Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

Yoshise, Akiko

doi:10.1007/978-1-4614-0769-0_12

Cited by 8 publications

(7 citation statements)

References 116 publications

(157 reference statements)

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“…QPCCs withK × K 1 × K 1 * = R n + are discussed in [6,7], for example. Mathematical programs with complementarity constraints with other cones are considered in [33,60], for example.…”

Section: Two Classes Of Conic Quadratic Problemsmentioning

confidence: 99%

“…We generalize the definition of QPCC in this paper to replace the nonnegative orthant by a convex cone. Yoshise [60] surveys research on complementarity problems over symmetric cones. Ding et al [33] investigate MPCCs over the semidefinite cone, and in particular they show that a semidefinite program with a rank constraint can be cast as an equivalent LPCC over the SDP cone.…”

Section: Introductionmentioning

confidence: 99%

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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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“…QPCCs withK × K 1 × K 1 * = R n + are discussed in [6,7], for example. Mathematical programs with complementarity constraints with other cones are considered in [33,60], for example.…”

Section: Two Classes Of Conic Quadratic Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On conic QPCCs, conic QCQPs and completely positive programs

2015

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“…China. 2 College of Advanced Vocational Technology, Shanghai University of Engineering Science, Shanghai, 201620, P.R. China.…”

Section: Theorem  For the Large-update Method Which Is Characterimentioning

confidence: 99%

New complexity analysis of interior-point methods for the Cartesian P ∗ ( κ ) -SCLCP

Wang

Yue

et al. 2013

J Inequal Appl

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In this paper, we give a unified analysis for both large-and small-update interior-point methods for the Cartesian P * (κ)-linear complementarity problem over symmetric cones based on a finite barrier. The proposed finite barrier is used both for determining the search directions and for measuring the distance between the given iterate and the μ-center for the algorithm. The symmetry of the resulting search directions is forced by using the Nesterov-Todd scaling scheme. By means of Euclidean Jordan algebras, together with the feature of the finite kernel function, we derive the iteration bounds that match the currently best known iteration bounds for large-and small-update methods. Furthermore, our algorithm and its polynomial iteration complexity analysis provide a unified treatment for a class of primal-dual interior-point methods and their complexity analysis. MSC: 90C33; 90C51

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“…Kanno et al [31] showed that the problem can be recast as a second-order cone linear complementarity problem. To this formulation, various methods developed in the field of mathematical optimization may be applicable (although existing methods do not have a proof of convergence for this formulation); see Yoshise [45] and Chen and Pan [15] for survey on the second-order cone complementarity problem and its numerical solutions. For example, a regularized smoothing Newton method proposed by Hayashi et al [24] was adopted in Kanno et al [31].…”

Section: Introductionmentioning

confidence: 99%

Primal-dual algorithm for quasi-static contact problem with Coulomb's friction

Kanno¹

2021

Preprint

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This paper presents a fast first-order method for solving the quasi-static contact problem with the Coulomb friction. It is known that this problem can be formulated as a second-order cone linear complementarity problem, for which regularized or semi-smooth Newton methods are widely used. As an alternative approach, this paper develops a method based on an accelerated primal-dual algorithm. The proposed method is easy to implement, as most of computation consists of additions and multiplications of vectors and matrices. Numerical experiments demonstrate that this method outperforms a regularized and smoothed Newton method for the second-order cone complementarity problem.

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Complementarity Problems Over Symmetric Cones: A Survey of Recent Developments in Several Aspects

Cited by 8 publications

References 116 publications

On conic QPCCs, conic QCQPs and completely positive programs

On conic QPCCs, conic QCQPs and completely positive programs

New complexity analysis of interior-point methods for the Cartesian P ∗ ( κ ) -SCLCP

Primal-dual algorithm for quasi-static contact problem with Coulomb's friction

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