On Doubly Positive Semidefinite Programming ... - Optimization Online
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On Doubly Positive Semidefinite Programming ... - Optimization Online
On Doubly Positive Semidefinite Programming. Relaxations. Dongdong Ge. â. Yinyu Ye. â . August 25, 2010. Abstract. Recently, researchers have been ...
On Doubly Positive Semidefinite Programming Relaxations Dongdong Ge
∗
Yinyu Ye†
August 25, 2010
Abstract 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.
1
Introduction
Consider the quadratically constrained quadratic programming problem Maximize xT Q0 x + cT0 x Subject to xT Qi x + cTi x = bi , i = 1, . . . , m, −e ≤ x ≤ e,
