2.1 Scientific Context

Critical problems of the 21st century like the search for highly energy efficient or even carbon-neutral, and cost-efficient systems, or the design of new molecules against extensively drug-resistant bacteria crucially rely on the resolution of challenging numerical optimization problems. Such problems typically depend on noisy experimental data or involve complex numerical simulations where derivatives are not useful or not available and the function is considered as a black-box.

Many of those optimization problems are in essence multiobjective—one needs to optimize simultaneously several conflicting objectives like minimizing the cost of an energy network and maximizing its reliability—and most of the challenging black-box problems are non-convex and non-smooth and they combine difficulties related to ill-conditioning, non-separability, and ruggedness (a term that characterizes functions that can be non-smooth but also noisy or multi-modal). Additionally, the objective function can be expensive to evaluate, that is, one function evaluation can take several minutes to hours (it can involve for instance a CFD simulation).

In this context, the use of randomness combined with proper adaptive mechanisms that notably satisfy certain invariance properties (affine invariance, invariance to monotonic transformations) has proven to be one key component for the design of robust global numerical optimization algorithms 54, 39.

The field of adaptive stochastic optimization algorithms has witnessed some important progress over the past 15 years. On the one hand, subdomains like medium-scale unconstrained optimization may be considered as “solved” (particularly, the CMA-ES algorithm, an instance of Evolution Strategy (ES) algorithms, stands out as state-of-the-art method) and considerably better standards have been established in the way benchmarking and experimentation are performed. On the other hand, multiobjective population-based stochastic algorithms became the method of choice to address multiobjective problems when a set of some best possible compromises is sought after. In all cases, the resulting algorithms have been naturally transferred to industry (the CMA-ES algorithm is now regularly used in companies such as Bosch, Total, ALSTOM, ...) or to other academic domains where difficult problems need to be solved such as physics, biology 59, geoscience 47, or robotics 50).

ES algorithms also attracted quite some attention in Machine Learning with the OpenAI article Evolution Strategies as a Scalable Alternative to Reinforcement Learning. It is shown that the training time for difficult reinforcement learning benchmarks could be reduced from 1 day (with standard RL approaches) to 1 hour using ES 56.1 Already ten years ago, another impressive application of CMA-ES, how “Computer Sim Teaches Itself To Walk Upright” (published at the conference SIGGRAPH Asia 2013) was presented in the press in the UK.

Several of these important advances around adaptive stochastic optimization algorithms rely to a great extent on works initiated or achieved by the founding members of RandOpt, particularly related to the CMA-ES algorithm and to the Comparing Continuous Optimizer (COCO) benchmarking platform.

Yet, the field of adaptive stochastic algorithms for black-box optimization is relatively young compared to the “classical optimization” field that includes convex and gradient-based optimization. For instance, the state-of-the art algorithms for unconstrained gradient based optimization like quasi-Newton methods (e.g. the BFGS method) date from the 1970s 38 while the stochastic derivative-free counterpart, CMA-ES dates from the early 2000s 40. Consequently, in some subdomains with important practical demands, not even the most fundamental and basic questions are answered:

• This is the case of constrained optimization where one needs to find a solution $x^{*}\in {ℝ}^n$ minimizing a numerical function ${min}_{x\in {ℝ}^n}f\left(x\right)$ while respecting a number of constraints $m$ typically formulated as $g_i\left(x^{*}\right)\le 0$ for $i=1,...,m$. Only somewhat recently, the fundamental requirement of linear convergence2, as in the unconstrained case, has been clearly stated 30.
• In multiobjective optimization, most of the research so far has been focusing on how to select candidate solutions from one iteration to the next one. The difficult question of how to generate effectively new solutions is not yet answered in a proper way and we know today that simply applying operators from single-objective optimization may not be effective with the current best selection strategies. As a comparison, in the single-objective case, the question of selection of candidate solutions was already solved in the 1980s and 15 more years were needed to solve the trickier question of an effective adaptive strategy to generate new solutions.
• With the current demand to solve larger and larger optimization problems (e.g. in the domain of deep learning), optimization algorithms that scale linearly (in terms of internal complexity, memory and number of function evaluations to reach an $ϵ$-ball around the optimum) with the problem dimension are nowadays in increasing demand. Not long ago, first proposals of how to reduce the quadratic scaling of CMA-ES have been made without a clear view of what can be achieved in the best case in practice. These later variants apply to optimization problems with thousands of variables. The question of designing randomized algorithms capable to handle problems with one or two orders of magnitude more variables effectively and efficiently is still largely open.
• For expensive optimization, standard methods are so called Bayesian optimization (BO) algorithms often based on Gaussian processes. Commonly used examples of BO algorithms are EGO 44, SMAC 42, Spearmint 57, or TPE 33 which are implemented in different libraries. Yet, our experience with a popular method like EGO is that many important aspects to come up with a good implementation rely on insider knowledge and are not standard across implementations. Two EGO implementations can differ for example in how they perform the initial design, which bandwidth for the Gaussian kernel is used, or which strategy is taken to optimize the expected improvement.

Additionally, the development of stochastic adaptive methods for black-box optimization has been mainly driven by heuristics and practice—rather than a general theoretical framework—validated by intensive computational simulations. Undoubtedly, this has been an asset as the scope of possibilities for design was not restricted by mathematical frameworks for proving convergence. In effect, powerful stochastic adaptive algorithms for unconstrained optimization like the CMA-ES algorithm emerged from this approach. At the same time, naturally, theory strongly lags behind practice. For instance, the striking performances of CMA-ES empirically observed contrast with how little is theoretically proven on the method. This situation is clearly not satisfactory. On the one hand, theory generally lifts performance assessment from an empirical level to a conceptual one, rendering results independent from the problem instances where they have been obtained. On the other hand, theory typically provides insights that change perspectives on some algorithm components. Also theoretical guarantees generally increase the trust in the reliability of a method and facilitate the task to make it accepted by wider communities.

Finally, as discussed above, the development of novel black-box algorithms strongly relies on scientific experimentation, and it is quite difficult to conduct proper and meaningful experimental analysis. This is well known for more than two decades now and summarized in this quote from Johnson in 1996

Since then, quite some progress has been made to set better standards in conducting scientific experiments and benchmarking. Yet, some domains still suffer from poor benchmarking standards and from the generic problem of the lack of reproducibility of results. For instance, in multiobjective optimization, it is (still) not rare to see comparisons between algorithms made by solely visually inspecting Pareto fronts after a fixed budget. In Bayesian optimization, good performance seems often to be due to insider knowledge not always well described in papers.

In the context of black-box numerical optimization previously described, the scientific positioning of the RandOpt ream is at the intersection between theory, algorithm design, and applications. Our vision is that the field of stochastic black-box optimization should reach the same level of maturity than gradient-based convex mathematical optimization. This entails major algorithmic developments for constrained, multiobjective and large-scale black-box optimization and major theoretical developments for analyzing current methods including the state-of-the-art CMA-ES.

The specificity in black-box optimization is that methods are intended to solve problems characterized by "non-properties"—non-linear, non-convex, non-smooth, non-Lipschitz. This contrasts with gradient-based optimization and poses on the one hand some challenges when developing theoretical frameworks but also makes it compulsory to complement theory with empirical investigations.

On the practical side, our ultimate goal is to provide software that is suitable for researchers and industry that need to solve practical optimization problems. We see theory also as a means for this end (rather than only an end in itself) and we also firmly belief that parameter tuning is part of the algorithm designer's task.

This shapes, on the one hand, four main scientific objectives for our team:

1. develop novel theoretical frameworks for guiding (a) the design of novel black-box methods and (b) their analysis, allowing to
2. provide proofs of key features of stochastic adaptive algorithms including the state-of-the-art method CMA-ES: linear convergence and learning of second order information.
3. develop stochastic numerical black-box algorithms following a principled design in domains with a strong practical need for much better methods namely constrained, multiobjective, large-scale and expensive optimization. Implement the methods such that they are easy to use. And finally, to
4. set new standards in scientific experimentation, performance assessment and benchmarking both for optimization on continuous or combinatorial search spaces. This should allow in particular to advance the state of reproducibility of results of scientific papers in optimization.

On the other hand, the above motivates our objectives with respect to dissemination and transfer:

1. develop software packages that people can directly use to solve their problems. This means having carefully thought out interfaces, generically applicable setting of parameters and termination conditions, proper treatment of numerical errors, catching properly various exceptions, etc.;
2. have direct collaborations with industrials;
3. publish our results both in applied mathematics and computer science bridging the gap between very often disjoint communities.

3.1 Developing Novel Theoretical Frameworks for Analyzing and Designing Adaptive Stochastic Algorithms

Stochastic black-box algorithms typically optimize non-convex, non-smooth functions. This is possible because the algorithms rely on weak mathematical properties of the underlying functions: the algorithms do not use the derivatives—hence the function does not need to be differentiable—and, additionally, often do not use the exact function value but instead how the objective function ranks candidate solutions (such methods are sometimes called function-value-free). (To illustrate a comparison-based update, consider an algorithm that samples $\lambda$ (with $\lambda$ an even integer) candidate solutions from a multivariate normal distribution. Let $x_1,...,x_{\lambda }$ in ${ℝ}^n$ denote those $\lambda$ candidate solutions at a given iteration. The solutions are evaluated on the function $f$ to be minimized and ranked from the best to the worse:

In the previous equation $i:\lambda$ denotes the index of the sampled solution associated to the $i$-th best solution. The new mean of the Gaussian vector from which new solutions will be sampled at the next iteration can be updated as

The previous update moves the mean towards the $\lambda /2$ best solutions. Yet the update is only based on the ranking of the candidate solutions such that the update is the same if $f$ is optimized or $g\circ f$ where $g:\mathrm{Im}\left(f\right)\to ℝ$ is strictly increasing. Consequently, such algorithms are invariant with respect to strictly increasing transformations of the objective function. This entails that they are robust and their performances generalize well.)

Additionally, adaptive stochastic optimization algorithms typically have a complex state space which encodes the parameters of a probability distribution (e.g. mean and covariance matrix of a Gaussian vector) and other state vectors. This state-space is a manifold. While the algorithms are Markov chains, the complexity of the state-space makes that standard Markov chain theory tools do not directly apply. The same holds with tools stemming from stochastic approximation theory or Ordinary Differential Equation (ODE) theory where it is usually assumed that the underlying ODE (obtained by proper averaging and limit for learning rate to zero) has its critical points inside the search space. In contrast, in the cases we are interested in, the critical points of the ODEs are at the boundary of the domain.

Last, since we aim at developing theory that on the one hand allows to analyze the main properties of state-of-the-art methods and on the other hand is useful for algorithm design, we need to be careful not to use simplifications that would allow a proof to be done but would not capture the important properties of the algorithms. With that respect one tricky point is to develop theory that accounts for invariance properties.

To face those specific challenges, we need to develop novel theoretical frameworks exploiting invariance properties and accounting for peculiar state-spaces. Those frameworks should allow researchers to analyze one of the core properties of adaptive stochastic methods, namely linear convergence on the widest possible class of functions.

We are planning to approach the question of linear convergence from three different complementary angles, using three different frameworks:

• the Markov chain framework where the convergence derives from the analysis of the stability of a normalized Markov chain existing on scaling-invariant functions for translation and scale-invariant algorithms 32. This framework allows for a fine analysis where the exact convergence rate can be given as an implicit function of the invariant measure of the normalized Markov chain. Yet it requires the objective function to be scaling-invariant. The stability analysis can be particularly tricky as the Markov chain that needs to be studied writes as ${\Phi }_{t+1}=F({\Phi }_t,W_{t+1})$ where $\{W_t:t>0\}$ are independent identically distributed and $F$ is typically discontinuous because the algorithms studied are comparison-based. This implies that practical tools for analyzing a standard property like irreducibility, that rely on investigating the stability of underlying deterministic control models 51, cannot be used. Additionally, the construction of a drift to prove ergodicity is particularly delicate when the state space includes a (normalized) covariance matrix as it is the case for analyzing the CMA-ES algorithm.
• The stochastic approximation or ODE framework. Those are standard techniques to prove the convergence of stochastic algorithms when an algorithm can be expressed as a stochastic approximation of the solution of a mean field ODE 35, 34, 48. What is specific and induces difficulties for the algorithms we aim at analyzing is the non-standard state-space since the ODE variables correspond to the state-variables of the algorithm (e.g. ${ℝ}^n\times {ℝ}_{>0}$ for step-size adaptive algorithms, ${ℝ}^n\times {ℝ}_{>0}\times S_{++}^n$ where $S_{++}^n$ denotes the set of positive definite matrices if a covariance matrix is additionally adapted). Consequently, the ODE can have many critical points at the boundary of its definition domain (e.g. all points corresponding to ${\sigma }_t=0$ are critical points of the ODE) which is not typical. Also we aim at proving linear convergence, for that it is crucial that the learning rate does not decrease to zero which is non-standard in ODE method.
• The direct framework where we construct a global Lyapunov function for the original algorithm from which we deduce bounds on the hitting time to reach an $ϵ$-ball of the optimum. For this framework as for the ODE framework, we expect that the class of functions where we can prove linear convergence are composite of $g\circ f$ where $f$ is differentiable and $g:\mathrm{Im}\left(f\right)\to ℝ$ is strictly increasing and that we can show convergence to a local minimum.

We expect those frameworks to be complementary in the sense that the assumptions required are different. Typically, the ODE framework should allow for proofs under the assumptions that learning rates are small enough while it is not needed for the Markov chain framework. Hence this latter framework captures better the real dynamics of the algorithm, yet under the assumption of scaling-invariance of the objective functions. Also, we expect some overlap in terms of function classes that can be studied by the different frameworks (typically convex-quadratic functions should be encompassed in the three frameworks). By studying the different frameworks in parallel, we expect to gain synergies and possibly understand what is the most promising approach for solving the holy grail question of the linear convergence of CMA-ES. We foresee for instance that similar approaches like the use of Foster-Lyapunov drift conditions are needed in all the frameworks and that intuition can be gained on how to establish the conditions from one framework to another one.

3.2 Algorithmic developments

We are planning on developing algorithms in the subdomains with strong practical demand for better methods of constrained, multiobjective, large-scale and expensive optimization.

Many of the algorithm developments, we propose, rely on the CMA-ES method. While this seems to restrict our possibilities, we want to emphasize that CMA-ES became a family of methods over the years that nowadays include various techniques and developments from the literature to handle non-standard optimization problems (noisy, large-scale, ...). The core idea of all CMA-ES variants—namely the mechanism to adapt a Gaussian distribution—has furthermore been shown to derive naturally from first principles with only minimal assumptions in the context of derivative-free black-box stochastic optimization 54, 39. This is a strong justification for relying on the CMA-ES premises while new developments naturally include new techniques typically borrowed from other fields. While CMA-ES is now a full family of methods, for visibility reasons, we continue to refer often to “the CMA-ES algorithm”.

3.3 Setting novel standards in scientific experimentation and benchmarking

Numerical experimentation is needed as a complement to theory to test novel ideas, hypotheses, the stability of an algorithm, and/or to obtain quantitative estimates. Optimally, theory and experimentation go hand in hand, jointly guiding the understanding of the mechanisms underlying optimization algorithms. Though performing numerical experimentation on optimization algorithms is crucial and a common task, it is non-trivial and easy to fall in (common) pitfalls as stated by J. N. Hooker in his seminal paper 41.

In the RandOpt team we aim at raising the standards for both scientific experimentation and benchmarking.

On the experimentation aspect, we are convinced that there is common ground over how scientific experimentation should be done across many (sub-)domains of optimization, in particular with respect to the visualization of results, testing extreme scenarios (parameter settings, initial conditions, etc.), how to conduct understandable and small experiments, how to account for invariance properties, performing scaling up experiments and so forth. We therefore want to formalize and generalize these ideas in order to make them known to the entire optimization community with the final aim that they become standards for experimental research.

Extensive numerical benchmarking, on the other hand, is a compulsory task for evaluating and comparing the performance of algorithms. It puts algorithms to a standardized test and allows to make recommendations which algorithms should be used preferably in practice. To ease this part of optimization research, we have been developing the Comparing Continuous Optimizers platform (COCO) since 2007 which allows to automatize the tedious task of benchmarking. It is a game changer in the sense that the freed time can now be spent on the scientific part of algorithm design (instead of implementing the experiments, visualization, statistical tests, etc.) and it opened novel perspectives in algorithm testing. COCO implements a thorough, well-documented methodology that is based on the above mentioned general principles for scientific experimentation.

Also due to the freely available data from 300+ algorithms benchmarked with the platform, COCO became a quasi-standard for single-objective, noiseless optimization benchmarking. It is therefore natural to extend the reach of COCO towards other subdomains (particularly constrained optimization, many-objective optimization) which can benefit greatly from an automated benchmarking methodology and standardized tests without (much) effort. This entails particularly the design of novel test suites and rethinking the methodology for measuring performance and more generally evaluating the algorithms. Particularly challenging is the design of scalable non-trivial testbeds for constrained optimization where one can still control where the solutions lies. Other optimization problem types, we are targeting are expensive problems (and the Bayesian optimization community in particular, see our AESOP project), optimization problems in machine learning (for example parameter tuning in reinforcement learning), and the collection of real-world problems from industry.

Another aspect of our future research on benchmarking is to investigate the large amounts of benchmarking data, we collected with COCO during the years. Extracting information about the influence of algorithms on the best performing portfolio, clustering algorithms of similar performance, or the automated detection of anomalies in terms of good/bad behavior of algorithms on a subset of the functions or dimensions are some of the ideas here.

Last, we want to expand the focus of COCO from automatized (large) benchmarking experiments towards everyday experimentation, for example by allowing the user to visually investigate algorithm internals on the fly or by simplifying the set up of algorithm parameter influence studies.

5.1 Footprint of research activities

We are concerned about CO2 footprint and discourage oversea conferences when far away. After the situation with respect to Covid went back to normal with respect to be able to travel, we are still dedicated to travel less than in the past and attend some conferences online.

5.2 Impact of research results

We develop general purpose optimization methods that apply in difficult optimization contexts where little is required on the function to be optimized. Application domains include optimization and design of renewable systems and climate change.

Our main method CMA-ES is transferred and widely used. The code stemming from the team is frequently downloaded (see Section 7). Among the usage of our method and our code, we find naturally problems in the domain of energy to capture carbon dioxide 52, 49, 53, solar energy 45, 46, or wind-thermal power systems 55.

Those publications witness the impact of our research results with respect to research questions and engineering design related to climate change and renewable energy.

7.1 New software

8.1 Analysis of Adaptive Stochastic Search Algorithms

Participants: Anne Auger, Nikolaus Hansen, Cheikh Touré, Armand Gissler.

External collaborators: Youhei Akimoto (Tsukuba University), Tobias Glasmachers (Ruhr University, Bochum)

We published the article where we prove the global linear convergence of the (1+1)-ES with one-fifth success rule as step-size adaptation on a class of functions that embed smooth strongly convex functions and positively homogeneous functions. Because of the invariance to monotonic transformations, the study holds for non-continuous and non-convex functions. Arguably, our study provides the first proof of the linear convergence of an adaptive evolution strategy without modifying its underlying updates (to make a proof work) on such a wide class of functions 13.

We have analyzed the linear convergence of a step-size adaptive $(\mu /\mu ,\lambda )$-ES on continuously differentiable scaling invariant functions 17. The study constitutes the first convergence analysis of this class of algorithm which is the underlying algorithmic scheme of the CMA-ES algorithms while we assume here only step-size adaptation. This study has been using the methodology based on Markov chain that we have been developing. We published our research on extending ODE theory (that connects convergence of a stochastic algorithm to the statility of an underlying ODE obtained when taking the limit for learning rate to zero) in order to prove the geometric convergence of an algorithm 14. We illustrate how to apply the theory to prove the linear convergence of a step-size adaptive ES.

We investigated theoretically and numerically the possible benefit of adapting the learning rate on the mean in adaptive Evolution Strategies 20, 27. We quantified in particular convergence rates under the assumption of optimal learning rate.

8.2 Multi-objective optimization

Participants: Eugénie Marescaux, Anne Auger, Dimo Brockhoff, Nikolaus Hansen.

8.3 Benchmarking: methodology and the Comparing Continuous Optimziers Platform (COCO)

Participants: Anne Auger, Dimo Brockhoff, Paul Dufossé, Nikolaus Hansen.

External collaborators: Olaf Mersmann (TH Köln), Tea Tušar (Jozef Stefan Institute)

Benchmarking is an important task in optimization in order to assess and compare the performance of algorithms as well as to motivate the design of better solvers. We are leading the benchmarking of derivative free solvers in the context of difficult problems: we have been developing methodologies and testbeds as well as assembled this into a platform automatizing the benchmarking process. This is a continuing effort that we are pursuing in the team.

The COCO platform, developed at Inria first in the TAO team and then in Randopt since 2007, aims at automatizing numerical benchmarking experiments and the visual presentation of their results. The platform consists of an experimental part to generate benchmarking data (in various programming languages) and a postprocessing module (in python), see Figure 2. At the interface between the two, we provide data sets from numerical experiments of 300+ algorithms and algorithm variants from various fields (quasi-Newton, derivative-free optimization, evolutionary computing, Bayesian optimization) and for various problem characteristics (noiseless/noisy optimization, single-/multi-objective optimization, continuous/mixed-integer, ...).

Visual overview of the COCO platform

We have been using the platform in the past to initiate workshop papers during the ACM-GECCO conference as well as to collect algorithm data sets from the entire optimization community (300+ so far over the different test suites). The next workshop in this series is going to take place in 2023 and we also held a workshop in 2022.

In this context, we constantly improve and extend the software and provide additional data from benchmarking experiments. In 2022, 14 new data sets on the existing test suites have been collected from the BBOB-2022 participants and 6 through continuous submissions. All of which are available via the COCO postprocessing module. Our own contribution was a comparison of diagonal decoding versions of CMA-ES 19.

After several years of development, we released the first constrained test suite 21 for COCO in 2022 and used it for the BBOB-2022 workshop. Overall, 9 data sets have been collected so far on this new test suite. We contributed by comparing different versions of the augmented Lagrangian version of CMA-ES 18.

Journal articles describing the bbob-biobj test suite 15 and the general performance assessment methodology within COCO 16 have been published in 2022, too.

9.1 Bilateral contracts with industry

• Contract with the company Storengy funding the PhD thesis of Mohamed Gharafi in the context of the CIROQUO project (2021–2024)
• Contract with Thales for the CIFRE PhD thesis of Paul Dufossé (2018–2022)
• Contract with Thales for the CIFRE PhD thesis of Tristan Marty (2023–2025)

10.1 National initiatives

• CIROQUO ("Consortium Industriel de Recherche en Optimisation et QUantification d'incertitudes pour les données Onéreuses"), participation as Inria Saclay/Ecole Polytechnique, together with six other academic and five industrial partners (2021–2024)

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

12.1 Major publications

• 1 articleY.Youhei Akimoto, A.Anne Auger, T.Tobias Glasmachers and D.Daiki Morinaga. Global Linear Convergence of Evolution Strategies on More Than Smooth Strongly Convex Functions.SIAM Journal on OptimizationJune 2022
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• 2 articleY.Youhei Akimoto, A.Anne Auger and N.Nikolaus Hansen. An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms.Stochastic Processes and their Applications1452022, 269-307
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• 3 articleY.Youhei Akimoto and N.Nikolaus Hansen. Diagonal Acceleration for Covariance Matrix Adaptation Evolution Strategies.Evolutionary Computation2832020, 405-435
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• 4 articleA.Anne Auger and N.Nikolaus Hansen. A SIGEVO impact award for a paper arising from the COCO platform.ACM SIGEVOlution134January 2021, 1-11
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• 5 articleD.Dimo Brockhoff, A.Anne Auger, N.Nikolaus Hansen and T.Tea Tušar. Using Well-Understood Single-Objective Functions in Multiobjective Black-Box Optimization Test Suites.Evolutionary Computation2022
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• 6 articleA.Alexandre Chotard and A.Anne Auger. Verifiable Conditions for the Irreducibility and Aperiodicity of Markov Chains by Analyzing Underlying Deterministic Models.Bernoulli251December 2018, 112-147
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• 7 inproceedingsN.Nikolaus Hansen. A Global Surrogate Assisted CMA-ES.GECCO 2019 - The Genetic and Evolutionary Computation ConferenceACMPrague, Czech RepublicJuly 2019, 664-672
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• 8 articleN.Nikolaus Hansen, A.Anne Auger, R.Raymond Ros, O.Olaf Mersmann, T.Tea Tušar and D.Dimo Brockhoff. COCO: A Platform for Comparing Continuous Optimizers in a Black-Box Setting.Optimization Methods and Software361ArXiv e-prints, arXiv:1603.087852020, 114-144
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• 9 articleY.Yann Ollivier, L.Ludovic Arnold, A.Anne Auger and N.Nikolaus Hansen. Information-Geometric Optimization Algorithms: A Unifying Picture via Invariance Principles.Journal of Machine Learning Research18182017, 1-65
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• 10 articleC.Cheikh Touré, A.Anne Auger and N.Nikolaus Hansen. Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions.Journal of Global Optimization2022
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• 11 articleC.Cheikh Touré, A.Armand Gissler, A.Anne Auger and N.Nikolaus Hansen. Scaling-invariant functions versus positively homogeneous functions.Journal of Optimization Theory and ApplicationsSeptember 2021
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• 12 inproceedingsC.Cheikh Touré, N.Nikolaus Hansen, A.Anne Auger and D.Dimo Brockhoff. Uncrowded Hypervolume Improvement: COMO-CMA-ES and the Sofomore framework.GECCO 2019 - The Genetic and Evolutionary Computation ConferencePart of this research has been conducted in the context of a research collaboration between Storengy and InriaPrague, Czech RepublicJuly 2019
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12.2 Publications of the year

12.3 Cited publications

• 28 inproceedingsY.Youhei Akimoto and N.Nikolaus Hansen. Online model selection for restricted covariance matrix adaptation.International Conference on Parallel Problem Solving from NatureSpringer2016, 3--13
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• 29 inproceedingsY.Youhei Akimoto and N.Nikolaus Hansen. Projection-based restricted covariance matrix adaptation for high dimension.Proceedings of the 2016 on Genetic and Evolutionary Computation ConferenceACM2016, 197--204
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• 30 inproceedingsD. V.D. V. Arnold and J.J. Porter. Towards au Augmented Lagrangian Constraint Handling Approach for the $(1+1)$-ES.Genetic and Evolutionary Computation ConferenceACM Press2015, 249-256
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• 31 inproceedingsA.Asma Atamna, A.Anne Auger and N.Nikolaus Hansen. Linearly Convergent Evolution Strategies via Augmented Lagrangian Constraint Handling.Foundation of Genetic Algorithms (FOGA)2017
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• 32 articleA.Anne Auger and N.Nikolaus Hansen. Linear Convergence of Comparison-based Step-size Adaptive Randomized Search via Stability of Markov Chains.SIAM Journal on Optimization2632016, 1589-1624
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• 33 inproceedingsJ.J. Bergstra, R.R. Bardenet, Y.Y. Bengio and B.B. Kégl. Algorithms for Hyper-Parameter Optimization.Neural Information Processing Systems (NIPS 2011)2011
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• 34 articleV.V.S. Borkar and S.S.P. Meyn. The O.D.E. Method for Convergence of Stochastic Approximation and Reinforcement Learning.SIAM Journal on Control and Optimization382January 2000
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• 35 bookletV. S.Vivek S Borkar. Stochastic approximation: a dynamical systems viewpoint.Cambridge University Press2008
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• 36 inproceedingsC. A.Carlos A Coello Coello. Constraint-handling techniques used with evolutionary algorithms.Proceedings of the 2008 Genetic and Evolutionary Computation ConferenceACM2008, 2445--2466
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• 37 inproceedingsG.Guillaume Collange, S.Stéphane Reynaud and N.Nikolaus Hansen. Covariance Matrix Adaptation Evolution Strategy for Multidisciplinary Optimization of Expendable Launcher Families.13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Proceedings2010
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• 38 bookJ. E.J. E. Dennis and R. B.R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations.Englewood Cliffs, NJPrentice-Hall1983
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• 39 incollectionN.Nikolaus Hansen and A.Anne Auger. Principled design of continuous stochastic search: From theory to practice.Theory and principled methods for the design of metaheuristicsSpringer2014, 145--180
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• 40 articleN.N. Hansen and A.A. Ostermeier. Completely Derandomized Self-Adaptation in Evolution Strategies.Evolutionary Computation922001, 159--195
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• 41 articleJ. N.John N Hooker. Testing heuristics: We have it all wrong.Journal of heuristics111995, 33--42
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• 42 inproceedingsF.F. Hutter, H.H. Hoos and K.K. Leyton-Brown. An Evaluation of Sequential Model-based Optimization for Expensive Blackbox Functions.GECCO (Companion) 2013Amsterdam, The NetherlandsACM2013, 1209--1216
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• 43 articleD. S.David S Johnson. A theoretician’s guide to the experimental analysis of algorithms.Data structures, near neighbor searches, and methodology: fifth and sixth DIMACS implementation challenges592002, 215--250
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• 44 articleD. R.Donald R Jones, M.Matthias Schonlau and W. J.William J Welch. Efficient global optimization of expensive black-box functions.Journal of Global optimization1341998, 455--492
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• 45 articleJ. H.Jérôme Henri Kämpf and D.Darren Robinson. A hybrid CMA-ES and HDE optimisation algorithm with application to solar energy potential.Applied Soft Computing922009, 738--745
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• 46 articleJ. H.Jérôme Henri Kämpf and D.Darren Robinson. Optimisation of building form for solar energy utilisation using constrained evolutionary algorithms.Energy and Buildings4262010, 807--814
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• 47 articleI.Iris Kriest, V.Volkmar Sauerland, S.Samar Khatiwala, A.Anand Srivastav and A.Andreas Oschlies. Calibrating a global three-dimensional biogeochemical ocean model (MOPS-1.0).Geoscientific Model Development1012017, 127
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• 48 bookH. J.Harold Joseph Kushner and G.George Yin. Stochastic approximation and recursive algorithms and applications.Applications of mathematicsNew YorkSpringer2003
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• 49 inproceedingsX.Xueying Lu, B.Benjamin Ganis and M. F.Mary F Wheeler. Optimal Design of CO2 Sequestration with Three-Way Coupling of Flow-Geomechanics Simulations and Evolution Strategy.SPE Reservoir Simulation ConferenceOnePetro2019
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• 50 inproceedingsP.Patrick MacAlpine, S.Samuel Barrett, D.Daniel Urieli, V.Victor Vu and P.Peter Stone. Design and Optimization of an Omnidirectional Humanoid Walk: A Winning Approach at the RoboCup 2011 3D Simulation Competition.Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence (AAAI)Toronto, Ontario, CanadaJuly 2012
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