arXiv subjects

Sébastien Le Digabel

Publications and source records attributed to Sébastien Le Digabel.

At least 19 recordsLinked to original sources

arXiv2026

An Inexact Modified Quasi-Newton Method for Nonsmooth Regularized Optimization

We introduce iR2N, a modified proximal quasi-Newton method for minimizing the sum of a smooth function $f$ and a lower semi-continuous prox-bounded function $h$, allowing inexact evaluations of $f$, its gradient, and the associated proximal operators. Both $f$ and $h$ may be nonconvex. iR2N is particularly suited to settings where proximal operators are computed via iterative procedures that can be stopped early, or where the accuracy of $f$ and $\nabla f$ can be controlled, leading to significant computational savings. At each iteration, the method approximately minimizes the sum of a quadratic model of $f$, a model of $h$, and an adaptive quadratic regularization term ensuring global convergence. Under standard accuracy assumptions, we prove global convergence in the sense that a first-order stationarity measure converges to zero, with worst-case evaluation complexity $O(ε^{-2})$. Numerical experiments with $\ell_p$ norms, $\ell_p$ total variation, and the indicator of the nonconvex pseudo $p$-norm ball illustrate the effectiveness and flexibility of the approach, and show how controlled inexactness can substantially reduce computational effort.

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arXiv2026

Parallel versions of the mesh adaptive direct search algorithm

This work surveys the different parallel variants of the mesh adaptive direct search (MADS) algorithm for constrained blackbox optimization. These problems can inherently imply high computational costs due to the possible large number of variables and multi-modality of the search space. In addition, the potential time-intensive nature and time heterogeneity of the blackboxes defining the problem prompts the need for efficient implementations. Parallelism emerges as an actionable solution to mitigate computation time, as modern computer systems rely on multi-core architecture. The reviewed methods employ diverse levels of parallelism and distinct parallel strategies to effectively tackle each aspect outlined above. The manuscript details the practical implementations, provides computational results, and offers insights into the advantages and limitations of each MADS parallel method.

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arXiv2026

Adaptive direct search algorithms with relaxable and quantifiable constraints

This work introduces ADS-PB, an extension of the Adaptive Direct Search (ADS) framework for solving constrained blackbox optimization problems. With ADS, iterates progress without relying on mesh structures or sufficient decrease conditions on the objective function value. Unlike the extreme barrier approach used in ADS, where only unrelaxable constraints are considered, the proposed method also handles quantifiable and relaxable constraints using a Progressive Barrier (PB) mechanism that exploits both constraint and objective function values. A convergence analysis of the proposed framework under mild assumptions is presented. The performance of the proposed method is assessed using sets of analytical and simulation-based constrained test problems and is compared with state-of-the-art blackbox optimization solvers, including the PB approach within the Mesh Adaptive Direct Search (MADS) framework.

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arXiv2026

Surrogate-based categorical neighborhoods for mixed-variable blackbox optimization

In simulation-based engineering, design choices are often obtained following the optimization of complex blackbox models. These models frequently involve mixed-variable domains with quantitative and categorical variables. Unlike quantitative variables, categorical variables lack an inherent structure, which makes them difficult to handle, especially in the presence of constraints. This work proposes a systematic approach to structure and model categorical variables in constrained mixed-variable blackbox optimization. Surrogate models of the objective and constraint functions are used to induce problem-specific categorical distances. From these distances, surrogate-based neighborhoods are constructed using notions of dominance from bi-objective optimization, jointly accounting for information from both the objective and the constraint functions. This study addresses the lack of automatic and constraint-aware categorical neighborhood construction in mixed-variable blackbox optimization. As a proof of concept, these neighborhoods are employed within CatMADS, an extension of the MADS algorithm for categorical variables. The surrogate models are Gaussian processes, and the resulting method is called CatMADS-GP. The method is benchmarked on the Cat-Suite collection of 60 mixed-variable optimization problems and compared against state-of-the-art solvers. Data profiles indicate that CatMADS-GP achieves superior performance for both unconstrained and constrained problems.

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arXiv2026

A penalty-interior point method combined with MADS for equality and inequality constrained optimization

This work introduces MADS-PIP, an efficient framework that integrates a penalty-interior point strategy into the mesh adaptive direct search (MADS) algorithm for solving nonsmooth blackbox optimization problems with general inequality and equality constraints. Inequality constraints are partitioned into two subsets: one treated via a logarithmic barrier applied to an aggregated interior constraint violation, and the other handled through an exterior quadratic penalty. All equality constraints are treated by the exterior penalty. A merit function defines a sequence of unconstrained subproblems, which are solved approximately using MADS, while a carefully designed update rule drives the penalty-barrier parameter to zero. In the nonsmooth setting, we establish convergence results ensuring feasibility for general constraints as well as Clarke stationarity for inequality-constrained problems. Computational experiments on both analytical test sets and challenging blackbox problems demonstrate that the proposed MADS-PIP algorithm is competitive with, and often outperforms, MADS with the progressive barrier strategy, particularly in the presence of equality constraints.

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arXiv2026

Multi-fidelity constraints in blackbox optimization

This work studies constrained blackbox optimization problems that cannot be solved in reasonable time due to prohibitive computational costs. This challenge is especially prevalent in industrial applications, where blackbox evaluations are costly. However, constraints can be evaluated at various fidelities at a lower computational cost. More specifically, this work targets situations in which the infeasibility of each individual constraint can be detected at lower fidelities, and where a large discrete number of fidelities are available. Moreover, highly discontinuous problems which may fail to evaluate are considered, such that direct search methods are preferred to model-based ones. To this effect, the Interruptible Direct Search (IDS) and the Dynamic Interruptible Direct Search (DIDS) algorithms are proposed to leverage feasibility assessments from various fidelity levels to avoid high cost evaluations. The results show highly increased performances from NOMAD when it is paired with IDS or DIDS.

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arXiv2025

Adaptive direct search algorithms for constrained optimization

Two families of directional direct search methods have emerged in derivative-free and blackbox optimization (DFO and BBO), each based on distinct principles: Mesh Adaptive Direct Search (MADS) and Sufficient Decrease Direct Search (SDDS). MADS restricts trial points to a mesh and accepts any improvement, ensuring none are missed, but at the cost of restraining the placement of trial points. SDDS allows greater freedom by evaluating points anywhere in the space, but accepts only those yielding a sufficient decrease in the objective function value, which may lead to discarding improving points. This work introduces a new class of methods, Adaptive Direct Search (ADS), which uses a novel acceptance rule based on the so-called punctured space, avoiding both meshes and sufficient decrease conditions. ADS enables flexible search while addressing the limitations of MADS and SDDS, and retains the theoretical foundations of directional direct search. Computational results in constrained and unconstrained settings highlight its performance compared to both MADS and SDDS.

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arXiv2025

CatMADS: Mesh Adaptive Direct Search for constrained blackbox optimization with categorical variables

Solving optimization problems in which functions are blackboxes and variables involve different types poses significant theoretical and algorithmic challenges. Nevertheless, such settings frequently occur in simulation-based engineering design and machine learning. This paper extends the Mesh Adaptive Direct Search (MADS) algorithm to address mixed-variable problems with categorical, integer and continuous variables. MADS is a robust derivative-free optimization framework with a well-established convergence analysis for constrained quantitative problems. CatMADS generalizes MADS by incorporating categorical variables through distance-induced neighborhoods. A detailed convergence analysis of CatMADS is provided, with flexible choices balancing computational cost and local optimality strength. Four types of mixed-variable local minima are introduced, corresponding to progressively stronger notions of local optimality. CatMADS integrates the progressive barrier strategy for handling constraints, and ensures Clarke stationarity. An instance of \catmads employs cross-validation to construct problem-specific categorical distances. This instance is compared to state-of-the-art solvers on 32 mixed-variable problems, half of which are constrained. Data profiles show that CatMADS achieves the best results, demonstrating that the framework is empirically efficient in addition to having strong theoretical foundations.

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arXiv2025

Efficient search strategies for constrained multiobjective blackbox optimization

Multiobjective blackbox optimization deals with problems where the objective and constraint functions are the outputs of a numerical simulation. In this context, no derivatives are available, nor can they be approximated by finite differences, which precludes the use of classical gradient-based techniques. The DMulti-MADS algorithm implements a state-of-the-art direct search procedure for multiobjective blackbox optimization based on the mesh adaptive direct search (MADS) algorithm. Since its conception, many search strategies have been proposed to improve the practical efficiency of the single-objective MADS algorithm. Inspired by this previous research, this work proposes the integration of two search heuristics into the DMulti-MADS algorithm. The first uses quadratic models, built from previously evaluated points, which act as surrogates for the true objectives and constraints, to suggest new promising candidates. The second exploits the sampling strategy of the Nelder-Mead algorithm to explore the decision space for new non-dominated points. Computational experiments on analytical problems and three engineering applications show that the use of such search steps considerably improves the performance of the DMulti-MADS algorithm.

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arXiv2025

A distance for mixed-variable and hierarchical domains with meta variables

Heterogeneous datasets emerge in various machine learning and optimization applications that feature different input sources, types or formats. Most models or methods do not natively tackle heterogeneity. Hence, such datasets are often partitioned into smaller and simpler ones, which may limit the generalizability or performance, especially when data is limited. The first main contribution of this work is a modeling framework that generalizes hierarchical, tree-structured, variable-size or conditional search frameworks. The framework models mixed-variable and hierarchical domains in which variables may be continuous, integer, or categorical, with some identified as meta when they influence the structure of the problem. The second main contribution is a novel distance that compares any pair of mixed-variable points that do not share the same variables, allowing to use whole heterogeneous datasets that reside in mixed-variable and hierarchical domains with meta variables. The contributions are illustrated through regression and classification experiments using simple distance-based models applied to datasets of hyperparameters with corresponding performance scores.

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arXiv2024

Fidelity and interruption control for expensive constrained multi-fidelity blackbox optimization

This work introduces a novel blackbox optimization algorithm for computationally expensive constrained multi-fidelity problems. When applying a direct search method to such problems, the scarcity of feasible points may lead to numerous costly evaluations spent on infeasible points. Our proposed fidelity and interruption controlled optimization algorithm addresses this issue by leveraging multi-fidelity information, allowing for premature interruption of an evaluation when a point is estimated to be infeasible. These estimations are controlled by a biadjacency matrix, for which we propose a construction. The proposed method acts as an intermediary component bridging any non multi-fidelity direct search solver and a multi-fidelity blackbox problem, giving the user freedom of choice for the solver. A series of computational tests are conducted to validate the approach. The results show a significant improvement in solution quality when an initial feasible starting point is provided. When this condition is not met, the outcomes are contingent upon specific properties of the blackbox.

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arXiv2024

solar: A solar thermal power plant simulator for blackbox optimization benchmarking

This work introduces solar, a collection of ten optimization problem instances for benchmarking blackbox optimization solvers. The instances present different design aspects of a concentrated solar power plant simulated by blackbox numerical models. The type of variables (discrete or continuous), dimensionality, and number and types of constraints (including hidden constraints) differ across instances. Some are deterministic, others are stochastic with possibilities to execute several replications to control stochasticity. Most instances offer variable fidelity surrogates, two are biobjective and one is unconstrained. The solar plant model takes into account various subsystems: a heliostats field, a central cavity receiver (the receiver), a molten salt thermal energy storage, a steam generator and an idealized power block. Several numerical methods are implemented throughout the solar code and most of the executions are time-consuming. Great care was applied to guarantee reproducibility across platforms. The solar tool encompasses most of the characteristics that can be found in industrial and real-life blackbox optimization problems, all in an open-source and stand-alone code.

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arXiv2023

Efficient Training Under Limited Resources

Training time budget and size of the dataset are among the factors affecting the performance of a Deep Neural Network (DNN). This paper shows that Neural Architecture Search (NAS), Hyper Parameters Optimization (HPO), and Data Augmentation help DNNs perform much better while these two factors are limited. However, searching for an optimal architecture and the best hyperparameter values besides a good combination of data augmentation techniques under low resources requires many experiments. We present our approach to achieving such a goal in three steps: reducing training epoch time by compressing the model while maintaining the performance compared to the original model, preventing model overfitting when the dataset is small, and performing the hyperparameter tuning. We used NOMAD, which is a blackbox optimization software based on a derivative-free algorithm to do NAS and HPO. Our work achieved an accuracy of 86.0 % on a tiny subset of Mini-ImageNet at the ICLR 2021 Hardware Aware Efficient Training (HAET) Challenge and won second place in the competition. The competition results can be found at haet2021.github.io/challenge and our source code can be found at github.com/DouniaLakhmiri/ICLR\_HAET2021.

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arXiv2023

Scaling Deep Networks with the Mesh Adaptive Direct Search algorithm

Deep neural networks are getting larger. Their implementation on edge and IoT devices becomes more challenging and moved the community to design lighter versions with similar performance. Standard automatic design tools such as \emph{reinforcement learning} and \emph{evolutionary computing} fundamentally rely on cheap evaluations of an objective function. In the neural network design context, this objective is the accuracy after training, which is expensive and time-consuming to evaluate. We automate the design of a light deep neural network for image classification using the \emph{Mesh Adaptive Direct Search}(MADS) algorithm, a mature derivative-free optimization method that effectively accounts for the expensive blackbox nature of the objective function to explore the design space, even in the presence of constraints.Our tests show competitive compression rates with reduced numbers of trials.

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arXiv2022

A general mathematical framework for constrained mixed-variable blackbox optimization problems with meta and categorical variables

A mathematical framework for modelling constrained mixed-variable optimization problems is presented in a blackbox optimization context. The framework introduces a new notation and allows solution strategies. The notation framework allows meta and categorical variables to be explicitly and efficiently modelled, which facilitates the solution of such problems. The new term meta variables is used to describe variables that influence which variables are acting or nonacting: meta variables may affect the number of variables and constraints. The flexibility of the solution strategies supports the main blackbox mixed-variable optimization approaches: direct search methods and surrogate-based methods (Bayesian optimization). The notation system and solution strategies are illustrated through an example of a hyperparameter optimization problem from the machine learning community.

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arXiv2022

Handling of constraints in multiobjective blackbox optimization

This work proposes the integration of two new constraint-handling approaches into the blackbox constrained multiobjective optimization algorithm DMulti-MADS, an extension of the Mesh Adaptive Direct Search (MADS) algorithm for single-objective constrained optimization. The constraints are aggregated into a single constraint violation function which is used either in a two-phase approach, where research of a feasible point is prioritized if not available before improving the current solution set, or in a progressive barrier approach, where any trial point whose constraint violation function values are above a threshold are rejected. This threshold is progressively decreased along the iterations. As in the single-objective case, it is proved that these two variants generate feasible and/or infeasible sequences which converge either in the feasible case to a set of local Pareto optimal points or in the infeasible case to Clarke stationary points according to the constraint violation function. Computational experiments show that these two approaches are competitive with other state-of-the-art algorithms.

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arXiv2021

Hierarchically constrained blackbox optimization

In blackbox optimization, evaluation of the objective and constraint functions is time consuming. In some situations, constraint values may be evaluated independently or sequentially. The present work proposes and compares two strategies to define a hierarchical ordering of the constraints and to interrupt the evaluation process at a trial point when it is detected that it will not improve the current best solution. Numerical experiments are performed on a closed-form test problem.

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arXiv2021

Quantifying uncertainty with ensembles of surrogates for blackbox optimization

This work is in the context of blackbox optimization where the functions defining the problem are expensive to evaluate and where no derivatives are available. A tried and tested technique is to build surrogates of the objective and the constraints in order to conduct the optimization at a cheaper computational cost. This work proposes different uncertainty measures when using ensembles of surrogates. The resulting combination of an ensemble of surrogates with our measures behaves as a stochastic model and allows the use of efficient Bayesian optimization tools. The method is incorporated in the search step of the mesh adaptive direct search (MADS) algorithm to improve the exploration of the search space. Computational experiments are conducted on seven analytical problems, two multi-disciplinary optimization problems and two simulation problems. The results show that the proposed approach solves expensive simulation-based problems at a greater precision and with a lower computational effort than stochastic models.

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