Prioritized optimization by Nash games : towards an adaptive multi-objective strategy

Jean-Antoine Désidéri,Régis Duvigneau

doi:10.1051/proc/202171106

Abstract

This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution xA*. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {x̅ε} (ε≥ 0), in a way such that: firstly, x̅0=xA* (compatibility), and secondly, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε. The theoretical results are recalled and the method is applied numerically to a Super-Sonic Business Jet (SSBJ) sizing problem to optimize the flight performance.

Highlights

In the engineering environment, the process of numerical optimization is complex due to several coupled elements that we outline in the example of the shape optimization of a civil aircraft wing or configuration
Many objective functions are potentially significant in the design optimization process and some may reveal, at the stage of an a posteriori validation, more critical than anticipated
If a preliminary phase of optimization accounted for the most essential design objectives, a second phase may turn out to be necessary to correct the design in view of reducing additional cost functions

Summary

Introduction

The process of numerical optimization is complex due to several coupled elements that we outline in the example of the shape optimization of a civil aircraft wing or configuration. Most of the functions of interest are not directly expressed in terms of the optimization variables, such as geometrical parameters, but are instead functionals of state variables through the partial-differential equations of fluid dynamics, elasticity and of all other physical disciplines to be accounted for, via a complex modeler-mesher-solver coupled sequence [7]. As a result, these functions are evaluated at high cost, and the calculation of the gradient is usually not trivial.

Mathematical problem definition

Preliminary calculations

Numerical implementation

Findings

Application to a SSBJ sizing problem for optimal flight performance