unconstrained optimization?

This paper proposes two effective nonmonotone trust-region frameworks for solving nonlinear unconstrained optimization problems while provide a new effective policy to update the trust-region radius. Conventional nonmonotone trust-region algorithms apply a specific nonmonotone ratio to accept  new trial step and update the trust-region radius. This paper recommends using the nonmonotone ratio only as an acceptance criterion for a new trial step. In contrast, the monotone ratio or a hybrid of monotone and nonmonotone ratios is proposed as a criterion for updating the trust-region radius. We investigate the global convergence to first- and second-order stationary points for the proposed approaches under certain classical assumptions.  Initial numerical results indicate that the proposed methods significantly enhance the performance of nonmonotone trust-region methods.

In an effort to make modification on the classical Hestenes--Stiefel method, Shengwei et al. proposed an efficient conjugate gradient method which possesses the sufficient descent condition when the line search fulfills the strong Wolfe conditions (by restricting the line search parameters). Here, we develop a three--term extension of the method which guarantees the sufficient descent condition independent to the line search. Also, we establish global convergence of the method using convexity assumption. At last, practical merits of the proposed method are investigated by numerical experiments on a set of CUTEr test functions. The results show numerical efficiency of the method.

In this paper, we solve unconstrained optimization problem using a free line search steepest descent method. First, we propose a double parameter scaled quasi Newton formula for calculating an approximation of the Hessian matrix. The approximation obtained from this formula is a positive definite matrix that is satisfied in the standard secant relation. We also show that the largest eigen value of this matrix is not greater than the number of variables of the problem. Then, using this double parameter scaled quasi Newton formula, an explicit formula for calculating the step length in the steepest descent method is presented and therefore, this method does not require the use of approximate methods for calculating step length. The numerical results obtained from the implementation of the algorithm in MATLAB software environment are presented for some optimization problems. These results show the efficiency of the proposed method in comparison with other existing methods.

Using search directions of a recent class of three--term conjugate gradient methods, modified versions of the Hestenes-Stiefel and Polak-Ribiere-Polyak methods are proposed which satisfy the sufficient descent condition. The methods are shown to be globally convergent when the line search fulfills the (strong) Wolfe conditions. Numerical experiments are done on a set of CUTEr unconstrained optimization test problems. They demonstrate efficiency of the proposed methods in the sense of the Dolan-More performance profile.

Nonlinear conjugate gradient method is well known in solving large-scale unconstrained optimization problems due to it’s low storage requirement and simple to implement. Research activities on it’s application to handle higher dimensional systems of nonlinear equations are just beginning. This paper presents a Threeterm Conjugate Gradient algorithm for solving Large-Scale systems of nonlinear equations by incoporating the hyperplane projection and Powel restart approach. We prove the global convergence of the proposed method with a derivative free line search under suitable assumtions. the numerical results are presented which show that the proposed method is promising.

This study concerns with a trust-region-based method for solving unconstrained optimization problems. The approach takes the advantages of the compact limited memory BFGS updating formula together with an appropriate adaptive radius strategy. In our approach, the adaptive technique leads us to decrease the number of subproblems solving, while utilizing the structure of limited memory quasi-Newton formulas helps to handle large-scale problems. Theoretical analysis indicates that the new approach preserves the global convergence to a first-order stationary point under classical assumptions. Moreover, the superlinear and the quadratic convergence rates are also established under suitable conditions. Preliminary numerical experiments show the effectiveness of the proposed approach for solving large-scale unconstrained optimization problems.

Based on an eigenvalue analysis, a new proof for the sufficient descent property of the modified Polak-Ribière-Polyak conjugate gradient method proposed by Yu et al. is presented.