KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems

Lü, Cheng; Fang, Shu‐Cherng; Jin, Qingwei; Wang, Zhenbo; Xing, Wenxun

doi:10.1137/100793955

Cited by 39 publications

(18 citation statements)

References 25 publications

(28 reference statements)

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“…Sturm and Zhang [57] discuss cones of nonnegative quadratic functions and impose a copositivity restriction on a more general set. More recently, the papers [44,45] have addressed quadratically constrained quadratic programs by examining cones of nonnegative quadratic functions, based on the programs' KKT conditions that are lifted along with the feasible region of the conic QCQPs to define a certain cone of symmetric matrices. It would be interesting to investigate the detailed connections of the latter papers with our work; this investigation is left for a future study.…”

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confidence: 99%

On conic QPCCs, conic QCQPs and completely positive programs

2015

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This paper studies several classes of nonconvex optimization problems defined over convex cones, establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs. The problems being studied include: a conic quadratically constrained quadratic program (QCQP), a conic quadratic program with complementarity constraints (QPCC), and rank constrained semidefinite programs. Our results do not make any boundedness assumptions on the feasible regions of the various problems considered. The first stage in the reformulation is to cast the problem as a conic QCQP with just one nonconvex constraint q(x) ≤ 0, where q(x) is nonnegative over the (convex) conic and linear constraints, so the problem fails the Slater constraint qualification. A conic QPCC has such a structure; we prove the converse, namely that any conic QCQP satisfying a constraint qualification can be expressed as an equivalent conic QPCC. of the reformulation lifts the problem to a completely positive program, and exploits and generalizes a result of Burer. We also show that a Frank-Wolfe type result holds for a subclass of this class of conic QCQPs. Further, we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper.Mathematics Subject Classification Primary 90C20; Secondary 65K05 · 90C11 · 90C22 · 90C25 · 90C26 · 90C33 · 90C46

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confidence: 99%

On conic QPCCs, conic QCQPs and completely positive programs

2015

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“…A similar yet different approach is taken in [35], where a conic exact reformulation of problem (2.1) is proposed, using another intractable cone and constructing tractable approximation hierarchies for this cone. The examples specified in [35] again reduce to the NND cone K 0 or its dual, the DNN cone K 0 .…”

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confidence: 99%

“…The examples specified in [35] again reduce to the NND cone K 0 or its dual, the DNN cone K 0 . However, for large n, even K 0 may involve too many (namely (n−1)n 2 ) linear inequalities to allow for efficient computation.…”

Section: Possible Algorithmic Implicationsmentioning

confidence: 99%

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Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Bomze¹

2015

SIAM J. Optim.

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We study nonconvex quadratic minimization problems under (possibly nonconvex) quadratic and linear constraints, characterizing both Lagrangian and semi-Lagrangian dual bounds in terms of conic optimization. While the Lagrangian dual is equivalent to the SDP relaxation (which has been known for quite a while, although the presented form, incorporating explicitly linear constraints, seems to be novel), we show that the semi-Lagrangian dual is equivalent to a natural copositive relaxation (and this has apparently not been observed before). This way, we arrive at conic bounds tighter than the usual Lagrangian dual (and thus than the SDP) bounds. Any of the known tractable inner approximations of the copositive cone can be used for this tightening, but in particular, the above-mentioned characterization with explicit linear constraints is a natural, much cheaper, relaxation than the usual zero-order approximation by doubly nonnegative (DNN) matrices and still improves upon the Lagrangian dual bounds. These approximations are based on LMIs on matrices of basically the original order plus additional linear constraints (in contrast to more familiar sum-of-squares or moment approximation hierarchies) and thus may have merits in particular for large instances where it is important to employ only a few inequality constraints (e.g., n instead of n(n−1) 2 for the DNN relaxation). Further, we specify sufficient conditions for tightness of the semi-Lagrangian relaxation and show that copositivity of the slack matrix guarantees global optimality for KKT points of this problem, thus significantly improving upon a well-known secondorder global optimality condition.

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“…Moreover, Sturm and Zhang [20] extended the copositive cone to the cone of nonnegative quadratic functions and reformulated a quadratic programming problem as a linear conic programming problem. Note that, the cone of nonnegative quadratic functions or forms over a general domain is uncomputable [13]. However, for some special domains, these cones are solvable subclasses.…”

Section: Ye Tian Qingwei Jin and Zhibin Dengmentioning

confidence: 99%

Quadratic optimization over a polyhedral cone

Tian¹,

Jin²,

Deng³

2015

JIMO

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In this paper, we study the polyhedral cone constrained homogeneous quadratic programming problem and provide an equivalent linear conic reformulation. Based on a union of second-order cones which covers the polyhedral cone, a sequence of computable linear conic programming problems are constructed to approximate the linear conic reformulation. The convergence of the sequential solutions is guaranteed as the number of second-order cones increases such that the union of the second-order cones gets close to the polyhedral cone. In order to relieve the computational burden and improve the efficiency, an adaptive scheme and valid inequalities derived by the reformulation-linearization technique are added to the proposed algorithm. Finally, the numerical results demonstrate the effectiveness of the algorithm.2010 Mathematics Subject Classification. Primary: 90C26, 90C59, 90C22; Secondary: 30E10.

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KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems

Cited by 39 publications

References 25 publications

On conic QPCCs, conic QCQPs and completely positive programs

On conic QPCCs, conic QCQPs and completely positive programs

Copositive Relaxation Beats Lagrangian Dual Bounds in Quadratically and Linearly Constrained Quadratic Optimization Problems

Quadratic optimization over a polyhedral cone

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