arXiv subjects

Charles Audet

Publications and source records attributed to Charles Audet.

27 records · Page 2Linked to original sources

arXiv2022

Tight bounds on the maximal perimeter of convex equilateral small polygons

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s \ge 4$, and show that their perimeters are within $O(1/n^4)$ of the maximal perimeter and exceed the previously best known values from the literature. In particular, for the first open case $n=16$, our result proves that Mossinghoff's equilateral hexadecagon is suboptimal.

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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

The equilateral small octagon of maximal width

A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s \ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24\%$ larger than the width of the regular octagon: $\cos(π/8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $s\ge 4$, whose widths are within $O(1/n^4)$ of the maximal width.

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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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arXiv2021

NOMAD version 4: Nonlinear optimization with the MADS algorithm

NOMAD is software for optimizing blackbox problems. In continuous development since 2001, it constantly evolved with the integration of new algorithmic features published in scientific publications. These features are motivated by real applications encountered by industrial partners. The latest major release of NOMAD, version 3, dates from 2008. Minor releases are produced as new features are incorporated. The present work describes NOMAD 4, a complete redesign of the previous version, with a new architecture providing more flexible code, added functionalities and reusable code. We introduce algorithmic components, which are building blocks for more complex algorithms, and can initiate other components, launch nested algorithms, or perform specialized tasks. They facilitate the implementation of new ideas, including the MegaSearchPoll component, warm and hot restarts, and a revised version of the PSD-MADS algorithm. Another main improvement of NOMAD 4 is the usage of parallelism, to simultaneously compute multiple blackbox evaluations, and to maximize usage of available cores. Running different algorithms, tuning their parameters, and comparing their performance for optimization is simpler than before, while overall optimization performance is maintained between versions 3 and 4. NOMAD is freely available at www.gerad.ca/nomad and the whole project is visible at github.com/bbopt/nomad.

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arXiv2019

Optimization of noisy blackboxes with adaptive precision

In derivative-free and blackbox optimization, the objective function is often evaluated through the execution of a computer program seen as a blackbox. It can be noisy, in the sense that its outputs are contaminated by random errors. Sometimes, the source of these errors is identified and controllable, in the sense that it is possible to reduce the standard deviation of the stochastic noise it generates. A common strategy to deal with such a situation is to monotonically diminish this standard deviation, to asymptotically make it converge to zero and ensure convergence of algorithms because the noise is dismantled. This work presents MpMads, an algorithm which follows this approach. However, in practice a reduction of the standard deviation increases the computation time, and makes the optimization process long. Therefore, a second algorithm called DpMads is introduced to explore another strategy, which does not force the standard deviation to monotonically diminish. Although these strategies are proved to be theoretically equivalents, tests on analytical problems and an industrial blackbox are presented to illustrate practical differences.

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arXiv2019

StoMADS: Stochastic blackbox optimization using probabilistic estimates

This work introduces StoMADS, a stochastic variant of the mesh adaptive direct-search (MADS) algorithm originally developed for deterministic blackbox optimization. StoMADS considers the unconstrained optimization of an objective function f whose values can be computed only through a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based on an algorithmic framework similar to that of MADS and uses random estimates of function values obtained from stochastic observations since the exact deterministic computable version of f is not available. Such estimates are required to be accurate with a sufficiently large but fixed probability and satisfy a variance condition. The ability of the proposed algorithm to generate an asymptotically dense set of search directions is then exploited to show convergence to a Clarke stationary point of f with probability one, using martingale theory.

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arXiv2016

Algorithmic construction of the subdifferential from directional derivatives

The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is $\R^1$ and $\R^2$, as well as in $\R^n$ where subdifferential is known to possess at most three vertices.

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