arXiv subjects

Anne Auger

Publications and source records attributed to Anne Auger.

At least 19 recordsLinked to original sources

arXiv2026

Pareto Set Characterization in Constrained Multiobjective Optimization and the COBI Problem Generator *

Benchmark problems play a central role in assessing the performance of numerical optimization algorithms. However, many existing constrained multiobjective optimization benchmark problems rely on overly restricted constructions or lack formal analysis of their optimal solution sets, limiting their relevance for systematic algorithm evaluation. In this work, we introduce a class of analytically tractable constrained multiobjective optimization problems whose Pareto sets can be formally characterized. The construction is based on convex-quadratic functions with positive definite Hessians, combined through multipeak formulations in which each objective is defined as the minimum over several convex-quadratic components. This approach preserves analytical structure while enabling multimodality (non-convexity), ill-conditioning and non-separability. The constraints are built as sublevel sets of multipeak functions giving rise to problems with potentially disconnected feasible regions. Building on these results, we propose COBI, a scalable generator of constrained bi-objective test problems designed for benchmarking derivative-free optimization algorithms. We provide a reference Python implementation that enables straightforward integration of COBI instances into benchmarking workflows.

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arXiv2025

On the irreducibility and convergence of a class of nonsmooth nonlinear state-space models on manifolds and their applications to zeroth-order optimization

In this paper, we analyze a large class of general nonlinear state-space models on a state-space X, defined by the recursion $ϕ_{k+1} = F(ϕ_k,α(ϕ_k,U_{k+1}))$, $k \in\mathbb N$, where $F,α$ are some functions and $\{U_{k+1}\}_{k\in\mathbb N}$ is a sequence of i.i.d. random variables. More precisely, we extend conditions under which this class of Markov chains is irreducible, aperiodic and satisfies important continuity properties, relaxing two key assumptions from prior works. First, the state-space X is supposed to be a smooth manifold instead of an open subset of a Euclidean space. Second, we only suppose that $F$ is locally Lipschitz continuous. We demonstrate the significance of our results through their application to Markov chains underlying optimization algorithms. These schemes belong to the class of evolution strategies with covariance matrix adaptation and step-size adaptation.

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arXiv2024

Irreducibility of nonsmooth state-space models with an application to CMA-ES

We analyze a stochastic process resulting from the normalization of states in the zeroth-order optimization method CMA-ES. On a specific class of minimization problems where the objective function is scaling-invariant, this process defines a time-homogeneous Markov chain whose convergence at a geometric rate can imply the linear convergence of CMA-ES. However, the analysis of the intricate updates for this process constitute a great mathematical challenge. We establish that this Markov chain is an irreducible and aperiodic T-chain. These contributions represent a first major step for the convergence analysis towards a stationary distribution. We rely for this analysis on conditions for the irreducibility of nonsmooth state-space models on manifolds. To obtain our results, we extend these conditions to address the irreducibility in different hyperparameter settings that define different Markov chains, and to include nonsmooth state spaces.

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arXiv2024

Asymptotic estimations of a perturbed symmetric eigenproblem

We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends to infinity, we bound the values of the coordinates of the eigenvectors of the perturbed matrix. Equivalently, in the coordinate system where the initial matrix is diagonal, we bound the rate of convergence of coordinates that tend to zero.

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arXiv2022

Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions

Evolution Strategies (ES) are stochastic derivative-free optimization algorithms whose most prominent representative, the CMA-ES algorithm, is widely used to solve difficult numerical optimization problems. We provide the first rigorous investigation of the linear convergence of step-size adaptive ES involving a population and recombination, two ingredients crucially important in practice to be robust to local irregularities or multimodality.We investigate convergence of step-size adaptive ES with weighted recombination on composites of strictly increasing functions with continuously differentiable scaling-invariant functions with a global optimum. This function class includes functions with non-convex sublevel sets and discontinuous functions. We prove the existence of a constant r such that the logarithm of the distance to the optimum divided by the number of iterations converges to r. The constant is given as an expectation with respect to the stationary distribution of a Markov chain-its sign allows to infer linear convergence or divergence of the ES and is found numerically.Our main condition for convergence is the increase of the expected log step-size on linear functions. In contrast to previous results, our condition is equivalent to the almost sure geometric divergence of the step-size on linear functions.

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arXiv2022

Global Linear Convergence of Evolution Strategies on More Than Smooth Strongly Convex Functions

Evolution strategies (ESs) are zeroth-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions are generated. Among different variants, CMA-ES is nowadays recognized as one of the state-of-the-art zeroth-order optimizers for difficult problems. Albeit ample empirical evidence that ESs with a step-size control mechanism converge linearly, theoretical guarantees of linear convergence of ESs have been established only on limited classes of functions. In particular, theoretical results on convex functions are missing, where zeroth-order and also first-order optimization methods are often analyzed. In this paper, we establish almost sure linear convergence and a bound on the expected hitting time of an \new{ES family, namely the $(1+1)_κ$-ES, which includes the (1+1)-ES with (generalized) one-fifth success rule} and an abstract covariance matrix adaptation with bounded condition number, on a broad class of functions. The analysis holds for monotonic transformations of positively homogeneous functions and of quadratically bounded functions, the latter of which particularly includes monotonic transformation of strongly convex functions with Lipschitz continuous gradient. As far as the authors know, this is the first work that proves linear convergence of ES on such a broad class of functions.

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arXiv2021

An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms

We consider stochastic algorithms derived from methods for solving deterministic optimization problems, especially comparison-based algorithms derived from stochastic approximation algorithms with a constant step-size. We develop a methodology for proving geometric convergence of the parameter sequence $\{θ_n\}_{n\geq 0}$ of such algorithms. We employ the ordinary differential equation (ODE) method, which relates a stochastic algorithm to its mean ODE, along with a Lyapunov-like function $Ψ$ such that the geometric convergence of $Ψ(θ_n)$ implies -- in the case of an optimization algorithm -- the geometric convergence of the expected distance between the optimum and the search point generated by the algorithm. We provide two sufficient conditions for $Ψ(θ_n)$ to decrease at a geometric rate: $Ψ$ should decrease "exponentially" along the solution to the mean ODE, and the deviation between the stochastic algorithm and the ODE solution (measured by $Ψ$) should be bounded by $Ψ(θ_n)$ times a constant. We also provide practical conditions under which the two sufficient conditions may be verified easily without knowing the solution of the mean ODE. Our results are any-time bounds on $Ψ(θ_n)$, so we can deduce not only the asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to a comparison-based stochastic algorithm with a constant step-size for optimization on continuous domains.

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arXiv2021

Scaling-invariant functions versus positively homogeneous functions

Scaling-invariant functions preserve the order of points when the points are scaled by the same positive scalar (with respect to a unique reference point). Composites of strictly monotonic functions with positively homogeneous functions are scaling-invariant with respect to zero. We prove in this paper that the reverse is true for large classes of scaling-invariant functions. Specifically, we give necessary and sufficient conditions for scaling-invariant functions to be composites of a strictly monotonic function with a positively homogeneous function. We also study sublevel sets of scaling-invariant functions generalizing well-known properties of positively homogeneous functions.

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arXiv2020

COCO: A Platform for Comparing Continuous Optimizers in a Black-Box Setting

We introduce COCO, an open source platform for Comparing Continuous Optimizers in a black-box setting. COCO aims at automatizing the tedious and repetitive task of benchmarking numerical optimization algorithms to the greatest possible extent. The platform and the underlying methodology allow to benchmark in the same framework deterministic and stochastic solvers for both single and multiobjective optimization. We present the rationales behind the (decade-long) development of the platform as a general proposition for guidelines towards better benchmarking. We detail underlying fundamental concepts of COCO such as the definition of a problem as a function instance, the underlying idea of instances, the use of target values, and runtime defined by the number of function calls as the central performance measure. Finally, we give a quick overview of the basic code structure and the currently available test suites.

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arXiv2019

Uncrowded Hypervolume Improvement: COMO-CMA-ES and the Sofomore framework

We present a framework to build a multiobjective algorithm from single-objective ones. This framework addresses the $p \times n$-dimensional problem of finding p solutions in an n-dimensional search space, maximizing an indicator by dynamic subspace optimization. Each single-objective algorithm optimizes the indicator function given $p - 1$ fixed solutions. Crucially, dominated solutions minimize their distance to the empirical Pareto front defined by these $p - 1$ solutions. We instantiate the framework with CMA-ES as single-objective optimizer. The new algorithm, COMO-CMA-ES, is empirically shown to converge linearly on bi-objective convex-quadratic problems and is compared to MO-CMA-ES, NSGA-II and SMS-EMOA.

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arXiv2019

COCO: The Large Scale Black-Box Optimization Benchmarking (bbob-largescale) Test Suite

The bbob-largescale test suite, containing 24 single-objective functions in continuous domain, extends the well-known single-objective noiseless bbob test suite, which has been used since 2009 in the BBOB workshop series, to large dimension. The core idea is to make the rotational transformations R, Q in search space that appear in the bbob test suite computationally cheaper while retaining some desired properties. This documentation presents an approach that replaces a full rotational transformation with a combination of a block-diagonal matrix and two permutation matrices in order to construct test functions whose computational and memory costs scale linearly in the dimension of the problem.

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arXiv2019

Drift Theory in Continuous Search Spaces: Expected Hitting Time of the (1+1)-ES with 1/5 Success Rule

This paper explores the use of the standard approach for proving runtime bounds in discrete domains---often referred to as drift analysis---in the context of optimization on a continuous domain. Using this framework we analyze the (1+1) Evolution Strategy with one-fifth success rule on the sphere function. To deal with potential functions that are not lower-bounded, we formulate novel drift theorems. We then use the theorems to prove bounds on the expected hitting time to reach a certain target fitness in finite dimension $d$. The bounds are akin to linear convergence. We then study the dependency of the different terms on $d$ proving a convergence rate dependency of $Θ(1/d)$. Our results constitute the first non-asymptotic analysis for the algorithm considered as well as the first explicit application of drift analysis to a randomized search heuristic with continuous domain.

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arXiv2019

Using Well-Understood Single-Objective Functions in Multiobjective Black-Box Optimization Test Suites

Several test function suites are being used for numerical benchmarking of multiobjective optimization algorithms. While they have some desirable properties, like well-understood Pareto sets and Pareto fronts of various shapes, most of the currently used functions possess characteristics that are arguably under-represented in real-world problems. They mainly stem from the easier construction of such functions and result in improbable properties such as separability, optima located exactly at the boundary constraints, and the existence of variables that solely control the distance between a solution and the Pareto front. Here, we propose an alternative way to constructing multiobjective problems-by combining existing single-objective problems from the literature. We describe in particular the bbob-biobj test suite with 55 bi-objective functions in continuous domain, and its extended version with 92 bi-objective functions (bbob-biobj-ext). Both test suites have been implemented in the COCO platform for black-box optimization benchmarking. Finally, we recommend a general procedure for creating test suites for an arbitrary number of objectives. Besides providing the formal function definitions and presenting their (known) properties, this paper also aims at giving the rationale behind our approach in terms of groups of functions with similar properties, objective space normalization, and problem instances. The latter allows us to easily compare the performance of deterministic and stochastic solvers, which is an often overlooked issue in benchmarking.

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arXiv2018

On Bi-Objective convex-quadratic problems

In this paper we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when both Hessian matrices are proportional. We then propose a novel set of convex-quadratic test problems, describe their theoretical properties and the algorithm abilities required by those test problems. This includes in particular testing the sensitivity with respect to separability, ill-conditioned problems, rotational invariance, and whether the Pareto set is aligned with the coordinate axis.

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arXiv2018

Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions

Quality gain is the expected relative improvement of the function value in a single step of a search algorithm. Quality gain analysis reveals the dependencies of the quality gain on the parameters of a search algorithm, based on which one can derive the optimal values for the parameters. In this paper, we investigate evolution strategies with weighted recombination on general convex quadratic functions. We derive a bound for the quality gain and two limit expressions of the quality gain. From the limit expressions, we derive the optimal recombination weights and the optimal step-size, and find that the optimal recombination weights are independent of the Hessian of the objective function. Moreover, the dependencies of the optimal parameters on the dimension and the population size are revealed. Differently from previous works where the population size is implicitly assumed to be smaller than the dimension, our results cover the population size proportional to or greater than the dimension. Numerical simulation shows that the asymptotically optimal step-size well approximates the empirically optimal step-size for a finite dimensional convex quadratic function.

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arXiv2017

Verifiable Conditions for the Irreducibility and Aperiodicity of Markov Chains by Analyzing Underlying Deterministic Models

We consider Markov chains that obey the following general non-linear state space model: $Φ_{k+1} = F(Φ_k, α(Φ_k, U_{k+1}))$ where the function $F$ is $C^1$ while $α$ is typically discontinuous and $\{U_k: k \in \mathbb{Z}_{> 0} \}$ is an independent and identically distributed process. We assume that for all $x$, the random variable $α(x, U_1)$ admits a density $p_x$ such that $(x, w) \mapsto p_x(w)$ is lower semi-continuous. We generalize and extend previous results that connect properties of the underlying deterministic control model to provide conditions for the chain to be $φ$-irreducible and aperiodic. By building on those results, we show that if a rank condition on the controllability matrix is satisfied for all $x$, there is equivalence between the existence of a globally attracting state for the control model and $φ$-irreducibility of the Markov chain. Additionally, under the same rank condition on the controllability matrix, we prove that there is equivalence between the existence of a steadily attracting state and the $φ$-irreducibility and aperiodicity of the chain. The notion of steadily attracting state is new. Those results hold under considerably weaker assumptions on the model than previous ones that would require $(x,u) \mapsto F(x,α(x,u))$ to be $C^\infty$ (while it can be discontinuous here). Additionally the establishment of a necessary and sufficient condition for the $φ$-irreducibility and aperiodicity without a structural assumption on the control set is novel---even for Markov chains where $(x,u) \mapsto F(x,α(x,u))$ is $C^\infty$. We illustrate that the conditions are easy to verify on a non-trivial and non-artificial example of Markov chain arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario.

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arXiv2017

Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles

We present a canonical way to turn any smooth parametric family of probability distributions on an arbitrary search space $X$ into a continuous-time black-box optimization method on $X$, the \emph{information-geometric optimization} (IGO) method. Invariance as a design principle minimizes the number of arbitrary choices. The resulting \emph{IGO flow} conducts the natural gradient ascent of an adaptive, time-dependent, quantile-based transformation of the objective function. It makes no assumptions on the objective function to be optimized. The IGO method produces explicit IGO algorithms through time discretization. It naturally recovers versions of known algorithms and offers a systematic way to derive new ones. The cross-entropy method is recovered in a particular case, and can be extended into a smoothed, parametrization-independent maximum likelihood update (IGO-ML). For Gaussian distributions on $\mathbb{R}^d$, IGO is related to natural evolution strategies (NES) and recovers a version of the CMA-ES algorithm. For Bernoulli distributions on $\{0,1\}^d$, we recover the PBIL algorithm. From restricted Boltzmann machines, we obtain a novel algorithm for optimization on $\{0,1\}^d$. All these algorithms are unified under a single information-geometric optimization framework. Thanks to its intrinsic formulation, the IGO method achieves invariance under reparametrization of the search space $X$, under a change of parameters of the probability distributions, and under increasing transformations of the objective function. Theory strongly suggests that IGO algorithms have minimal loss in diversity during optimization, provided the initial diversity is high. First experiments using restricted Boltzmann machines confirm this insight. Thus IGO seems to provide, from information theory, an elegant way to spontaneously explore several valleys of a fitness landscape in a single run.

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arXiv2016

Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains

In this paper, we consider comparison-based adaptive stochastic algorithms for solving numerical optimisation problems. We consider a specific subclass of algorithms that we call comparison-based step-size adaptive randomized search (CB-SARS), where the state variables at a given iteration are a vector of the search space and a positive parameter, the step-size, typically controlling the overall standard deviation of the underlying search distribution.We investigate the linear convergence of CB-SARS on\emph{scaling-invariant} objective functions. Scaling-invariantfunctions preserve the ordering of points with respect to their functionvalue when the points are scaled with the same positive parameter (thescaling is done w.r.t. a fixed reference point). This class offunctions includes norms composed with strictly increasing functions aswell as many non quasi-convex and non-continuousfunctions. On scaling-invariant functions, we show the existence of ahomogeneous Markov chain, as a consequence of natural invarianceproperties of CB-SARS (essentially scale-invariance and invariance tostrictly increasing transformation of the objective function). We thenderive sufficient conditions for \emph{global linear convergence} ofCB-SARS, expressed in terms of different stability conditions of thenormalised homogeneous Markov chain (irreducibility, positivity, Harrisrecurrence, geometric ergodicity) and thus define a general methodologyfor proving global linear convergence of CB-SARS algorithms onscaling-invariant functions. As a by-product we provide aconnexion between comparison-based adaptive stochasticalgorithms and Markov chain Monte Carlo algorithms.

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