8714 articles  [version française]

inria-00389811, version 3

## Multiple-Gradient Descent Algorithm (MGDA)

Jean-Antoine Désidéri () 1

N° RR-6953 (2009)

Abstract: In a previous report [Split of Territories, INRIA Research Report 6108], a methodology for the numerical treatment of a two-objective optimization problem, possibly subject to equality constraints, was proposed. The method was devised to be adapted to cases where an initial design-point is known and such that one of the two disciplines, considered to be preponderant, or fragile, and said to be the *primary discipline*, achieves a local or global optimum at this point. Then, a particular split of the design variables was proposed to accomplish a *competitive-optimization* phase by a Nash game, whose equilibrium point realizes an improvement of a *secondary discipline*, while causing the least possible degradation of the primary discipline from the initial optimum. In this new report, the initial design point and the number of disciplines are arbitrary. Certain theoretical results are established and they lead us to define a preliminary *cooperative-optimization* phase throughout which all the criteria improve, by a so-called *Multiple-Gradient Descent Algorithm (MGDA)*, which generalizes to $n$ disciplines ($n \geq 2$) the classical steepest-descent method. This phase is conducted until a design-point on the Pareto set is reached; then, the optimization is interrupted or continued in a subsequent competitive phase by a generalization of the former approach by territory splitting and Nash game.

• 1:  OPALE (INRIA Sophia Antipolis / INRIA Grenoble Rhône-Alpes)
• INRIA – CNRS : UMR6621 – Université Nice Sophia Antipolis (UNS)
• Domain : Mathematics/Numerical Analysis
• Keywords : Optimum--shape design – concurrent engineering – multi-criterion optimization – split of territory – Nash and Stackelberg game strategies – Pareto optimality – descent direction – steepest-descent direction
• Internal note : RR-6953
• Comment : In this report – the problem of minimizing simultaneously n smooth and unconstrained criteria is considered. A descent direction common to all the criteria is identified – knowing all the gradients. An algorithm is defined in which the optimization process is carried out in two phases : one that is cooperative yielding to the Pareto front – and the other optional and competitive.
• Available versions :  v1 (2009-05-29) v2 (2009-06-05) v3 (2012-11-05)

• inria-00389811, version 3
• oai:hal.inria.fr:inria-00389811
• From:
• Submitted on: Monday, 5 November 2012 10:59:40
• Updated on: Monday, 5 November 2012 13:56:28