Youssef Diouane

Globally convergent evolution strategies

In this paper we show how to modify a large class of evolution strategies (ES’s) for unconstrained optimization to rigorously achieve a form of global convergence, meaning convergence to stationary points independently of the starting point. The type of ES under consideration recombines the parent points by means of a weighted sum, around which the offspring points are computed by random generation. One relevant instance of such an ES is covariance matrix adaptation ES (CMA-ES). The modifications consist essentially of the reduction of the size of the steps whenever a sufficient decrease condition on the function values is not verified. When such a condition is satisfied, the step size can be reset to the step size maintained by the ES’s themselves, as long as this latter one is sufficiently large. We suggest a number of ways of imposing sufficient decrease for which global convergence holds under reasonable assumptions (in particular density of certain limit directions in the unit sphere). Given a limited budget of function evaluations, our numerical experiments have shown that the modified CMA-ES is capable of further progress in function values. Moreover, we have observed that such an improvement in efficiency comes without weakening significantly the performance of the underlying method in the presence of several local minimizers.

On the use of the energy norm in trust-region and adaptive cubic regularization subproblems

We consider solving unconstrained optimization problems by means of two popular globalization techniques: trust-region (TR) algorithms and adaptive regularized framework using cubics (ARC). Both techniques require the solution of a so-called “subproblem” in which a trial step is computed by solving an optimization problem involving an approximation of the objective function, called “the model”. The latter is supposed to be adequate in a neighborhood of the current iterate. In this paper, we address an important practical question related with the choice of the norm for defining the neighborhood. More precisely, assuming here that the Hessian B of the model is symmetric positive definite, we propose the use of the so-called “energy norm”—defined by

A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis

\Vert x\Vert _B= \sqrt{x^TBx}

A Parallel Evolution Strategy for Acoustic Full-Waveform Inversion

‖x‖B=xTBx for all

A parallel evolution strategy for an earth imaging problem in geophysics

x \in \mathbb {R}^n

A Stochastic Levenberg-Marquardt Method Using Random Models with Application to Data Assimilation.

x∈Rn—in both TR and ARC techniques. We show that the use of this norm induces remarkable relations between the trial step of both methods that can be used to obtain efficient practical algorithms. We furthermore consider the use of truncated Krylov subspace methods to obtain an approximate trial step for large scale optimization. Within the energy norm, we obtain line search algorithms along the Newton direction, with a special backtracking strategy and an acceptability condition in the spirit of TR/ARC methods. The new line search algorithm, derived by ARC, enjoys a worst-case iteration complexity of