A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

Pan, Shaohua; Chen, Jein Shan

doi:10.1007/s10589-008-9166-9

Cited by 45 publications

(22 citation statements)

References 26 publications

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“…Here, e is not the same in different problems. We compare the results with those in [19]. The numerical results show that the proposed algorithms have almost the same efficiency as that in [19].…”

Section: Numerical Illustration and Concluding Remarksmentioning

confidence: 91%

“…We compare the results with those in [19]. The numerical results show that the proposed algorithms have almost the same efficiency as that in [19]. Table 3 Numerical results of Algorithms I and II for four test problems Test nb nb-L1 nb-L2 nb-L2-bessel problem x = s = 0.05e, [19] x = 2e, y = 0, [19] x = s = 0.45e, [19] x = s = 0.17e, [19] I II I II I II I II k 52 52 28 55 55 87 28 28 / 22 22 9 Time/s …”

Section: Numerical Illustration and Concluding Remarksmentioning

confidence: 97%

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Two new predictor-corrector algorithms for second-order cone programming

Zeng

Bai

Jian

et al. 2011

Appl. Math. Mech.-Engl. Ed.

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Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming (SOCP) are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton (AHO) directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O(rln(ε0/ε)) is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.

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Section: Numerical Illustration and Concluding Remarksmentioning

confidence: 91%

Section: Numerical Illustration and Concluding Remarksmentioning

confidence: 97%

Two new predictor-corrector algorithms for second-order cone programming

Zeng

Bai

Jian

et al. 2011

Appl. Math. Mech.-Engl. Ed.

View full text Add to dashboard Cite

show abstract

“…A popular SOC-complementarity function is the Fischer-Burmeister function, which is semismooth [33] and defined as…”

Section: Neural Network Model Based On Smoothed Fischer-burmeister Fumentioning

confidence: 99%

“…The following lemma gives the gradient of φ ε FB . Since the proofs can be found in [14,33,37], we here omit them.…”

Section: Neural Network Model Based On Smoothed Fischer-burmeister Fumentioning

confidence: 99%

Neural networks for solving second-order cone constrained variational inequality problem

Sun

Chen

2010

Comput Optim Appl

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In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality (SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-KuhnTucker (KKT) conditions of the SOCCVI problem. The first neural network uses the Fischer-Burmeister (FB) function to achieve an unconstrained minimization which is a merit function of the Karush-Kuhn-Tucker equation. We show that the merit function is a Lyapunov function and this neural network is asymptotically stable. The second neural network is introduced for solving a projection formulation whose solutions coincide with the KKT triples of SOCCVI problem. Its Lyapunov stability and global convergence are proved under some conditions. Simulations are provided to show effectiveness of the proposed neural networks.

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“…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 equations and solve the smooth equations approximately by using Newton method at each iteration.…”

mentioning

confidence: 99%

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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A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

Abstract: Second-order cone, Complementarity, B-subdifferential, Semismooth, Newton’s method,

Cited by 45 publications

References 26 publications

Two new predictor-corrector algorithms for second-order cone programming

Two new predictor-corrector algorithms for second-order cone programming

Neural networks for solving second-order cone constrained variational inequality problem

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

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