Elias Salomao Helou

Técnicas de regularização são ferramentas importantes para a solução numérica de problemas mal-postos (no sentido de Hadamard). O presente trabalho apresenta um método para a seleção do parâmetro de regularização através da minimização de um estimador não tendencioso para o erro preditivo do regularizador. A técnica proposta generaliza métodos existentes através da possibilidade da utilização de preditores mais gerais baseados em divergências de Bregman. O método é aplicável a uma ampla gama de técnicas de regularização lineares ou não-lineares e a diversos modelos estocásticos para o ruído.

Explicitly nonsmooth reformulations of mathematical programs with equilibrium constraints (MPEC) described by smooth functions are investigated. A sequential optimality condition ($\epsilon $-ANOC) previously developed for nonsmooth optimization problems is adapted to suit the reformulations under analysis. Relationships between $\epsilon $-ANOC and sequential optimality conditions especially crafted for MPEC are established. It is revealed that approaches developed for nonsmooth optimization improve the understanding of smooth problems with ill-conditioned geometries, which is the case for MPEC. Moreover, based on the availability of algorithms capable of generating $\epsilon $-ANOC sequences, illustrative experiments are reported in which difficult problems are successfully solved.

A stochastic incremental subgradient algorithm for the minimization of a sum of convex functions is introduced. The method sequentially uses partial subgradient information and the sequence of partial subgradients is determined by a general Markov chain. This makes it suitable to be used in networks where the path of information flow is stochastically selected. We prove convergence of the algorithm to a weighted objective function where the weights are given by the Cesro limiting probablity distribution of the Markov chain. Unlike previous works in the literature, the Cesro limiting distribution is general (not necessarily uniform), allowing for general weighted objective functions and flexibility in the method.

Unbiased estimators are introduced for averaged Bregman divergences which generalize Stein's Unbiased (Predictive) Risk Estimator, and the minimization of these estimators is proposed as a regularization parameter selection method for regularization of inverse problems. Numerical experiments are presented in order to show the performance of the proposed technique. Experimental results indicate a useful occurence of a concentration of measure phenomena and some implications of this hypothesis are analyzed.

The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done by inserting target-function-reducing perturbations into a feasibility-seeking algorithm while retaining its feasibility-seeking ability and without paying a high computational price. A superiorized algorithm that employs component-wise target function reduction steps is presented. This enables derivative-free superiorization (DFS), meaning that superiorization can be applied to target functions that have no calculable partial derivatives or subgradients. The numerical behavior of our derivative-free superiorization algorithm is illustrated on a data set generated by simulating a problem of image reconstruction from projections. We present a tool (we call it a proximity-target curve) for deciding which of two iterative methods is "better" for solving a particular problem. The plots of proximity-target curves of our experiments demonstrate the advantage of the proposed derivative-free superiorization algorithm.