Symmetric Cone Complementarity Problems Research Articles

English

In this paper, we propose a smoothing Levenberg-Marquardt method for the symmetric cone complementarity problem. Based on a smoothing function, we turn this problem into a system of nonlinear equations and then solve the equations by the method proposed. Under the condition of Lipschitz continuity of the Jacobian matrix and local error bound, the new method is proved to be globally convergent and locally superlinearly/quadratically convergent. Numerical experiments are also employed to show that the method is stable and efficient.

A new projected Barzilai–Borwein method for the symmetric cone complementarity problem

This paper presents a new projected Barzilai–Borwein method for the complementarity problem over symmetric cone by applying the Barzilai–Borwein-like steplengths to the projected method. A new descent direction is employed and a non-monotone line search is used in the method in order to guarantee the global convergence. The projected Barzilai–Borwein method is proved to be globally convergent under some suitable conditions. Some preliminary computational results are also reported which confirm the good theoretical properties of the proposed method.

A new nonmonotone smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$-property

We present a new smoothing Newton method for the symmetric cone complementarity problem with the Cartesian $$P_0$$ -property. The new method is based on a new smoothing function and a nonmonotone line search which contains a monotone line search as a special case. It is proved that the new method is globally and locally superlinearly/quadratically convergent under mild conditions. Preliminary numerical results are also reported which indicate the proposed method is promising.

Predictor-corrector inexact smoothing algorithm for symmetric cone complementarity problems with Cartesian P0-property

In this paper we consider the symmetric cone complementarity problem with Cartesian P0-property (denoted by P0-SCCP) which includes the well-known monotone symmetric cone complementarity problem. We propose a predictor-corrector inexact smoothing algorithm for solving the P0-SCCP and prove that the method is globally and locally quadratically convergent under suitable assumptions. Especially, we prove that our algorithm can generate a bounded iteration sequence when the solution set of the P0-SCCP is nonempty and bounded, or the solution set of the monotone SCCP is nonempty. Moreover, the proposed algorithm solves the linear systems in both predictor step and corrector step only approximately by using an inexact Newton method. Hence when one solves large-scale SCCPs, our algorithm can save much computation work compared to existing smoothing-type algorithms. Numerical results confirm these good theoretical properties.

A projection and contraction method for symmetric cone complementarity problem

ABSTRACTThis paper considers the symmetric cone complementarity problem. A new projection and contraction method is presented which only requires some projection calculations and functional computations. It is proved that the iteration sequence produced by the proposed method converges to a solution of the symmetric cone complementarity problem under the condition that the underlying transformation is monotone. Numerical experiments also show the effectiveness of this method.

Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP

The smoothing algorithms have been successfully applied to solve the symmetric cone complementarity problem (denoted by SCCP), which in general have the global and local superlinear/quadratic convergence if the solution set of the SCCP is nonempty and bounded. Huang, Hu and Han [Science in China Series A: Mathematics, 52: 833–848, 2009] presented a nonmonotone smoothing algorithm for solving the SCCP, whose global convergence is established by just requiring that the solution set of the SCCP is nonempty. In this paper, we propose a new nonmonotone smoothing algorithm for solving the SCCP by modifying the version of Huang-Hu-Han’s algorithm. We prove that the modified nonmonotone smoothing algorithm not only is globally convergent but also has local superlinear/quadratical convergence if the solution set of the SCCP is nonempty. This convergence result is stronger than those obtained by most smoothing-type algorithms. Finally, some numerical results are reported.

A new smoothing and regularization Newton method for the symmetric cone complementarity problem

Based on a regularized Chen–Harker–Kanzow–Smale (CHKS) smoothing function, we propose a new smoothing and regularization Newton method for solving the symmetric cone complementarity problem. By using the theory of Euclidean Jordan algebras, we establish the global and local quadratic convergence of the method on certain assumptions. The proposed method uses a nonmonotone line search technique which includes the usual monotone line search as a special case. In addition, our method treats both the smoothing parameter $$\mu $$ and the regularization parameter $$\varepsilon $$ as independent variables. Preliminary numerical results are reported which indicate that the proposed method is effective.

Differentiability v.s. convexity for complementarity functions

It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity problem (SOCCP) as special cases. Moreover, there is also so-called generalized complementarity problem (GCP) in infinite dimensional space. Among the existing NCP-functions, it was observed that there are no differentiable and convex NCP-functions. In particular, Miri and Effati (J Optim Theory Appl 164:723–730, 2015) show that convexity and differentiability cannot hold simultaneously for an NCP-function. In this paper, we further establish that such result also holds for general complementarity functions associated with the GCP.

A generalized smoothing Newton method for the symmetric cone complementarity problem

In this paper, a concept of regulation functions is proposed, and some related properties and examples are explored. Based on this regulation function and some smoothing complementarity functions, we present a family of smoothing Newton methods to solve the symmetric cone complementarity problem. This algorithm allows a unified convergence analysis for some smoothing Newton methods. We show that the resulting Newton equation is well-defined and solvable, and provides a theory of global convergence.

Solvability based on E-property for the nonlinear symmetric cone complementarity problem

In this paper, we introduce some concepts of E-properties for nonlinear transformations defined on Euclidean Jordan algebras, such as E-property, quad E-property, uniform Jordan E-property, E0-property. And then we study the implications among these E-properties. In particular, the quad E-property is equivalent to the E-property. The K-copositive property implies the E0-property. The uniform Jordan E-property is proved to have the E0 and R0-properties. Based on the E-properties, we get some sufficient conditions to guarantee the solution existence for the nonlinear symmetric cone complementarity problem.

A smoothing Newton method for symmetric cone complementarity problem

We first extend a new class of smoothing functions, which contains the well-known Chen-Harker-Kanzow-Smale smoothing function and Huang-Han-Chen smoothing function as special cases, for the nonlinear complementarity problem to the symmetric cone complementarity problem (SCCP). And then we present a smoothing Newton algorithm for the SCCP based on the new class of smoothing functions. Both the existence of Newton directions and the boundedness of the level set are showed for the SCCP with the Cartesian $$P_0$$ -property, which contains the monotone SCCP as a special case. The global linear convergence and locally superlinear convergence are established under a nonsingular assumption. Some numerical results for second order cone complementarity problems, a special case of SCCP, show that the proposed algorithm is effective.

A smoothing Newton method for symmetric cone complementarity problems

Recently, there has been much interest in studying optimization problems over symmetric cones. This paper uses Euclidean Jordan algebras as a basic tool to construct a new smoothing function for symmetric cone complementarity problems. It is showed that this new function has similar structure and some good properties as the widely used symmetric perturbed Chen–Harker–Kanzow–Smale smooth function. In particularly, based on the function, we obtain global convergence and locally superlinear convergence of the smoothing Newton algorithm under two weaker assumptions respectively. Some numerical results for second-order cone complementarity problems are also reported.

A Smoothing Newton Algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems

Recently, the study of symmetric cone complementarity problems has been a hot topic in the literature. Many numerical methods have been proposed for solving such a class of problems. Among them, the problems concerned are generally monotonic. In this paper, we consider symmetric cone linear complementarity problems with a class of non-monotonic transformations. A smoothing Newton algorithm is extended to solve this class of non-monotonic symmetric cone linear complementarity problems; and the algorithm is proved to be well-defined. In particular, we show that the algorithm is globally and locally quadratically convergent under mild assumptions. The preliminary numerical results are also reported.

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.

Error bounds for symmetric cone complementarity problems

In this paper, we investigate the issue of error bounds for symmetric cone complementarity problems (SCCPs). In particular, we show that the distance between an arbitrary point in Euclidean Jordan algebra and the solution set of the symmetric cone complementarity problem can be bounded above by some merit functions such as Fischer-Burmeister merit function, the natural residual function and the implicit Lagrangian function. The so-called $R_0$-type conditions, which are new and weaker than existing ones in the literature, are assumed to guarantee that such merit functions can provide local and global error bounds for SCCPs. Moreover, when SCCPs reduce to linear cases, we demonstrate such merit functions cannot serve as global error bounds under general monotone condition, which implicitly indicates that the proposed $R_0$-type conditions cannot be replaced by $P$-type conditions which include monotone condition as special cases.

Level-boundedness of the Penalized NR Function for Symmetric Cone Complementarity Problem

Level-boundedness of the Penalized NR Function for Symmetric Cone Complementarity Problem

The Hölder continuity of Löwner’s operator in Euclidean Jordan algebras

Lowner’s operator in Euclidean Jordan algebras, defined via the spectral decomposition of the elements of a scalar function, has been widely used in various optimization problems over Euclidean Jordan algebras. In this note, we shall show that Lowner’s operator in Euclidean Jordan algebras is Holder continuous if and only if the underlying scalar function is Holder continuous. Such a property will be useful in designing solution methods for symmetric cone programming and symmetric cone complementarity problem.

Lipschitz Continuity of the Solution Mapping of Symmetric ConeComplementarity Problems

This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform Cartesian P‐property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultra P‐property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultra P‐property, the Cartesian P‐property, and the Lipschitz continuity of the solutions are all equivalent to each other.

On the Uniform Nonsingularity Property for Linear Transformations on Euclidean Jordan Algebras

In a recent paper, Chua and Yi introduced the so-called uniform nonsingularity property for a nonlinear transformation on a Euclidean Jordan algebra and showed that it implies the global uniqueness property in the context of symmetric cone complementarity problems. In a related paper, Chua, Lin, and Yi raise the question of converse. In this paper, we show that, for linear transformations, the uniform nonsingularity property is inherited by principal subtransformations and, on simple algebras, it is invariant under the action of cone automorphisms. Based on these results, we answer the question of Chua, Lin, and Yi in the negative.

A regularized smoothing Newton method for solving the symmetric cone complementarity problem

The symmetric cone complementarity problem (denoted by SCCP) is a class of equilibrium optimization problems, and it contains the standard linear/nonlinear complementarity problem (LCP/NCP), the second-order cone complementarity problem (SOCCP) and the semidefinite complementarity problem (SDCP) as special cases. In this paper, we present a regularized smoothing Newton algorithm for SCCP by making use of Euclidean Jordan algebraic technique. Under suitable conditions, we obtain global convergence and local quadratic convergence of the proposed algorithm. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.