Hidden convexity of quadratic systems and its application to quadratic programming
Nguyen Quang Huy, Nguyen Huy Hung, Tran Van Nghi, Hoang Ngoc Tuan, Nguyen Van Tuyen
TL;DR
The paper studies hidden convexity in quadratic systems and its impact on quadratic programming by introducing weaker assumptions (H1)-(H4) that guarantee convexity of the image-set $U(f,g_1,...,g_m)$. It develops sufficient conditions for convexity, analyzes hidden convexity for a trust-region problem with linear inequalities, and provides a complete proof for the convexity of a system of two quadratic functions. It derives necessary and sufficient S-lemma conditions for quadratic inequalities and establishes global optimality and strong duality results for QCQP under the proposed framework. These results broaden the applicability of convex reformulations to nonconvex quadratic problems, enabling robust optimality guarantees and duality-based solution methods.
Abstract
In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math. Program. 147, 171--206, 2014}]. We also provide a complete proof of the hidden convexity of a system of two quadratic functions in [{\it J. Glob. Optim. 56, 1045--1072, 2013}]. Furthermore, necessary and sufficient conditions for the S-lemma concerning systems of quadratic inequalities are investigated. Finally, we derive necessary and sufficient global optimality conditions and strong duality results for quadratic programming.