An inexact restoration approach to optimization problems with multiobjective constraints under weighted-sum scalarization

Assessing the reliability of general-purpose Inexact Restoration methods

“…The next algorithm is essentially the "Flexible Inexact Restoration Algorithm with Sharp Lagrangian" introduced in [9], with the particular choices of Algorithm 3.1 for the restoration procedure and for the computation of the optimization direction. The only difference between the algorithm proposed in [9] and the algorithm below is that in the line search of the latter one we ask for a sufficient decrease of the Lagrangian instead of the simple decrease required in [9].…”

“…The only difference between the algorithm proposed in [9] and the algorithm below is that in the line search of the latter one we ask for a sufficient decrease of the Lagrangian instead of the simple decrease required in [9]. Note that in [9] the algorithm was introduced with the specific purpose of minimizing an objective functions with multiobjective constraints. Given a penalty parameter θ ∈ [0, 1], we consider, for all x ∈ Ω and λ ∈ IR m , the merit function [28] given by Φ(x, λ, θ) = θL(x, λ)…”

“…Global convergence theories for modern Inexact Restoration methods were given in [29,28,15,20,12] and [9]. In [29] the theory is based in trust regions and a quadratic penalty merit function.…”

“…Fischer and Friedlander [12] proved global convergence theorems based on line searches and exact penalty functions. A global convergence approach that employs the sharp Lagrangian and line searches was defined in [9]. Local and superlinear convergence of an Inexact Restoration algorithm for general problems was proved in [7] and a general local framework that includes composite-step methods was given in [17].…”

“…The second method will be a variation of the local method that aims to improve the global convergence performance and has the same local convergence properties as the first one. The third method uses the basic tools of the first two but is globally convergent thanks to the employment of line searches and sharp Lagrangians, as in [9]. The forth one is a hybrid combination of the second and the third methods, designed to improve its computational performance.…”