend these results by proving that nonglobal minimizers of the TRS, or a certificate of their absence, can also be calculated efficiently by computing the two rightmost eigenpairs of an eigenproblem. We demonstrate the usefulness and scalability of the algorithm in a series of experiments that often outperform state-of-the-art approaches; these include calculation of high-quality search directions arising in Sequential Quadratic Programming on problems of the CUTEst collection, and Sparse Principal Component Analysis on a large text corpus problem (70 million nonzeros) that can help organize documents in a user interpretable way.