NDLI: Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wu, Baiyi Zheng, Xiaojin Sun, Xiaoling Li, Duan
Copyright Year	2017
Abstract	We consider in this paper a class of semicontinuous quadratic programming problems, which arises in many real-world applications such as production planning, portfolio selection, and subset selection in regression. We build upon the idea of the quadratic convex reformulation approach, i.e., adding to the original objective function an additional convex term that is zero at all points of the feasible region. Exploiting the structure of semicontinuous variables, we derive the most general set of quadratic functions that can be added. We also incorporate the state-of-the-art perspective reformulation into our new framework. Within this framework, the set of optimal reformulations with the tightest continuous relaxation can be found by solving a semidefinite programming problem. We also explore the relationship among different reformulations within our framework and simplify the search for optimal ones in certain cases with the help of second-order cone programming. Among the proposed set of optimal reformulations, we can choose one with or without the perspective component. If the perspective component is present, the new reformulation can be solved by the perspective cut algorithm in the literature. If the perspective component is not present, the new reformulation is a standard mixed-integer quadratic programming (MIQP) problem and can be plugged into and solved by any MIQP solvers. Our preliminary numerical tests in both portfolio selection and subset selection problems have demonstrated the effectiveness of our new reformulations, and the comparison results indicate that the performance of the new reformulation without the perspective component, when solved in general MIQP solvers, is competitive to the perspective cut approach.
Sponsorship	Research Grants Council, University Grants Committee. Patrick Huen Wing Ming Chair Professorship of Systems Engineering and Engineering Management. National Natural Science Foundation of China
Starting Page	1531
Ending Page	1553
Page Count	23
File Format	PDF
ISSN	10526234
DOI	10.1137/15M1012232
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-08-03
Access Restriction	Subscribed
Subject Keyword	subset selection Semi-infinite programming mixed-integer quadratic programming quadratic convex reformulation semicontinuous quadratic program portfolio selection Management decision making, including multiple objectives semidefinite program Mixed integer programming Quadratic programming perspective cut reformulation
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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