\( \def\vepsi{\varepsilon} \def\bold#1{{\bf #1}} \) \( \def\Hcal{{\mathcal{H}}} \def\bold#1{{\bf #1}} \) \( \def\RR{{\bf R}} \def\bold#1{{\bf #1}} \) \( \def\LL{{\bf L}} \def\bold#1{{\bf #1}} \) \( \def\xv{{\bf x}} \def\bold#1{{\bf #1}} \) \( \def\uv{{\bf u}} \def\bold#1{{\bf #1}} \) \( \def\qv{{\bf q}} \def\bold#1{{\bf #1}} \) \( \def\vv{{\bf v}} \def\bold#1{{\bf #1}} \) \( \def\Xv{{\bf X}} \def\bold#1{{\bf #1}} \) \( \def\cv{{\bf c}} \def\bold#1{{\bf #1}} \) \( \def\gv{{\bf g}} \def\bold#1{{\bf #1}} \) \( \def\wv{{\bf w}} \def\bold#1{{\bf #1}} \) \( \def\Omegamat{{\bf \Omega}} \def\bold#1{{\bf #1}} \) \( \def\Smat{{\bf S}} \def\bold#1{{\bf #1}} \) \( \def\Pmat{{\bf P}} \def\bold#1{{\bf #1}} \) \( \def\Hmat{{\bf H}} \def\bold#1{{\bf #1}} \) \( \def\Imat{{\bf I}} \def\bold#1{{\bf #1}} \) \( \def\Gmat{{\bf G}} \def\bold#1{{\bf #1}} \) \( \def\Gammamat{{\bf \Gamma}} \def\bold#1{{\bf #1}} \) \( \def\Qmat{{\bf Q}} \def\bold#1{{\bf #1}} \) \( \def\Rmat{{\bf R}} \def\bold#1{{\bf #1}} \) \( \def\omegas{{\bf \omega^{\star}}} \def\bold#1{{\bf #1}} \) \( \def\hf{{\frac{1}{2}}} \) \( \def\nablau{\nabla_{\uv}^{2}} \def\bold#1{{\bf #1}} \) \( \def\nablav{\nabla_{\vv}^{2}} \def\bold#1{{\bf #1}} \) \( \def\nablaw{\nabla_{\wv}^{2}} \def\bold#1{{\bf #1}} \) \( \def\nablavv{\nabla_{\vv\vv}^{2}} \def\bold#1{{\bf #1}} \) \( \def\xbar{\overset{-}{\xv}} \def\bold#1{{\bf #1}} \) \( \def\ubar{\overset{-}{\uv}} \def\bold#1{{\bf #1}} \) \( \def\vbar{\overset{-}{\vv}} \def\bold#1{{\bf #1}} \)

MGDA Platform

Multiple Gradient Descent Algorithm for Multi Objective Differentiable Optimization.

Stochastic formulations

Abstract

For stochastic problems that are formulated through cost functions expressed as expectations of random functions, the classical stochastic gradient algorithm can be combined efficiently with MGDA. This approach has been developed at Onera with F. Poirion and Q. Mercier, and demonstrated in problems of data variability and reliability.

Bibliography

Q. Mercier. Optimisation multicritère sous incertitude: un algorithme de descente ( Multiobjective optimization under uncertainties: a descent algorithm ), University Côte d’Azur, École Doctorale Sciences Fondamentales et Appliquées, 10 October 2018

F. Poirion, Q. Mercier, and J.-A. Désidéri. Descent algorithm for nonsmooth stochastic multiobjective optimization. Computational Optimization and Applications, 68(2):317-331, 2017.

Q. Mercier, F. Poirion, and J.-A. Désidéri. A stochastic multiple gradient descent algorithm. European Journal of Operational Research, ELSEVIER Publish., 271(3):808-817, 16 December 2018.

Q. Mercier, F.Poirion, and J.-A. Désidéri. Non-convex multiobjective optimization under uncertainty: a descent algorithm. Application to sandwich plate design and reliability. Engineering Optimization, 51(5):733-752, 2019.