SMOOTHING NEWTON ALGORITHM FOR THE CIRCULAR CONE PROGRAMMING WITH A NONMONOTONE LINE SEARCH

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming(CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone(SOC), we reformulate the CCP problem as the second-order cone problem(SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.

Smoothing Newton-Like Method for the Solution of Nonlinear Systems of Equalities and Inequalities

In this paper,we present a smoothing Newton-like method for solving nonlinear systems of equalities and inequalities.By using the so-called max function,we transfer the inequalities into a system of semismooth equalities.Then a smoothing Newton-like method is proposed for solving the reformulated system,which only needs to solve one system of linear equations and to perform one line search at each iteration. The global and local quadratic convergence are studied under appropriate assumptions. Numerical examples show that the new approach is effective.

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

A GLOBAL LINEAR AND LOCAL QUADRATIC SINGLE-STEP NONINTERIOR CONTINUATION METHOD FOR MONOTONE SEMIDEFINITE COMPLEMENTARITY PROBLEMS

A noninterior continuation method is proposed for semidefinite complementarity problem （SDCP）. This method improves the noninterior continuation methods recently developed for SDCP by Chen and Tseng. The main properties of our method are： （i） it is well d.efined for the monotones SDCP; （ii） it has to solve just one linear system of equations at each step; （iii） it is shown to be both globally linearly convergent and locally quadratically convergent under suitable assumptions.

Smoothing Inexact Newton Method for Solving P_0-NCP Problems

Based on a smoothing symmetric disturbance FB-function,a smoothing inexact Newton method for solving the nonlinear complementarity problem with P0-function was proposed.It was proved that under mild conditions,the given algorithm performed global and superlinear convergence without strict complementarity.For the same linear complementarity problem(LCP),the algorithm needs similar iteration times to the literature.However,its accuracy is improved by at least 4 orders with calculation time reduced by almost 50%,and the iterative number is insensitive to the size of the LCP.Moreover,fewer iterations and shorter time are required for solving the problem by using inexact Newton methods for different initial points.

Non-interior Continuation Algorithm for Solving System of Inequalities over Symmetric Cones

As a basic mathematical structure,the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems.In this paper,a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone.It is shown that the proposed algorithm is globally convergent and well-defined.Moreover,it can start from any point and only needs to solve one system of linear equations at most at each iteration.Under suitable assumptions,global linear and local quadratic convergence is established with Euclidean Jordan algebras.Numerical results indicate that the algorithm is efficient.The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100,,1 000 respectively and the problems of each size were generated randomly for 10 times.The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations.It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the secondorder cones quickly.Moreover,a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones,and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.

GLOBAL LINEAR AND QUADRATIC ONE-STEP SMOOTHING NEWTON METHOD FOR VERTICAL LINEAR COMPLEMENTARITY PROBLEMS

A one_step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so_called aggregation function. The proposed algorithm has the following good features: (ⅰ) It solves only one linear system of equations and does only one line search at each iteration; (ⅱ) It is well_defined for the vertical linear complementarity problem with vertical block P 0 matrix and any accumulation point of iteration sequence is its solution.Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P 0+R 0 matrix; (ⅲ) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (ⅲ).

A HYBRID METHOD FOR SOLVING VARIATIONAL INEQUALITY PROBLEMS

By using Fukushima's differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993.In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix Δ F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty,polyhedral,closed and convex is proposed.Armijo type line search and trust region strategies as well as Fukushima's differentiable merit function are incorporated into the method.It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al.,the method reduces to the basic Newton method and hence the rate of convergence is quadratic.Computational experiences show the efficiency of the proposed method.

A SQP METHOD FOR GENERAL NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper,the nonlinear complementarity problem is transformed into the least squares problem with nonnegative constraints,and a SQP algorithm for this reformulation based on a damped Gauss Newton type method is presented.It is shown that the algorithm is globally and locally superlinearly (quadratically) convergent without the assumption of monotonicity.

WONG-ZAKAI APPROXIMATIONS FOR STOCHASTIC VOLTERRA EQUATIONS

In this paper,we shall prove a Wong-Zakai approximation for stochastic Volterra equations under appropriate assumptions.We may apply it to a class of stochastic differential equations with the kernel of fractional Brownian motion with Hurst parameter H∈(1/2,1)and subfractional Brownian motion with Hurst parameter H∈(1/2,1).As far as we know,this is the first result on stochastic Volterra equations in this topic.

Convergence of a Non-interior Continuation Algorithm for the Monotone SCCP

It is well known that the symmetric cone complementarity problem（SCCP） is a broad class of optimization problems which contains many optimization problems as special cases.Based on a general smoothing function,we propose in this paper a non-interior continuation algorithm for solving the monotone SCCP.The proposed algorithm solves at most one system of linear equations at each iteration.By using the theory of Euclidean Jordan algebras,we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions.

A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS

Based on the work of paper [1], we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations F(x) = 0, where F(x) : Rn - Rn is continuously differentiable and F'(x) is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense, and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if ||F(x)||2 provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.

Superlinear/Quadratic One-step Smoothing Newton Method for P_0-NCP

We propose a one–step smoothing Newton method for solving the non-linearcomplementarity problem with P 0–function (P_0–NCP) based on the smoothing symmetric perturbedFisher function (for short, denoted as the SSPF–function). The proposed algorithm has to solve onlyone linear system of equations and performs only one line search per iteration. Without requiringany strict complementarity assumption at the P_0–NCP solution, we show that the proposed algorithmconverges globally and superlinearly under mild conditions. Furthermore, the algorithm has localquadratic convergence under suitable conditions. The main feature of our global convergence resultsis that we do not assume a priori the existence of an accumulation point. Compared to the previousliteratures, our algorithm has stronger convergence results under weaker conditions.

Neighborhood-following algorithms for linear programming

In this paper, we present neighborhood-following algorithms for linear programming. When the neighborhood is a wide neighborhood, our algorithms are wide neighborhood primal-dual interior point algorithms. If the neighborhood degenerates into the central path, our algorithms also degenerate into path-following algorithms. We prove that our algorithms maintain the O9√nL) -iteration complexity still, while the classical wide neighborhood primal-dual interior point algorithms have only the O(nL-iteration complexity. We also proved that the algorithms are quadratic convergence if the optimal vertex is nondegenerate. Finally, we show some computational results of our algorithms.

A TRUST REGION-TYPE METHOD FOR SOLVINGMONOTONE VARIATIONAL INEQUALITY

Presents a study which proposed to introduce a trust region-type modification of Newton method for the monotone inequality problem using merit function. Concepts of monotone mapping; Proof of convergence of algorithm variational inequality trust region; Results.

A TRUST-REGION ALGORITHM FOR NONLINEAR INEQUALITY CONSTRAINED OPTIMIZATION

This paper presents a new trust-region algorithm for n-dimension nonlinear optimization subject to m nonlinear inequality constraints. Equivalent KKT conditions are derived, which is the basis for constructing the new algorithm. Global convergence of the algorithm to a first-order KKT point is established under mild conditions on the trial steps, local quadratic convergence theorem is proved for nondegenerate minimizer point. Numerical experiment is presented to show the effectiveness of our approach.

Predictor-Corrector Smoothing Methods for Monotone LCP

In this paper, we analyze the global and local convergence properties of two predictor-corrector smoothing methods, which are based on the framework of the method in [1], for monotone linear complementarity problems (LCPs). The difference between the algorithm in [1] and our algorithms is that the neighborhood of smoothing central path in our paper is different to that in [1]. In addition, the difference between Algorithm 2.1 and the algorithm in [1] exists in the calculation of the predictor step. Comparing with the results in [1], the global and local convergence of the two methods can be obtained under very mild conditions. The global convergence of the two methods do not need the boundness of the inverse of the Jacobian. The superlinear convergence of Algorithm 2.1′ is obtained under the assumption of nonsingularity of generalized Jacobian of φ(x, y) at the limit point and Algorithm 2.1 obtains superlinear convergence under the assumption of strict complementarity at the solution. The effciency of the two methods is tested by numerical experiments.

AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.

GENERAL CENTRAL PATH AND THE LARGEST STEP GENERAL CENTRAL PATH FOLLOWING ALGORITHM FOR LINEAR PROGRAMMING

In this paper, we propose a general path following method, in which the starting point can be any feasible interior pair and each iteration uses a step with the largest possible reduction in duality gap. The algorithm maintains the O (nL) ineration complexity It enjoys quadratic convergence if the optimal vertex is nondegenerate.

AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

Presents information on a study which analyzed an interior trust-region-based algorithm for linearly constrained minimization problems. Optimality conditions for the linearly constrained minimization problem presented; Vectors for each updating step in the algorithm proposed; Establishment of the convergence properties of the proposed algorithm.