Second-order Cone Complementarity Problem Research Articles

One-parametric class of smoothing functions for (WSOCCP) and its Jacobian consistency

In this paper, we are interested in the study of weighted second-order cone complementarity problem (WSOCCP) and the smoothing function associated with it. A weighted second-order cone complementarity problem is to find a pair of vectors ( x , y ) that belong to the intersection of a manifold and a second-order cone such that the product of the vectors in a certain algebra, x ∘ y equals a given weight vector w. It is obvious to see that if w is zero, we get a second-order cone complementarity problem (SOCCP). Thus, (WSOCCP) provides the essential framework for a broader class of problems arising from real applications (science and engineering) more conveniently. This paper demonstrates the Jacobian consistency of the one-parametric class of smoothing functions. Furthermore, we provide an estimation of the distance between the subgradient of the one-parametric class of the weighted SOC complementarity functions and the gradient of its associated smoothing function. This estimation facilitates to adjust an appropriate parameter in smoothing methods.

Inexact sequential injective algorithm for weakly univalent vector equation and its application to regularized smoothing Newton algorithm for mixed second-order cone complementarity problems

<p style='text-indent:20px;'>It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this paper, we first propose an inexact sequential injective algorithm (ISIA) for a vector equation, and show the global convergence under weak univalence assumption. Roughly speaking, the ISIA generates the sequence of inexact solutions of approximate vector equations, which consist of the injectives converging to the original vector-valued function. Although the ISIA is simple and conceptual, it can be a prototype to many other algorithms such as a smoothing Newton algorithm, semismooth Newton algorithm, etc. Next, we apply the ISIA prototype to the regularized smoothing Newton algorithm (ReSNA) for mixed second-order cone complementarity problems (MSOCCPs). Exploiting the ISIA convergence scheme, we prove that the ReSNA is globally convergent under Cartesian <inline-formula><tex-math id="M1">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula> assumption.</p>

A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

Chance-constrained approach for decentralized supply chain network under uncertain cost

<abstract><p>A decentralized supply chain network under uncertain cost is studied to obtain the optimal decisions of the enterprises in a situation in which the cost is uncertain. The supply chain network members adopt a chance-constrained approach to make decisions. The second-order cone-constrained variational inequality problem is used to construct the chance-constrained supply chain network equilibrium model. Then, the existence and uniqueness properties of the proposed equilibrium model are discussed under some mild assumptions. For the discontinuous functions in the feasible region of the model, the proposed model is converted to a second-order cone complementarity problem. The numerical results show that the uncertainty and risk attitude of retailers and manufacturers have different effects on supply chain network members. When the risk attitude is high, a small change in the risk attitude will significantly change all decisions of supply chain members. If the supply chain member is affected by the uncertainty positively, its profit will increase as its risk attitude increases. Moreover, it is appropriate to adopt a chance-constrained approach when the supply chain members can estimate the distributions of the competitor's strategies.</p></abstract>

PRIMAL-DUAL ALGORITHM FOR QUASI-STATIC CONTACT PROBLEM WITH COULOMB'S FRICTION

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.

New convergence analysis of a class of smoothing Newton-type methods for second-order cone complementarity problem

<abstract><p>In this paper we propose a class of smoothing Newton-type methods for solving the second-order cone complementarity problem (SOCCP). The proposed method design is based on a special regularized Chen-Harker-Kanzow-Smale (CHKS) smoothing function. When the solution set of the SOCCP is nonempty, our method has the following convergence properties: (ⅰ) it generates a bounded iteration sequence; (ⅱ) the value of the merit function converges to zero; (ⅲ) any accumulation point of the generated iteration sequence is a solution of the SOCCP; (ⅳ) it has the local quadratic convergence rate under suitable assumptions. Some numerical results are reported.</p></abstract>

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

A smoothing quasi-Newton method for solving general second-order cone complementarity problems

Recently, there are much interests in studying smoothing Newton method for solving montone second-order cone complementarity problem (SOCCP) or SOCCPs with Cartesian $$P/P_0$$ -property. In this paper, we propose a smoothing quasi-Newon method for solving general SOCCP. We show that the proposed method is well-defined without any additional assumption and has global convergence under standard conditions. Moreover, under the Jacobian nonsingularity assumption, the method is shown to have local superlinear or quadratic convergence rate. Our preliminary numerical experiments show the method could be very effective for solving SOCCPs.

Smoothing Homotopy Method for Solving Second-Order Cone Complementarity Problem

In this paper, based on Chen–Harker–Kanzow–Smale smooth function, we obtain a smoothing homotopy method to solve the second-order cone complementarity problem. The global convergence is ensured under certain non-monotonicity condition for the defined mapping [Formula: see text]. The numerical results illustrate that this method is feasible.

Improved convergence results for an inexact smoothing method for the second-order cone complementarity problem

For the second-order cone complementarity problem (denoted by SOCCP), this paper proves that an inexact smoothing method proposed by Rui and Xu [Applied Mathematics and Computation, 241: 167–182, 2014] is globally and locally superlinearly/quadratically convergent under an assumption that the solution set of the SOCCP is nonempty, without requiring it to be bounded. This convergence result improves those established by Rui and Xu. It is also stronger than those obtained by most exact smoothing methods for the SOCCP. Some numerical results are reported.

A New Complementarity Function and Applications in Stochastic Second-Order Cone Complementarity Problems

This paper considers the so-called expected residual minimization (ERM) formulation for stochastic second-order cone complementarity problems, which is based on a new complementarity function called termwise residual complementarity function associated with second-order cone. We show that the ERM model has bounded level sets under the stochastic weak $$R_0$$ -property. We further derive some error bound results under either the strong monotonicity or some kind of constraint qualifications. Then, we apply the Monte Carlo approximation techniques to solve the ERM model and establish a comprehensive convergence analysis. Furthermore, we report some numerical results on a stochastic second-order cone model for optimal power flow in radial networks.

A new model for solving stochastic second-order cone complementarity problem and its convergence analysis

In this paper, we mainly consider the stochastic second-order cone complementarity problem (SSOCCP). Due to the existence of stochastic variable, the SSOCCP may have no solutions. In order to deal with this problem, we first regard the merit function of the stochastic second-order cone complementarity problem as the loss function and then present a low-risk deterministic model that is a conditional value-at-risk (CVaR) model. However, there may be two difficulties for solving the CVaR model directly: One is that the objective function is a non-smoothing function. The other is that the objective function contains expectation. (In general, the value of expectation is not easy to be calculated.) In view of these two problems, we present the approximation problems of the model by using a smoothing method and a sample average approximation technique. Furthermore, we give the convergence results of global optimal solutions and the convergence results of stationary points of the approximation problems, respectively.

Discovery of new complementarity functions for NCP and SOCCP

It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of Fischer–Burmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions can also be used in many other contexts.

Expected residual minimization formulation for a class of stochastic linear second-order cone complementarity problems

This paper considers a class of stochastic linear second-order cone complementarity problems (SLSOCCP). Noticing that the SLSOCCP does not have a solution suitable to all realizations in general, we present a deterministic formulation, called the expected residual minimization (ERM) formulation, for it. The coercive property of the ERM problem and the robustness of its solutions are discussed. Due to the existence of expectation in the ERM problem, we employ the Monte Carlo approximation techniques to approximate the ERM problem and show that, under mild conditions, this approximation approach possesses exponential convergence rate. Then, we extend the above results to a general mixed SLSOCCP. Furthermore, we apply the theoretical results to a stochastic optimal power flow model in radial network and report some numerical dispatching experiments for real-world Southern California Edison 47-bus network.

Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems

The error bound property for a solution set defined by a set-valued mapping refers to an inequality that bounds the distance between vectors closed to a solution of the given set by a residual function. The error bound property is a Lipschitz-like/calmness property of the perturbed solution mapping, or equivalently the metric subregularity of the underlining set-valued mapping. It has been proved to be extremely useful in analyzing the convergence of many algorithms for solving optimization problems, as well as serving as a constraint qualification for optimality conditions. In this paper, we study the error bound property for the solution set of a very general second-order cone complementarity problem (SOCCP). We derive some sufficient conditions for error bounds of SOCCP which is verifiable based on the initial problem data.

Expected Value and Sample Average Approximation Method for Solving Stochastic Second-Order Cone Complementarity Problems

ABSTRACTIn this paper, we consider the stochastic second-order cone complementarity problems (SSOCCP). We first formulate the SSOCCP contained expectation as an optimization problem using the so-called second-order cone complementarity function. We then use sample average approximation method and smoothing technique to obtain the approximation problems for solving this reformulation. In theory, we show that any accumulation point of the global optimal solutions or stationary points of the approximation problems are global optimal solution or stationary point of the original problem under suitable conditions. Finally, some numerical examples are given to explain that the proposed methods are feasible.

Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for SOCCP

The Jacobian consistency of smoothing functions plays an important role for achieving the rapid convergence of Newton methods or Newton-like methods with an appropriate parameter control. In this paper, we study the properties, derive the computable formula for the Jacobian matrix and prove the Jacobian consistency of a one-parametric class of smoothing Fischer–Burmeister functions for second-order cone complementarity problems proposed by Tang et al. (Comput Appl Math 33:655–669, 2014). Then we apply its Jacobian consistency to a smoothing Newton method with the appropriate parameter control presented by Chen et al. (Math Comput 67:519–540, 1998), and show the global convergence and local quadratic convergence of the algorithm for solving the SOCCP under rather weak assumptions.

Exact formulas for the proximal/regular/limiting normal cone of the second-order cone complementarity set

The proximal, regular and limiting normal cones to the second-order cone complementarity set play important roles in studying mathematical programs with second-order cone complementarity constraints, second-order cone programs, and the second-order cone complementarity problems. It is needed in the first-order optimality conditions for mathematical programs with second-order cone complementarity constraint, the second-order subdifferential criteria in characterizing the full stability for second-order cone programs and second-order cone complementarity problems, as well as in the characterizing the pseudo-Lipschitz continuity of the solution mapping to parametric second-order cone complementarity problems. In this paper we establish explicit formulas for the proximal, regular, and limiting normal cone of the second-order cone complementarity set.

二阶锥三角函数及其相关公式

二阶锥函数在求解二阶锥规划及二阶锥互补问题的算法中有广泛应用。本文在约当代数的理论基础上推导得出三角函数在二阶锥上的表达式，然后给出了二阶锥三角函数的若干相关公式。 The function associated with the second-order cones (SOCs) is wildly used in solving second-order cone programming and second-order cone complementarity problems. In this article, on the basis of Euclidean Jordan algebra, the simple formulas of trigonometric functions associated with SOC are derived and some correlation formulas of trigonometric function are also presented.