Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation

Second order cone constrained convex relaxations for nonconvex quadratically constrained quadratic programming

“…In this section, we will introduce an artificial linear valid inequality for problem (P), which was first proposed by Zheng et al [31]. We then propose a new relaxation by introducing RLT, SOC-RLT and GSRT constraints associated with this new linear valid inequality and show its dominance over the decomposition-approximation method in [31]. Adopting the setting in [31] in the following of this section, we consider problem (P) with nonnegativity constraint x ≥ 0.…”

“…We illustrate below the different kinds of valid inequalities generated by RLT-like technique, i.e., linearizing the product of the left hand side yields the valid inequalities on the right hand side, and also indicate in the list the sections (or subsections) in which different RLT-like techniques are developed, [3] and this paper) Section 5, where L represents a linear inequality constraint, SOC(convex) (SOC(nonconvex), respectively) represents an SOC constraint generated from a convex (nonconvex, respectively) constraint, M( 0) represents an LMI, HSOC represents the valid inequalities generated by linearizing the Hadamard product of two valid LMIs (expressed in (34) later in the paper) in [31] and KSOC represents the valid inequalities generated by linearizing the Kronecker product of two valid LMIs first derived in [3].…”

“…In Section 2, we first review existing convex relaxations with various valid inequalities in the literature and then propose our novel GSRT constraints. In Section 3, we apply RLT-like techniques to additional redundant linear constraint and demonstrate a dominance relationship of our method over the method in [31]. We propose in Section 4 another class of valid inequalities, SST constraints, by linearizing the product of two SOC constraints.…”

Optimal Information Blending with Measurements in the L2 Sphere

“…Since X is bounded,γ ν < ∞. A lemma established in [44] shows that for any x ∈ X , diag(ν)xx ⊤ diag(ν) γ ν diag(ν)diag(x) . Recall that S 0 iff P SP ⊤ 0, where P can be any invertible matrix.…”

Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs

“…For instances, convexification approach [10], interval newton method [11], simplicial branch-and-bound algorithm [12], semidefinite relaxation approach [13], matrix cone decomposition and polyhedral approximation algorithm [14], duality bound algorithm [15], robust optimization algorithm [16], rectangle branch-and-bound algorithms [17][18][19][20][21][22][23][24][25][26][27][28]. However, to our knowledge, although some algorithms can be also used to solve the QCQP, due to the complication of the investigated problem, it is rather challenging to globally solve the QCQP problem.…”