Raimundo J. B. de Sampaio

An adaptive approach of conic trust-region method for unconstrained optimization problems

In this paper, an adaptive trust region method based on the conic model for unconstrained optimization problems is proposed and analyzed. We establish the global and superlinear convergence results of the method. Numerical tests are reported that confirm the efficiency of the new method.

An adaptive conic trust-region method for unconstrained optimization

It is well known that among the current methods for unconstrained optimization problems the quasi-Newton methods with global strategy may be the most efficient methods, which have local superlinear convergence. However, when the iterative point is far away from the solution of the problem, quasi-Newton method may proceed slowly for the general unconstrained optimization problems. In this article an adaptive conic trust-region method for unconstrained optimization is presented. Not only the gradient information but also the values of the objective function are used to construct the local model at the current iterative point. Moreover, we define a concept of super steepest descent direction and embed its information into the local model. The amount of computation in each iteration of this adaptive algorithm is the same as that of the standard quasi-Newton method with trust region. Some numerical results show that the modified method requires fewer iterations than the standard methods to reach the solution of the optimization problem. Global and local convergence of the method is also analyzed.

A Barzilai and Borwein scaling conjugate gradient method for unconstrained optimization problems

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.

Numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters

In this paper, a numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters is presented. We compare the numerical efficiency of two classes of nonmonotone trust region (NTR) algorithms in the context of unconstrained optimization. We examine the sensitivity of the algorithms to the parameters related to the nonmonotone technique and the initial trust region radius. We show that the numerical efficiency of nonmonotone trust region algorithms can be improved by choosing appropriate parameters. Based on extensive numerical tests, some efficient ranges of these parameters for nonmonotone trust region algorithms are recommended.

Using Penalty in Mathematical Decomposition for Production-Planning to Accommodate Clearing Function Constraints of Capacity

The idea of using clearing functions in Linear Programming models for production planning to represent the nonlinear dependence between workload and lead times in productive systems may result in a large nonlinear convex model. Nevertheless, this convex programming model is not considered directly, but approximated by using linear programming models which sometimes results in a large linear programming problem, requiring mathematical decomposition for efficient solution. The classic method of decomposition, however, does not function properly in the presence of the restriction of capacity provided by the clearing function, frustrating the efforts to introduce lead times into the models. In this chapter we provide a strategy to modify the classical decomposition approach using a penalty function for the subproblems, which circumvents the pointed drawbacks.

Generalization: One technique of computational and applied mathematical methodology ☆

In many researches and applications in applied mathematics and engineering, we need to generalize the existing results, algorithms or methods in order to apply them to different situations or considerations. Generalization will allow certain particular problems to be solved more efficiently. Here we use some examples in scientific computing to demonstrate the importance of generalization technique for some researches, and how to generalize results or to improve conditions.