Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Abstract

Summary A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.

Design optimization under uncertainties of a mesoscale implant in biological tissues using a probabilistic learning algorithm

“…In this paper, the probabilistic learning used for solving the optimization problem under uncertainties refers to an algorithm that is based on the use of the two following methodologies. (i) The first one is the novel methodology that is proposed in [25,88] for solving the CNOPPU, which consists in using -an optimizer that is adapted to the nature of the optimization problem under consideration, -a dataset that is made up of a small number of points generated by the optimizer for which only a small number of expensive evaluations of functions is carried out.…”

“…(ii) The second ingredient is a smoothing technique [25,88] for estimating J(w 0 ) and c(w 0 ) at any point w 0 in C w , by using only the additional realizations generated from the given dataset. The points w 0 are those that are generated for the optimization algorithm.…”

“…(ii) As previously explained, the CNOPPU is solved with a combination of the algorithms recently introduced in [25,85,87], which will allow us to solve the design optimization of a mesoscale implant in a cortical bone with a reasonable numerical cost. The algorithm proposed is different from those existing and does not introduce any approximations such as surface responses, surrogate models, etc.…”

Probabilistic Learning on Manifold for Optimization Under Uncertainties

“…The development of mathematical and algorithmic constructs that promote learning with successive optimization steps continues to be a key challenge in that regard. In the framework of this paper, "optimization under uncertainties" is related to the novel methodology that is proposed in [3] • for solving a probabilistic nonconvex constrained optimization (an OUU),…”

“…The second ingredient is a smoothing technique that allows for estimating the mathematical expectations in the computation of f (w 0 ) and c(w 0 ) at any point w 0 in C w , by using only the given dataset and the additional samples [3].…”