Hidden conic quadratic representation of some nonconvex quadratic optimization problems

• Published in: Mathematical programming Vol. 143; no. 1-2; pp. 1 - 29
• Main Authors: Ben-Tal, Aharon, den Hertog, Dick
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2014; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Conics; Full Length Paper; Inequality; Linear programming; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Optimization; Quadratic programming; Studies; Theoretical; 90C20; Robust optimization; Conic Quadratic Program; Hidden convexity; Implementation error; 90C26; Simultaneous diagonalizability; S-lemma
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-013-0710-8

The problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property, even when the quadratic forms are indefinite. The equivalent convex problem is a semidefinite one, and the equivalence is based on the celebrated S-lemma. In this paper, we show that when the quadratic forms are simultaneously diagonalizable (SD), it is possible to derive an equivalent convex problem, which is a conic quadratic (CQ) one, and as such is significantly more tractable than a semidefinite problem. The SD condition holds for free for many problems arising in applications, in particular, when deriving robust counterparts of quadratic, or conic quadratic, constraints affected by implementation error. The proof of the hidden CQ property is constructive and does not rely on the S-lemma. This fact may be significant in discovering hidden convexity in some nonquadratic problems.