A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems

Hayashi, Shunsuke; Yamashita, Nobuo; Fukushima, Masao

doi:10.1137/s1052623403421516

Cited by 190 publications

(184 citation statements)

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“…Our result generalizes corresponding results of [8] and [12] (from second-order cones and positive semi-definite cones respectively) to symmetric cones. Interesting enough, the expression of the Clarke generalized Jacobian of x + is linked to rank-one matrices.…”

Section: Introductionsupporting

confidence: 87%

“…Our next result generalizes Lemma 2.6 of [9] (for B-subdifferential), Theorem 3.7 of [12] and Proposition 4.8 of [8] (for Clarke generalized Jacobian).…”

Section: Clarke Generalized Jacobian Of π Ksupporting

confidence: 66%

“…This is influenced by the recent work [8,9,12] in the special settings of second-order cones and positive semi-definite cones, where Kanzow, Ferenczi and Fukushima [9] gave an expression for the B-subdifferential of the projection onto second-order cones, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones; Malick and Sendov [12] worked out the Clarke generalized Jacobian of the projection onto the cone of symmetric positive semi-definite matrices. We generalize the above results to symmetric cones.…”

Section: Clarke Generalized Jacobian Of π Kmentioning

confidence: 99%

“…Note that the projection onto an arbitrary closed convex set is Lipschitz continuous. In the case of second-order cone, we know that the projection operator is strongly semismooth, see, e.g., [2,3,8]. In particular, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones.…”

Section: Introductionmentioning

confidence: 99%

“…In the case of second-order cone, we know that the projection operator is strongly semismooth, see, e.g., [2,3,8]. In particular, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones. Malick and Sendov [12] studied the differentiability properties of the projection onto cone of positive semi-definite matrices and worked out its Clarke generalized Jacobian.…”

Section: Introductionmentioning

confidence: 99%

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Clarke Generalized Jacobian of the Projection onto Symmetric Cones

2009

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Section: Introductionsupporting

confidence: 87%

“…Our next result generalizes Lemma 2.6 of [9] (for B-subdifferential), Theorem 3.7 of [12] and Proposition 4.8 of [8] (for Clarke generalized Jacobian).…”

Section: Clarke Generalized Jacobian Of π Ksupporting

confidence: 66%

Section: Clarke Generalized Jacobian Of π Kmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Clarke Generalized Jacobian of the Projection onto Symmetric Cones

2009

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A new one-step smoothing newton method for the second-order cone complementarity problem

Fang

Han

2010

Math. Meth. Appl. Sci.

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A new one-step smoothing Newton method for the second-order cone complementarity problem Liang Fang a * † and Congying Han b Communicated by J. CashIn this paper, we present a new one-step smoothing Newton method for solving the second-order cone complementarity problem (SOCCP). Based on a new smoothing function, the SOCCP is approximated by a family of parameterized smooth equations. At each iteration, the proposed algorithm only need to solve one system of linear equations and perform only one Armijo-type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the SOCCP solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm. Copyright © 2010 John Wiley & Sons, Ltd.Keywords: second-order cone complementarity; smoothing Newton method; Jordan product; coerciveness; global convergence IntroductionConsider the following second-order cone complementarity problem (SOCCP) of finding a vector x ∈ R n such thatwhere ·, · represents the Euclidean inner product, f : R n → R n is a continuously differentiable P 0 -function, Q is the Cartesian product of second-order cones (SOCs), i.e.with n 1 +n 2 +···+n m = n, and the n i -dimensional SOC Q n i is defined bywherex = (x 2 ,x 3 , ···,x n ) ∈ R n i −1 . For simplicity, the problem defined by (1), (2) and (3) is written by P 0 −SOCCP(f ). It is well known that Q n i is a closed, convex, self-dual cone with nonempty interior given by int(Q n i ):={(x 1 ,x) ∈ R×R n i −1 |x 1 > x } and the boundary given by:bd(Q n i ):={(x 1 ,x) ∈ R×R n i −1 |x 1 = x }.We may often drop the Subscripts if the dimension is evident from the context. Here and below, · refers to the Euclidean norm, n i is the dimension of Q n i , Q 1 denoting the set of nonnegative reals R + , and for abbreviation, we write x = (x 1 ,x) ∈ R n−1 instead of (x 1 ,x T ) T . It can be easily seen that P 0 -SOCCP(f ) is a natural extension of the nonlinear complementarity problem (NCP) over the nonnegative orthant cone R n + with n 1 = n 2 =···=n m = 1. Without loss of generality, in this paper, we assume that m = 1 and n 1 = n in the subsequent analysis, since our analysis can be easily extended to general cases. Recently there has been strong interests in smoothing Newton methods for solving the SOCCPs [1--6]. One motivation is that SOCCPs have wide applications in many fields [7,8]. The SOCCPs have been utilized as a general framework for quadratic programming, linear complementarity, and many other mathematical programming problems. Different concepts have been developed to treat SOCCPs. Some approaches employ a reformulation of P 0 −SOCCP(f ) as an unconstrained smooth minimization problem or a system of nonlinear equations, and different methods have been developed to treat them [1,2,5,6,9]. The idea of smoothing Newton method is to use a smooth function to reformulate the problem concerned as a family of parameterized smooth equ...

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On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones

Zhang

2022

Numerical Linear Algebra App

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There are many studies on the well-known modulus-based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance.Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second-order cones (SOCLCPs). It also works on single second-order cone and the non-negative orthant. On this basis, we further propose an iteration-independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state-of-the-art solvers SCS and SuperSCS that solve the equivalent SOC programming.

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A Combined Smoothing and Regularization Method for Monotone Second-Order Cone Complementarity Problems

Cited by 190 publications

References 21 publications

Clarke Generalized Jacobian of the Projection onto Symmetric Cones

Clarke Generalized Jacobian of the Projection onto Symmetric Cones

A new one-step smoothing newton method for the second-order cone complementarity problem

On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones

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