Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems

In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45). The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.     

A New Class of Complementarity Function and the Boundedness of Its Merit Function for Symmetric Cone Complementarity Problem

In this paper, we introduce a new class of two-parametric penalized function, which includes the penalized minimum function and the penalized Fischer-Burmeister flmc- tion over symmetric cone complementarity problems. We propose that this class of function is a class of complementarity functions（C-function）. Moreover, its merit function has bounded level set under a weak condition.     

A new class of smoothing complementarity functions over symmetric cones

A popular approach to solving the complementarity problem is to reformulate it as an equivalent system of smooth equations via a smoothing complementarity function. In this paper, first we propose a new class of smoothing complementarity functions, which contains the natural residual smoothing function and the Fischer–Burmeister smoothing function for symmetric cone complementarity problems. Then we give some unified formulae of the Fréchet derivatives associated with Jordan product. Finally, the derivative of the new proposed class of smoothing complementarity functions is deduced over symmetric cones.     

A one-parametric class of merit functions for the symmetric cone complementarity problem

In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227-251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115-135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R₀₂-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293-327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function, Math. Program. 107 (2006) 547-553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework.     

Smoothing algorithms for complementarity problems over symmetric cones

There recently has been much interest in studying optimization problems over symmetric cones. In this paper, we first investigate a smoothing function in the context of symmetric cones and show that it is coercive under suitable assumptions. We then extend two generic frameworks of smoothing algorithms to solve the complementarity problems over symmetric cones, and prove the proposed algorithms are globally convergent under suitable assumptions. We also give a specific smoothing Newton algorithm which is globally and locally quadratically convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for second-order cone complementarity problems are reported.     

A Damped Gauss-Newton Method for the Second-Order Cone Complementarity Problem

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method. S. Pan’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. J.-S. Chen is member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. J.-S. Chen’s work is partially supported by National Science Council of Taiwan.     

A new C-function for symmetric cone complementarity problems

Recently, there has been much interest in studying optimization problems over symmetric cones and second-order cone. This paper uses Euclidean Jordan algebras as a basic tool to introduce a new C-function to symmetric cone complementarity problems. Then we show that the function is coercive, strongly semismooth and its Jacobian is also strongly semismooth.     

A full Nesterov–Todd step infeasible-interior-point algorithm for Cartesian P*(κ) horizontal linear complementarity problems over symmetric cones

Euclidean Jordan algebra is a commonly used tool in designing interior-point algorithms for symmetric cone programs. In this paper, we present a full Nesterov–Todd (NT) step infeasible interior-point algorithm for horizontal linear complementarity problems over Cartesian product of symmetric cones. Since the algorithm uses only full-NT feasibility and centring steps, it has the advantage that no line searches are needed. The complexity result obtained here for symmetric cones using NT directions coincides with the best bound obtained for horizontal linear complementarity problems.     

A Class of Polynomial Interior Point Algorithms for the Cartesian P-Matrix Linear Complementarity Problem over Symmetric Cones

In this paper, we present a new class of polynomial interior point algorithms for the Cartesian P-matrix linear complementarity problem over symmetric cones based on a parametric kernel function, which determines both search directions and the proximity measure between the iterate and the center path. The symmetrization of the search directions used in this paper is based on the Nesterov and Todd scaling scheme. By using Euclidean Jordan algebras, we derive the iteration bounds that match the currently best known iteration bounds for large- and small-update methods.     

Merit functions for semi-definite complemetarity problems

Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems. This research is supported by National Science Foundation Grant CCR-9311621.     

隐互补问题的极小化变形及其稳定点

1 引言 互补问题在最优化中有着广泛的应用,例如线性规划中的对偶问题,非线性规划中求稳定点的KKT条件以及变分不等式的求解都可以转化为互补问题,另外,某些均衡网络设计问题、信号最优化问题以及交通配置等问题也可利用互补问题来求解.     

A new class of complementarity functions for symmetric cone complementarity problems

A popular approach to solving the complementarity problem is to reformulate it as an equivalent equation system via a complementarity function. In this paper, we propose a new class of functions, which contains the penalized natural residual function and the penalized Fischer–Burmeister function for symmetric cone complementarity problems. We show that this class of functions is indeed a class of complementarity functions. We finally prove that the merit function of this new class of complementarity functions is coercive.     

An arc-search predictor-corrector infeasible-interior-point algorithm for P∗(κ)-SCLCPs

Numerical Algorithms - In this paper, we propose a new arc-search predictor-corrector infeasible-interior-point algorithm for linear complementarity problems over symmetric cones with the Cartesian...     

Improved smoothing Newton methods for symmetric cone complementarity problems

There recently has been much interest in smoothing Newton method for solving nonlinear complementarity problems. We extend such method to symmetric cone complementarity problems (SCCP). In this paper, we first investigate a one-parametric class of smoothing functions in the context of symmetric cones, which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Then we propose a smoothing Newton method for the SCCP based on the one-parametric class of smoothing functions. For the proposed method, besides the classical step length, we provide a new step length and the global convergence is obtained. Finally, preliminary numerical results are reported, which show the effectiveness of the two step lengthes in the algorithm and provide efficient domains of the parameter for the complementarity problems.     

对称锥互补问题的一个惩罚NR函数(英文)

本文建立了一个对称锥互补问题的惩罚自然剩余函数,并且证明了单调情形下其相应势函数的水平有界性.     

Smoothing Newton algorithm for symmetric cone complementarity problems based on a one-parametric class of smoothing functions

In this paper, we introduce a one-parametric class of smoothing functions in the context of symmetric cones which contains two symmetric perturbed smoothing functions as special cases, and show that it is coercive under suitable assumptions. Based on this class of smoothing functions, a smoothing Newton algorithm is extended to solve the complementarity problems over symmetric cones, and it is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. The theory of Euclidean Jordan algebras is a basic tool which is extensively used in our analysis. Preliminary numerical results for randomly generated second-order cone programs and several practical second-order cone programs from the DIMACS library are reported.     

Kernel-Based Interior-Point Methods for Monotone Linear Complementarity Problems over Symmetric Cones

We present an interior-point method for monotone linear complementarity problems over symmetric cones (SCLCP) that is based on barrier functions which are defined by a large class of univariate functions, called eligible kernel functions. This class is fairly general and includes the classical logarithmic function, the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both large-step and short-step versions of the method for ten frequently used eligible kernel functions. For some of them we match the best known iteration bound for large-step methods, while for short-step methods the best iteration bound is matched for all cases. The paper generalizes results of Lesaja and Roos (SIAM J. Optim. 20(6):3014–3039, 2010) from P _∗(κ)-LCP over the non-negative orthant to monotone LCPs over symmetric cones.     

A descent method for a reformulation of the second-order cone complementarity problem

Analogous to the nonlinear complementarity problem and the semi-definite complementarity problem, a popular approach to solving the second-order cone complementarity problem (SOCCP) is to reformulate it as an unconstrained minimization of a certain merit function over Rⁿ${\mathbb{R}}^n$. In this paper, we present a descent method for solving the unconstrained minimization reformulation of the SOCCP which is based on the Fischer–Burmeister merit function (FBMF) associated with second-order cone [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327], and prove its global convergence. Particularly, we compare the numerical performance of the method for the symmetric affine SOCCP generated randomly with the FBMF approach [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Programming 104 (2005) 293–327]. The comparison results indicate that, if a scaling strategy is imposed on the test problem, the descent method proposed is comparable with the merit function approach in the CPU time for solving test problems although the former may require more function evaluations.     

The convergence of a modified smoothing-type algorithm for the symmetric cone complementarity problem

The symmetric cone complementarity problem (denoted by SCCP) is a broad class of optimization problems, which contains the semidefinite complementarity problem, the second-order cone complementarity problem, and the nonlinear complementarity problem. In this paper we first extend the smoothing function proposed by Huang et al. (Sci. China 44:1107–1114, 2001) for the nonlinear complementarity problem to the context of symmetric cones and show that it is coercive under suitable assumptions. Based on this smoothing function, a smoothing-type algorithm, which is a modified version of the Qi-Sun-Zhou method (Qi et al. in Math. Program. 87:1–35, 2000), is proposed for solving the SCCP. By using the theory of Euclidean Jordan algebras, we prove that the proposed algorithm is globally and locally quadratically convergent under suitable assumptions. Preliminary numerical results for some second-order cone complementarity problems are reported which indicate that the proposed algorithm is effective.     

On merit functions for <Emphasis Type="Italic">p</Emphasis>-order cone complementarity problem

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with p-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving p-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the p-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving p-order cone complementarity problem.