A stochastic multiple gradient descent algorithm

Quentin Mercier (1) , Fabrice Poirion (1) , Jean-Antoine Desideri (2)

1 DMAS, ONERA, Université Paris Saclay (COmUE) [Châtillon]
2 ACUMES - Analysis and Control of Unsteady Models for Engineering Sciences

In this article, we propose a new method for multiobjective optimization problems in which the objective functions are expressed as expectations of random functions. The present method is based on an extension of the classical stochastic gradient algorithm and a deterministic multiobjective algorithm, the Multiple Gradient Descent Algorithm (MGDA). In MGDA a descent direction common to all specified objective functions is identified through a result of convex geometry. The use of this common descent vector and the Pareto stationarity definition into the stochastic gradient algorithm makes the algorithm able to solve multiobjective problems. The mean square and almost sure convergence of this new algorithm are proven considering the classical stochastic gradient algorithm hypothesis. The algorithm efficiency is illustrated on a set of benchmarks with diverse complexity and assessed in comparison with two classical algorithms (NSGA-II, DMS) coupled with a Monte Carlo expectation estimator.

Mots clés: OPTIMISATION MULTIOBJECTIF; INCERTITUDE; MULTIPLE OBJECTIVE PROGRAMMING; MULTIOBJECTIVE STOCHASTIC OPTIMIZATION; STOCHASTIC GRADIENT ALGORITHM; MULTIPLE GRADIENT DESCENT ALGORITHM; COMMON DESCENT VECTOR

Domaines: Matériaux

Quentin Mercier, Fabrice Poirion, Jean-Antoine Desideri. A stochastic multiple gradient descent algorithm. European Journal of Operational Research, 2018, pp.10. ⟨10.1016/j.ejor.2018.05.064⟩. ⟨hal-01833165⟩