Probabilistic nonconvex constrained optimization with fixed number of function evaluations

Roger Ghanem (1) , Christian Soize (2)

1 USC - University of Southern California
2 MSME - Laboratoire de Modélisation et Simulation Multi Echelle

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 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 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.

Mots clés: Probabilistic optimization; Nonconvex constrained optimization; Diffusion maps; Optimization under uncertainty; MCMC generator; Probability distribution on manifolds

Domaines: Sciences de l'ingénieur [physics]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]; Statistiques [stat]

Roger Ghanem, Christian Soize. Probabilistic nonconvex constrained optimization with fixed number of function evaluations. International Journal for Numerical Methods in Engineering, 2018, 113 (4), pp.719-741. ⟨10.1002/nme.5632⟩. ⟨hal-01576263⟩