Derivative-free optimization and filter methods to solve nonlinear constrained problems

In real optimization problems, usually the analytical expression of the objective function is not known, nor its derivatives, or they are complex. In these cases it becomes essential to use optimization methods where the calculation of the derivatives, or the verification of their existence, is not necessary: the Direct Search Methods or Derivative-free Methods are one solution. When the problem has constraints, penalty functions are often used. Unfortunately the choice of the penalty parameters is, frequently, very difficult, because most strategies for choosing it are heuristics strategies. As an alternative to penalty function appeared the filter methods. A filter algorithm introduces a function that aggregates the constrained violations and constructs a biobjective problem. In this problem the step is accepted if it either reduces the objective function or the constrained violation. This implies that the filter methods are less parameter dependent than a penalty function. In this work, we present a new direct search method, based on simplex methods, for general constrained optimization that combines the features of the simplex method and filter methods. This method does not compute or approximate any derivatives, penalty constants or Lagrange multipliers. The basic idea of simplex filter algorithm is to construct an initial simplex and use the simplex to drive the search. We illustrate the behavior of our algorithm through some examples. The proposed methods were implemented in Java.

Multiple Roots of Systems of Equations by Repulsion Merit Functions

In this paper we address the problem of computing multiple roots of a system of nonlinear equations through the global optimization of an appropriate merit function. The search procedure for a global minimizer of the merit function is carried out by a metaheuristic, known as harmony search, which does not require any derivative information. The multiple roots of the system are sequentially determined along several iterations of a single run, where the merit function is accordingly modified by penalty terms that aim to create repulsion areas around previously computed minimizers. A repulsion algorithm based on a multiplicative kind penalty function is proposed. Preliminary numerical experiments with a benchmark set of problems show the effectiveness of the proposed method.

Activity-Dependent Neurorehabilitation Beyond Physical Trainings: "Mental Exercise" Through Mirror Neuron Activation

The activity dependent brain repair mechanism has been widely adopted in many types of neurorehabilitation. The activity leads to target specific and non-specific beneficial effects in different brain regions, such as the releasing of neurotrophic factors, modulation of the cytokines and generation of new neurons in adult hood. However physical exercise program clinically are limited to some of the patients with preserved motor functions; while many patients suffered from paralysis cannot make such efforts. Here the authors proposed the employment of mirror neurons system in promoting brain rehabilitation by "observation based stimulation". Mirror neuron system has been considered as an important basis for action understanding and learning by mimicking others. During the action observation, mirror neuron system mediated the direct activation of the same group of motor neurons that are responsible for the observed action. The effect is clear, direct, specific and evolutionarily conserved. Moreover, recent evidences hinted for the beneficial effects on stroke patients after mirror neuron system activation therapy. Finally some music-relevant therapies were proposed to be related with mirror neuron system.