SMGDA: an uncertainty based multi-objective optimization approach. Illustration to an airplane composite material

Progress in structural design often requires the resolution of various conflicting design objectives: decreasing the structure mass while increasing safety, etc. Multi-objective optimization is a powerful numerical tool capable of resolving such objectives. Contrary to the single objective case, there exists a continuous set of parameters which are solutions of a given multi-objective problem in the sense that there exist no other designs that can ameliorate all the objectives at the same time. Such a set of optimal points is called the Pareto frontier. Introducing uncertain parameters in the objective function significantly affects the decision making process, the Pareto frontier becoming itself uncertain. Under a probabilistic framework where uncertainty is introduced through a random variable, a first stage before solving the new optimization problem is to specify the mathematical meaning given to the probabilistic optimization problem and then to develop appropriate numerical procedures leading to the solutions, or the corresponding Pareto frontier. Here we shall consider to solve the stochastic multi-objective problem written in terms of the objective function expectation. We propose a new algorithm based on an extension of the deterministic multi-gradient descent algorithm to the stochastic context together with an extension of the Robbins-Monro algorithm. Mean square and almost sure convergence of the algorithm can be proven. An illustration to an aerospace composite material will be given. (C) 2017 The Authors. Published by Elsevier Ltd.