TY - JOUR

T1 - Some results on condition numbers in convex multiobjective optimization

AU - Bianchi, Monica

AU - Miglierina, Enrico

AU - Molho, Elena

AU - Pini, Rita

PY - 2012

Y1 - 2012

N2 - Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem defined via m convex C^1,1 objective functions on a given closed ball in ℝ^n . Two approaches are proposed: the first one adopts a local point of view around a given solution point, whereas the second one considers the solution set as a whole. A comparison between the two notions of well-conditioned problem is developed. We underline that both the condition numbers introduced in the present work reduce to the same condition number proposed by Zolezzi in 2003, in the special case of the scalar optimization problem considered there. A pseudodistance between functions is defined such that the condition number provides an upper bound on how far from a well–conditioned function f a perturbed function g can be chosen in order that g is well–conditioned too. For both the local and the global approach an extension of classical Eckart–Young distance theorem is proved, even if only a special class of perturbations is considered.

AB - Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem defined via m convex C^1,1 objective functions on a given closed ball in ℝ^n . Two approaches are proposed: the first one adopts a local point of view around a given solution point, whereas the second one considers the solution set as a whole. A comparison between the two notions of well-conditioned problem is developed. We underline that both the condition numbers introduced in the present work reduce to the same condition number proposed by Zolezzi in 2003, in the special case of the scalar optimization problem considered there. A pseudodistance between functions is defined such that the condition number provides an upper bound on how far from a well–conditioned function f a perturbed function g can be chosen in order that g is well–conditioned too. For both the local and the global approach an extension of classical Eckart–Young distance theorem is proved, even if only a special class of perturbations is considered.

KW - Condition number

KW - Eckart–Young theorem

KW - Multiobjective optimization

KW - Sensitivity analysis

KW - Condition number

KW - Eckart–Young theorem

KW - Multiobjective optimization

KW - Sensitivity analysis

UR - http://hdl.handle.net/10807/31291

UR - http://www.springerlink.com/content/d1k51026460v14h2/

U2 - 10.1007/s11228-012-0207-2

DO - 10.1007/s11228-012-0207-2

M3 - Article

SN - 1877-0533

VL - 21

SP - 47

EP - 65

JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

ER -