Sanyang Liu

A smoothing trust-region Newton-CG method for minimax problem

This paper presents a smooth approximate method with a new smoothing technique and a standard unconstrained minimization algorithm in the solution to the finite minimax problems. The new smooth approximations only replace the original problem in some neighborhoods of the kink points with a twice continuously differentiable function, its gradient and Hessian matrix are the combination of the first and the second order derivative of the original functions respectively. Compared to the other smooth functions such as the exponential penalty function, the remarkable advantage of the new smooth function is that the combination coefficients of its gradient and the Hessian matrix have sparse properties. Furthermore, the maximal possible difference value between the optimal values of the smooth approximate problem and the original one is determined by a fixed parameter selected previous. An algorithm to solve the equivalent unconstrained problem by using the trust-region Newton conjugate gradient method is proposed in the solution process. Finally, some numerical examples are reported to compare the proposed algorithm with SQP algorithm that implements in MATLAB toolbox and the algorithm in [E. Polak, J.O. Royset, R.S. Womersley, Algorithms with adaptive smoothing for finite minimax problems, Journal of Optimization Theory and Applications 119 (3) (2003) 459–484] based on the exponential penalty function, the numerical results prove that the proved algorithm is efficient. 2008 Elsevier Inc. All rights reserved.

Integral average method for oscillation of second order partial differential equations with delays

Using a generalized Riccati transformation, some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average. These results can be considered as generalizations and improvements of the results due to Kamenev in ordinary differential equations cases.

Traveling waves for delayed non-local diffusion equations with crossing-monostability

Abstract This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ ⩾ 0 .

One-step smoothing Newton method for solving the mixed complementarity problem with a P0 function

The mixed complementarity problem (denote by MCP(F)) can be reformulated as the solution of a smooth system of equations. In the paper, based on a perturbed mid function, we propose a new smoothing function, which has an important property, not satisfied by many other smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P0 function are discussed. Then we presented a one-step smoothing Newton algorithm to solve the MCP with a P0 function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the rate of convergence of the method is proved under some suitable assumptions.

A new modified one-step smoothing Newton method for solving the general mixed complementarity problem

In last decades, there has been much effort on the solution and the analysis of the mixed complementarity problem (MCP) by reformulating MCP as an unconstrained minimization involving an MCP function. In this paper, we propose a new modified one-step smoothing Newton method for solving general (not necessarily P0) mixed complementarity problems based on well-known Chen-Harker-Kanzow-Smale smooth function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established.

A new branch and bound method with pretreatment for the binary quadratic programming

A new branch and bound algorithm with pretreatment for the binary quadratic programming is presented in this paper. Firstly, we use pretreatment method to decrease the size of the binary quadratic programming. Then, based on the pretreatment method, A new branch and bound algorithm is proposed, which give the new method for the initial solution, the new bounding method, and the new pruning regulation. Numerical experiments demonstrate the algorithm is simple, fast, and efficient.

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.

On positive definite solutions of nonlinear matrix equation Xs-A*X -t A=Q

In this paper, the nonlinear matrix equation Xs-A∗X-tA=QXs-A∗X-tA=Q is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given ‖A‖·‖Q-1‖s+ts<st. In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.

On positive definite solutions of nonlinear matrix equation Xs-A∗X-tA=QXs-A∗X-tA=Q ☆

In this paper, the nonlinear matrix equation Xs-A∗X-tA=QXs-A∗X-tA=Q is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given ‖A‖·‖Q-1‖s+ts<st. In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates.