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**1 - 2**of**2**### Direct Search Based on Probabilistic Descent

, 2015

"... Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning s ..."

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Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional variable space. In addition, to ensure the global convergence of these algorithms, the positive spanning sets used throughout the iterations are required to be uniformly non-degenerate in the sense of having a positive (cosine) measure bounded away from zero. However, recent numerical results indicated that randomly generating the polling direc-tions without imposing the positive spanning property can improve the performance of these methods, especially when the number of directions is chosen considerably less than n+ 1. In this paper, we analyze direct-search algorithms when the polling directions are prob-abilistic descent, meaning that with a certain probability at least one of them is of descent type. Such a framework enjoys almost-sure global convergence. More interestingly, we will show a global decaying rate of 1/ k for the gradient size, with overwhelmingly high proba-bility, matching the corresponding rate for the deterministic versions of the gradient method or of direct search. Our analysis helps to understand numerical behavior and the choice of the number of polling directions.

### A Parallel Evolution Strategy for an Earth Imaging Problem in Geophysics

, 2015

"... In this paper we propose a new way to compute a rough approximation solution, to be later used as a warm starting point in a more refined optimization process, for a challenging global optimization problem related to Earth imaging in geophysics. The warm start con-sists of a velocity model that appr ..."

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In this paper we propose a new way to compute a rough approximation solution, to be later used as a warm starting point in a more refined optimization process, for a challenging global optimization problem related to Earth imaging in geophysics. The warm start con-sists of a velocity model that approximately solves a full-waveform inverse problem at low frequency. Our motivation arises from the availability of massively parallel computing plat-forms and the natural parallelization of evolution strategies as global optimization methods for continuous variables. Our first contribution consists of developing a new and efficient parametrization of the velocity models to significantly reduce the dimension of the original optimization space. Our second contribution is to adapt a class of evolution strategies to the specificity of the physical problem at hands where the objective function evaluation is known to be the most expen-sive computational part. A third contribution is the development of a parallel evolution strategy solver, taking advantage of a recently proposed modification of these class of evolu-tionary methods that ensures convergence and promotes better performance under moderate