Search

In this paper, an efficient modified Newton-type algorithm is proposed for nonlinear unconstrained optimization problems. The modified Hessian is a convex combination of the identity matrix (for steepest descent algorithm) and the Hessian matrix (for Newton algorithm). The coefficients of the convex combination are dynamically chosen in every iteration. The algorithm is proved to be globally and quadratically convergent for (convex and non-convex) nonlinear functions. Efficient implementation is described. Numerical test on widely used CUTEr test problems is conducted for the new algorithm. The test results are compared with those obtained by MATLAB optimization toolbox function fminunc. The test results are also compared with those obtained by some established and state $$-$$ of-the $$-$$ art algorithms, such as a limited memory BFGS, a descent and conjugate gradient algorithm, and a limited memory and descent conjugate gradient algorithm. The comparisons show that the new algorithm is promising. [ABSTRACT FROM AUTHOR]

In this paper, we reformulate the variational inequality problem as an equivalent smooth non-linear equation system by introducing the Chen-Harker-Kanzow-Smale smoothing function. A new smoothing inexact Newton algorithm is proposed to solve the smooth equations. In each iteration, the corresponding linear system is solved approximately. We prove that the proposed algorithm converges globally and superlinearly under mild conditions. Preliminary numerical results indicate that the method is effective. [ABSTRACT FROM AUTHOR]

In this paper, we present a flexible nonmonotone filter method for solving nonlinear constrained optimization problems which are common models in industry. This new method has more flexibility for the acceptance of the trial step compared to the traditional filter methods, and requires less computational costs compared with the monotone-type methods. Moreover, we use a self-adaptive parameter to adjust the acceptance criteria, so that Maratos effect can be avoided a certain degree. Under reasonable assumptions, the proposed algorithm is globally convergent. Numerical tests are presented that confirm the efficiency of the approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we try to solve the semidefinite program with box constraints. Since the traditional projection method for constrained optimization with box constraints is not suitable to the semidefinite constraints, we present a new algorithm based on the feasible direction method. In the paper, we discuss two cases: the objective function in semidefinite programming is linear and nonlinear, respectively. We establish the convergence of our algorithm, and report the numerical experiments which show the effectiveness of the algorithm. [Copyright &y& Elsevier]

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we combine the conjugate gradient method with the Barzilai and Borwein gradient method, and propose a Barzilai and Borwein scaling conjugate gradient method for nonlinear unconstrained optimization problems. The new method does not require to compute and store matrices associated with Hessian of the objective functions, and has an advantage of less computational efforts. Moreover, the descent direction property and the global convergence are established when the line search fulfills the Wolfe conditions. The limited numerical experiments and comparisons show that the proposed algorithm is potentially efficient. [ABSTRACT FROM AUTHOR]

In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly convergent algorithm is proposed. The initial iteration point can be chosen arbitrarily for the algorithm. At each iteration, the new algorithm solves one quadratic programming subproblem which is always feasible, and one or two systems of linear equations with a common coefficient matrix. Moreover, the coefficient matrix is uniformly nonsingular. After finite iterations, the iteration points can always enter the feasible set of the problem, and the search direction is obtained by solving one quadratic programming subproblem and only one system of linear equations. The new algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some numerical results are reported to show that the algorithm is promising. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new penalty-free-type method for nonlinear equality constrained problems. The new algorithm uses trust region framework and feasibility safeguarding technique. Moreover, it has no choice of penalty parameter and penalty function as a merit function, and it does not use the filter technique to avoid the penalty function either. We analyze the global convergence of the main algorithm under the standard assumptions. The preliminary numerical tests are reported. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate a class of nonlinear complementarity problems arising from the discretization of the free boundary problem, which was recently studied by Sun and Zeng [Z. Sun, J. Zeng, A monotone semismooth Newton type method for a class of complementarity problems, J. Comput. Appl. Math. 235 (5) (2011) 1261–1274]. We propose a new non-interior continuation algorithm for solving this class of problems, where the full-Newton step is used in each iteration. We show that the algorithm is globally convergent, where the iteration sequence of the variable converges monotonically. We also prove that the algorithm is globally linearly and locally superlinearly convergent without any additional assumption, and locally quadratically convergent under suitable assumptions. The preliminary numerical results demonstrate the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, a new feasible primal–dual interior point algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only two or three reduced systems of linear equations with the same coefficient matrix. The searching direction is feasible and the object function is monotone decreasing. The proposed algorithm is proved to possess global and superlinear convergence under mild conditions. Finally, some numerical experiments with the algorithm are reported. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a one-step smoothing Newton method for the solution of nonlinear complementarity problems with -functions. The proposed algorithm is based on the smoothing symmetric perturbed Fischer function, and it is proven to generate bounded iteration sequence and to possess strongly global and local convergence properties under weaker conditions. Preliminary numerical results indicate that the proposed algorithm is promising. [Copyright &y& Elsevier]

The paper explores a trust-region active-set algorithm for general nonlinear optimization with nonlinear equality and inequality constraints. In this algorithm, an active-set strategy is used together with trust-region methods to compute the trial step. penalty functions are employed to obtain the global convergence. The global convergence of this algorithm is proved under standard conditions. The numerical tests show the efficiency of the proposed algorithm. [Copyright &y& Elsevier]

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well. [ABSTRACT FROM AUTHOR]

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex (or even uniformly convex). We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems. [ABSTRACT FROM AUTHOR]