Local-meta-model CMA-ES for partially separable functions

Abstract

In this paper, we propose a new variant of the covariance matrix adaptation evolution strategy with local meta-models (lmm-CMA) for optimizing partially separable functions. We propose to exploit partial separability by building at each iteration a meta-model for each element function (or sub-function) using a full quadratic local model. After introducing the approach we present some first experiments using element functions with dimensions 2 and 4. Our results demonstrate that, as expected, exploiting partial separability leads to an important speedup compared to the standard CMA-ES. We show on the tested functions that the speedup increases with increasing dimensions for a fixed dimension of the element function. On the standard Rosenbrock function the maximum speedup of λ is reached in dimension 40 using element functions of dimension 2. We show also that higher speedups can be achieved by increasing the population size. The choice of the number of points used to build the meta-model is also described and the computational cost is discussed.

Well placement optimization with the covariance matrix adaptation evolution strategy and meta-models

“…The GA leads to well configurations dispersed over a large zone. The maximum value of NPV obtained by the GA is equal to $1.86 × 10 10 , and it corresponds to a well configuration close to a well configuration obtained by CMA-ES with an NPV $2.05 × 10 10 .…”

“…We believe, however, that performance over CMA-ES could be more significantly improved by exploiting, within the algorithm, knowledge, and relevant information about the optimization problem at hand, such as the problem structure. Some first steps in that direction have been conduced in [10,11] where the fact that the objective function can be split into local components referring to each of the wells where each depends on a smaller number of parameters (i.e., partial separability of the objective function) is exploited. Another approach could be to exploit some a priori information such as well allocation factors and connectivity using the work developed in [15].…”

“…CMA-ES succeeds in defining in 11 runs of the 14 performed the same potential zone to place the producer and the injector. This region gives an NPV between $1.99 × 10 10 and $2.05 × 10 10 Despite the large number of local optima, CMA-ES succeeds in providing satisfactory results on 93% of the runs, if we consider that a run is satisfactory if it gives an NPV greater or equal to $1.95 × 10 10 .…”

Reevaluating Exponential Crossover in Differential Evolution

“…The Rosenbrock and Schwefels 1.2 functions are well-known, classical benchmark functions that have been widely used to evaluate EA's [22]. The Block-rotated Ellipsoid [1] is a partially separable function designed to only have dependencies between z i and , and for distributed functions, P is a randomly generated ordering such as P = (6, 1, 3, ...). The 2 × 2 rotation matrix R i is uniformly generated according to the method of [15] and α = 1e+6.…”

Two Gaussian Approaches to Black-Box Optomization

“…This is actually a specific consequence of another possibility of using information from CMA-ES for the GPreplacing the original space of d-dimensional vectors, R d , by the space of their principal components with respect to C (g) . In this context, it is worth recalling that in [2,3,15], Mahalanobis instead of Euclidean distance was used when combining CMA-ES with quadratic response surface models in [2,3,15]. In the approach presented there, the space of the principal components with respect to C (g) is used, together with an estimate of density, to locally weight the model predictions with respect to the considered input.…”