On Continuation Methods for Non-Linear Bi-Objective Optimization: Certified Interval-Based Approach

The global optimization of constrained Non-Linear Bi-Objective Optimization problems (MO) aims at covering their Pareto-optimal front which is in general a manifold in R^2. Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarizing framework which transforms the bi-objective problem into a parametric mono-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. In this paper, we provide a survey on continuation techniques in global optimization methods for MO, which allow discovering large portions of locally Pareto-optimal solutions. We also propose a rigorous active set management strategy on top of a previously proposed certified continuation method based on interval analysis, and illustrate it on a challenging bi-objective problem

Computation of Khun-Tucker triples in optimum design problems in the presence of parametric singularities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76242/1/AIAA-1994-4416-414.pd

Towards a full higher order AD-based continuation and bifurcation framework

International audienceSome of the theoretical aspects of continuation and bifurcation methods devoted to the solution for nonlinear parametric systems are presented in a higher-order automatic differentiation (HOAD) framework. Besides benefits in terms of generality and ease of use, HOAD is used to assess fold and simple bifurcations points. In particular, the formation of a geometric series in successive Taylor coefficients allows for the implementation of an efficient detection and branch switching method at simple bifurcation points. Some comparisons with the Auto and MatCont continuation software are proposed. Strengths are then exemplified on a classical case study in structural mechanics

A Noninterior Path following Algorithm for Solving a Class of Multiobjective Programming Problems

Multiobjective programming problems have been widely applied to various engineering areas which include optimal design of an automotive engine, economics, and military strategies. In this paper, we propose a noninterior path following algorithm to solve a class of multiobjective programming problems. Under suitable conditions, a smooth path will be proven to exist. This can give a constructive proof of the existence of solutions and lead to an implementable globally convergent algorithm. Several numerical examples are given to illustrate the results of this paper