Second order optimality conditions for differentiable multiobjective problems

Abstract

Second order optimality conditions for differentiable multiobjective problems RAIRO. Recherche opérationnelle, tome 34, n o 4 (2000), p. 411-426 <http://www.numdam.org/item?id=RO_2000__34_4_411_0> © AFCET, 2000, tous droits réservés. L'accès aux archives de la revue « RAIRO. Recherche opérationnelle » implique l'accord avec les conditions générales d'utilisation (http://www. numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ RAIRO Opérations Research RAIRO Oper. Res. 34 (2000) 411-426 SECOND ORDER OPTIMALITY CONDITIONS FOR DIFFERENTIABLE MULTIOBJECTIVE PROBLEMS (*) by Giancarlo BIGI (*) and Marco CASTELLANÏ ( 2 )Communicated by Franco GIANNESSI Abstract. -A second order optimality condition for multiobjective optimization with a set constraint is developed; this condition is expressed as the impossibility of nonhomogeneous linear Systems. When the constraint is given in terms of inequalities and equalities, it can be turned into a John type multipliers rule, using a nonhomo geneous Motzkin Theorem of the Alternative. Adding weak second order regularity assumptions, Karush, Kuhn-Tucker type conditions are therefore deduced.

First and Second Order Sufficient Conditions for Strict Minimality in Multiobjective Programming

“…This paper is, to a large extent, complementary of Ref. [4] (similar results are given in Ref. [10,Chap.…”

“…When first order conditions do not give a satisfactory response, it is usual to consider second order conditions to improve the information provided by those. Though in real (or scalar) programming there are numerous papers devoted to study of second order conditions since the appearance of the first paper around the end of 60, in vector programming they are more scarce, and only their study has been intensified in the last decade (see Smale, [15,16] Wan, [19] Hoffmann and Kornstaedt, [9] Van Geldrop, [18] Linnemann, [12] Ben-Tal and Zowe, [3] Wang, [20] Liu, [13] Cambini et al, [6] Cambini, [7] Bolintine´anu and El Maghri, [5] Aghezzaf and Hachimi, [1] Jime´nez [10] and Bigi and Castellani [4] ).…”

Second-order optimality conditions for cone constrained multi-objective optimization

“…Results about second-order necessary optimality conditions and second-order sufficient optimality conditions for multi-objective programming can be found in [7], [13], [1], [4] and [3]. For the abstract constrained optimization problem (1), the second-order tangent set in variational analysis (see [5]) is used to develop second-order necessary optimality conditions in [4] and second-order sufficient optimality conditions in [3].…”

Existence theorems for second-order radial epiderivatives

“…In the last years, the second-order optimality condition and higher-order optimality conditions have been widely investigated in set-valued optimization [3], [4], [5], [6], [9], [11], [12], [13], [14], [16], [18], [25], [23], [21], [28], [31]. It can be seen that a second-order contingent set, introduced by Aubin and Frankowska [2], and a second-order asymptotic contingent cone, introduced by Penot [28], play a important role in establishing second-order optimality conditions.…”