This paper proposes an alternate method for nding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-o among the various con icting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property w h i c h the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational e ciency of continuation-type algorithms. This is an improvement o ver continuation techniques for tracing the trade-o curve since continuation strategies cannot easily be extended to handle more than two objectives.
Abstract. This paper introduces the Mesh Adaptive Direct Search (MADS) class of algorithms for nonlinear optimization. MADS extends the Generalized Pattern Search (GPS) class by allowing local exploration, called polling, in a dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions spans the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many constraints for GPS. The MADS class of algorithms extend this result; the set of refining directions may even be dense in R n , although we give an example where it is not.We present an implementable instance of MADS, and we illustrate and compare it with GPS on some test problems. We also illustrate the limitation of our results with examples.
This paper contains a new convergence analysis for the Lewis and Torczon generalized pattern search (GPS) class of methods for unconstrained and linearly constrained optimization. This analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems. Specifically, even if the objective function is discontinuous or extended valued, the methods find a limit point with some minimizing properties. Simple examples show that the strength of the optimality conditions at a limit point does not depend only on the algorithm, but also on the directions it uses, and on the smoothness of the objective at the limit point in question. The contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm, and it gives previous results as corollaries.
This paper is an attempt to motivate and justify quasi-Newton methods as useful modifications of Newton's method for general and gradient nonlinear systems of equations. References are given to ample numerical justification; here we give an overview of many of the important theoretical results and each is accompanied by sufficient discussion to make the results and henc6 the methods plausible.
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