arXiv subjects
Publications and source records attributed to Anne Auger.
We present a budget-free experimental setup and procedure for benchmarking numericaloptimization algorithms in a black-box scenario. This procedure can be applied with the COCO benchmarking platform. We describe initialization of and input to the algorithm and touch upon therelevance of termination and restarts.
We present an any-time performance assessment for benchmarking numerical optimization algorithms in a black-box scenario, applied within the COCO benchmarking platform. The performance assessment is based on runtimes measured in number of objective function evaluations to reach one or several quality indicator target values. We argue that runtime is the only available measure with a generic, meaningful, and quantitative interpretation. We discuss the choice of the target values, runlength-based targets, and the aggregation of results by using simulated restarts, averages, and empirical distribution functions.
This document details the rationales behind assessing the performance of numerical black-box optimizers on multi-objective problems within the COCO platform and in particular on the biobjective test suite bbob-biobj. The evaluation is based on a hypervolume of all non-dominated solutions in the archive of candidate solutions and measures the runtime until the hypervolume value succeeds prescribed target values.
This paper analyzes a (1, $λ$)-Evolution Strategy, a randomized comparison-based adaptive search algorithm, optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using cumulative step-size adaptation. We exhibit for each case a Markov chain describing the behaviour of the algorithm. Stability of the chain implies, by applying a law of large numbers, either convergence or divergence of the algorithm. Divergence is the desired behaviour. In the constant step-size case, we show stability of the Markov chain and prove the divergence of the algorithm. In the cumulative step-size adaptation case, we prove stability of the Markov chain in the simplified case where the cumulation parameter equals 1, and discuss steps to obtain similar results for the full (default) algorithm where the cumulation parameter is smaller than 1. The stability of the Markov chain allows us to deduce geometric divergence or convergence , depending on the dimension, constraint angle, population size and damping parameter, at a rate that we estimate. Our results complement previous studies where stability was assumed.
This paper analyses a $(1,λ)$-Evolution Strategy, a randomised comparison-based adaptive search algorithm, on a simple constraint optimisation problem. The algorithm uses resampling to handle the constraint and optimizes a linear function with a linear constraint. Two cases are investigated: first the case where the step-size is constant, and second the case where the step-size is adapted using path length control. We exhibit for each case a Markov chain whose stability analysis would allow us to deduce the divergence of the algorithm depending on its internal parameters. We show divergence at a constant rate when the step-size is constant. We sketch that with step-size adaptation geometric divergence takes place. Our results complement previous studies where stability was assumed.
In the context of unconstraint numerical optimization, this paper investigates the global linear convergence of a simple probabilistic derivative-free optimization algorithm (DFO). The algorithm samples a candidate solution from a standard multivariate normal distribution scaled by a step-size and centered in the current solution. This solution is accepted if it has a better objective function value than the current one. Crucial to the algorithm is the adaptation of the step-size that is done in order to maintain a certain probability of success. The algorithm, already proposed in the 60's, is a generalization of the well-known Rechenberg's $(1+1)$ Evolution Strategy (ES) with one-fifth success rule which was also proposed by Devroye under the name compound random search or by Schumer and Steiglitz under the name step-size adaptive random search. In addition to be derivative-free, the algorithm is function-value-free: it exploits the objective function only through comparisons. It belongs to the class of comparison-based step-size adaptive randomized search (CB-SARS). For the convergence analysis, we follow the methodology developed in a companion paper for investigating linear convergence of CB-SARS: by exploiting invariance properties of the algorithm, we turn the study of global linear convergence on scaling-invariant functions into the study of the stability of an underlying normalized Markov chain (MC). We hence prove global linear convergence by studying the stability (irreducibility, recurrence, positivity, geometric ergodicity) of the normalized MC associated to the $(1+1)$-ES. More precisely, we prove that starting from any initial solution and any step-size, linear convergence with probability one and in expectation occurs. Our proof holds on unimodal functions that are the composite of strictly increasing functions by positively homogeneous functions with degree $α$ (assumed also to be continuously differentiable). This function class includes composite of norm functions but also non-quasi convex functions. Because of the composition by a strictly increasing function, it includes non continuous functions. We find that a sufficient condition for global linear convergence is the step-size increase on linear functions, a condition typically satisfied for standard parameter choices. While introduced more than 40 years ago, we provide here the first proof of global linear convergence for the $(1+1)$-ES with generalized one-fifth success rule and the first proof of linear convergence for a CB-SARS on such a class of functions that includes non-quasi convex and non-continuous functions. Our proof also holds on functions where linear convergence of some CB-SARS was previously proven, namely convex-quadratic functions (including the well-know sphere function).
The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies the CSA-ES on composites of strictly increasing functions with affine linear functions through the investigation of its underlying Markov chains. Rigorous results on the change and the variation of the step size are derived with and without cumulation. The step-size diverges geometrically fast in most cases. Furthermore, the influence of the cumulation parameter is studied.
In the well placement problem, as well as in other field development optimization problems, geological uncertainty is a key source of risk affecting the viability of field development projects. Well placement problems under geological uncertainty are formulated as optimization problems in which the objective function is evaluated using a reservoir simulator on a number of possible geological realizations. In this paper, we present a new approach to handle geological uncertainty for the well placement problem with a reduced number of reservoir simulations. The proposed approach uses already simulated well configurations in the neighborhood of each well configuration for the objective function evaluation. We use thus only one single reservoir simulation performed on a randomly chosen realization together with the neighborhood to estimate the objective function instead of using multiple simulations on multiple realizations. This approach is combined with the stochastic optimizer CMA-ES. The proposed approach is shown on the benchmark reservoir case PUNQ-S3 to be able to capture the geological uncertainty using a smaller number of reservoir simulations. This approach is compared to the reference approach using all the possible realizations for each well configuration, and shown to be able to reduce significantly the number of reservoir simulations (around 80%).
The CSA-ES is an Evolution Strategy with Cumulative Step size Adaptation, where the step size is adapted measuring the length of a so-called cumulative path. The cumulative path is a combination of the previous steps realized by the algorithm, where the importance of each step decreases with time. This article studies the CSA-ES on composites of strictly increasing with affine linear functions through the investigation of its underlying Markov chains. Rigorous results on the change and the variation of the step size are derived with and without cumulation. The step-size diverges geometrically fast in most cases. Furthermore, the influence of the cumulation parameter is studied.
The Information-Geometric Optimization (IGO) has been introduced as a unified framework for stochastic search algorithms. Given a parametrized family of probability distributions on the search space, the IGO turns an arbitrary optimization problem on the search space into an optimization problem on the parameter space of the probability distribution family and defines a natural gradient ascent on this space. From the natural gradients defined over the entire parameter space we obtain continuous time trajectories which are the solutions of an ordinary differential equation (ODE). Via discretization, the IGO naturally defines an iterated gradient ascent algorithm. Depending on the chosen distribution family, the IGO recovers several known algorithms such as the pure rank-μupdate CMA-ES. Consequently, the continuous time IGO-trajectory can be viewed as an idealization of the original algorithm. In this paper we study the continuous time trajectories of the IGO given the family of isotropic Gaussian distributions. These trajectories are a deterministic continuous time model of the underlying evolution strategy in the limit for population size to infinity and change rates to zero. On functions that are the composite of a monotone and a convex-quadratic function, we prove the global convergence of the solution of the ODE towards the global optimum. We extend this result to composites of monotone and twice continuously differentiable functions and prove local convergence towards local optima.
Optimum implementation of non-conventional wells allows us to increase considerably hydrocarbon recovery. By considering the high drilling cost and the potential improvement in well productivity, well placement decision is an important issue in field development. Considering complex reservoir geology and high reservoir heterogeneities, stochastic optimization methods are the most suitable approaches for optimum well placement. This paper proposes an optimization methodology to determine optimal well location and trajectory based upon the Covariance Matrix Adaptation - Evolution Strategy (CMA-ES) which is a variant of Evolution Strategies recognized as one of the most powerful derivative-free optimizers for continuous optimization. To improve the optimization procedure, two new techniques are investigated: (1). Adaptive penalization with rejection is developed to handle well placement constraints. (2). A meta-model, based on locally weighted regression, is incorporated into CMA-ES using an approximate ranking procedure. Therefore, we can reduce the number of reservoir simulations, which are computationally expensive. Several examples are presented. Our new approach is compared with a Genetic Algorithm incorporating the Genocop III technique. It is shown that our approach outperforms the genetic algorithm: it leads in general to both a higher NPV and a significant reduction of the number of reservoir simulations.
In this paper, the performances of the quasi-Newton BFGS algorithm, the NEWUOA derivative free optimizer, the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), the Differential Evolution (DE) algorithm and Particle Swarm Optimizers (PSO) are compared experimentally on benchmark functions reflecting important challenges encountered in real-world optimization problems. Dependence of the performances in the conditioning of the problem and rotational invariance of the algorithms are in particular investigated.
In the context of inference with expectation constraints, we propose an approach based on the "loopy belief propagation" algorithm LBP, as a surrogate to an exact Markov Random Field MRF modelling. A prior information composed of correlations among a large set of N variables, is encoded into a graphical model; this encoding is optimized with respect to an approximate decoding procedure LBP, which is used to infer hidden variables from an observed subset. We focus on the situation where the underlying data have many different statistical components, representing a variety of independent patterns. Considering a single parameter family of models we show how LBP may be used to encode and decode efficiently such information, without solving the NP hard inverse problem yielding the optimal MRF. Contrary to usual practice, we work in the non-convex Bethe free energy minimization framework, and manage to associate a belief propagation fixed point to each component of the underlying probabilistic mixture. The mean field limit is considered and yields an exact connection with the Hopfield model at finite temperature and steady state, when the number of mixture components is proportional to the number of variables. In addition, we provide an enhanced learning procedure, based on a straightforward multi-parameter extension of the model in conjunction with an effective continuous optimization procedure. This is performed using the stochastic search heuristic CMAES and yields a significant improvement with respect to the single parameter basic model.
This paper deals with the identification of the flux for a system of conservation laws in the specific example of analytic chromatography. The fundamental equations of chromatographic process are highly non linear. The state-of-the-art Evolution Strategy, CMA-ES (the Covariance Matrix Adaptation Evolution Strategy), is used to identify the parameters of the so-called isotherm function. The approach was validated on different configurations of simulated data using either one, two or three components mixtures. CMA-ES is then applied to real data cases and its results are compared to those of a gradient-based strategy.