Effective algorithms for separable nonconvex quadratic programming with one quadratic and box constraints

Hezhi Luo (), Xianye Zhang, Huixian Wu () and Weiqiang Xu ()
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Hezhi Luo: Zhejiang Sci-Tech University
Xianye Zhang: Zhejiang Sci-Tech University
Huixian Wu: Hangzhou Dianzi University
Weiqiang Xu: Zhejiang Sci-Tech University

Computational Optimization and Applications, 2023, vol. 86, issue 1, No 6, 199-240

Abstract: Abstract We consider in this paper a separable and nonconvex quadratic program (QP) with a quadratic constraint and a box constraint that arises from application in optimal portfolio deleveraging (OPD) in finance and is known to be NP-hard. We first propose an improved Lagrangian breakpoint search algorithm based on the secant approach (called ILBSSA) for this nonconvex QP, and show that it converges to either a suboptimal solution or a global solution of the problem. We then develop a successive convex optimization (SCO) algorithm to improve the quality of suboptimal solutions derived from ILBSSA, and show that it converges to a KKT point of the problem. Second, we develop a new global algorithm (called ILBSSA-SCO-BB), which integrates the ILBSSA and SCO methods, convex relaxation and branch-and-bound framework, to find a globally optimal solution to the underlying QP within a pre-specified $$\epsilon $$ ϵ -tolerance. We establish the convergence of the ILBSSA-SCO-BB algorithm and its complexity. Preliminary numerical results are reported to demonstrate the effectiveness of the ILBSSA-SCO-BB algorithm in finding a globally optimal solution to large-scale OPD instances.

Keywords: Separable nonconvex quadratic program; Lagrangian breakpoint search; Successive convex optimization; Convex relaxation; Branch-and-bound; 90C20; 90C22; 90C26 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00485-0

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