### ON DOUBLY POSITIVE SEMIDEFINITE PROGRAMMING RELAXATIONS

Taoran Fu1, Dongdong Ge2, Yinyu Ye3

1. 1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China;
2. School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China;
3. Department of Management Science and Engineering, Stanford University, Stanford, CA 94305
• Received:2017-05-23 Revised:2017-06-12 Online:2018-05-15 Published:2018-05-15
• Supported by:

The second author's research was supported by Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and by National Natural Science Foundation of China (NSFC) Project 11471205; the third author's research was supported in part by NSF GOALI 0800151.

Taoran Fu, Dongdong Ge, Yinyu Ye. ON DOUBLY POSITIVE SEMIDEFINITE PROGRAMMING RELAXATIONS[J]. Journal of Computational Mathematics, 2018, 36(3): 391-403.

Recently, researchers have been interested in studying the semidefinite programming (SDP) relaxation model, where the matrix is both positive semidefinite and entry-wise nonnegative, for quadratically constrained quadratic programming (QCQP). Comparing to the basic SDP relaxation, this doubly-positive SDP model possesses additional O(n2) constraints, which makes the SDP solution complexity substantially higher than that for the basic model with O(n) constraints. In this paper, we prove that the doubly-positive SDP model is equivalent to the basic one with a set of valid quadratic cuts. When QCQP is symmetric and homogeneous (which represents many classical combinatorial and nonconvex optimization problems), the doubly-positive SDP model is equivalent to the basic SDP even without any valid cut. On the other hand, the doubly-positive SDP model could help to tighten the bound up to 36%, but no more. Finally, we manage to extend some of the previous results to quartic models.

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