Stability for Properly Quasiconvex Vector Optimization Problem
J Optim Theory Appl (2012) 155:492–506 DOI 10.1007/s10957-012-0079-5
Stability for Properly Quasiconvex Vector Optimization Problem C.S. Lalitha · Prashanto Chatterjee
Received: 17 August 2011 / Accepted: 30 April 2012 / Published online: 17 May 2012 © Springer Science+Business Media, LLC 2012
Abstract The aim of this paper is to study the stability aspects of various types of solution set of a vector optimization problem both in the given space and in its image space by perturbing the objective function and the feasible set. The Kuratowski– Painlevé set-convergence of the sets of minimal, weak minimal and Henig proper minimal points of the perturbed problems to the corresponding minimal set of the original problem is established assuming the objective functions to be (strictly) properly quasi cone-convex. Keywords Kuratowski–Painlevé convergence · Proper quasi cone-convexity · Efficiency · Weak efficiency · Proper efficiency
1 Introduction The analysis of the solution sets of a vector optimization problem under certain perturbations, either of the feasible set or of the objective function led to the study of stability theory. This topic has been of great interest in vector optimization. Some of the books dealing with this issue are [1–3]. Among the earlier papers dealing with stability analysis, the reader is referred to [4–6] and the references therein. By using various set-convergence notions and their epigraphical versions, Attouch and Riahi
Communicated by Marcin Studniarski. C.S. Lalitha Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi 110021, India e-mail: [email protected] P. Chatterjee () Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi 110007, India e-mail: [email protected]
J Optim Theory Appl (2012) 155:492–506
493
[7] studied semicontinuity properties, which led to further investigation in this direction by Chen and Huang [8]. Further, in [9] Huang discussed the stability results of the set of efficient solutions of vector-valued and set-valued optimization problems, where the data of the approximate (perturbed) problems converge to the data of the original problem in the sense of Kuratowski–Painlevé. Oppezzi and Rossi [10, 11] used the notion of gamma-convergence for sequences of vector-valued functions and applied it to stability theory. Lucchetti and Miglierina [12] studied stability under perturbations of both the objective function and the feasible region for convex vector optimization problem whereas Miglierina and Molho [13] presented stability analysis of minimal frontiers of sectionwise connected sets. Crespi, Papalia and Rocca [14] extended the study further by considering cone quasiconvex functions. These papers deal mainly with the Kuratowski–Painlevé convergence of the sets of minimal and efficient solutions. A well known notion closely related to stability is the concept of well-posedness. One of the first attempts made in this direction, for vector optimization problems, was by Loridan [15]. The well-posedness notion considered in [14] extended the notion of well-posedness considered in [16] to study cone quasiconvex vector optimization problems. In vector optimization, there are several solution concepts depending on the partial order induced by the cone, mainly weak efficiency and efficiency. However, it was observed by Kuhn and Tucker [17] and later by Geoffrion [18] that certain efficient points have an abnormal behaviour and may not be characterized by a scalar problem. Hence, refining the notion of efficiency, the concept of proper efficiency was introduced by these authors and in a more general framework by Henig [19]. Miglierina and Molho [20] proved that properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set. Lalitha and Chatterjee [21] established a stability result in terms of the upper semicontinuity of the perturbed Henig proper efficient solution set-valued map. In [12], it was observed that the convergence fails to hold for the set of Henig proper minimal solutions in the case of convex vector optimization problem. The aim of the present paper is to investigate stability results for various types of minimal solution for a vector optimization problem. As in [12] and [14], we perturb the objective function by generating a sequence of functions converging to it in the sense of continuous convergence and also the feasible set by generating a sequence of sets converging to it in the sense of Kuratowski–Painlevé, thereby establishing the stability results for properly quasiconvex vector optimization problem. Under (strict) proper quasiconvexity assumptions, the Kuratowski–Painlevé set-convergence results are established for the sets of minimal, weak minimal and Henig proper minimal points and for the sets of efficient, weak efficient and Henig proper efficient points in the given space. Section 2 deals with the preliminaries required in the sequel. In Sect. 3, we recall the notion of (strict) proper quasi cone-convexity and other related notions like (strict) cone-convexity, quasi cone-convexity, natural quasiconvexity and coneconvexlikeness. In Sect. 4, we establish the Kuratowski–Painlevé set-convergence of the various solution sets in the image space of the perturbed problems to the corresponding solution set of the original problem. In Sect. 5, we discuss these results in the given space for sets of efficient, weak efficient and Henig proper efficient points. In the last section, we provide some concluding remarks.
494
J Optim Theory Appl (2012) 155:492–506
2 Preliminaries Let P be a closed, convex, pointed cone in Rp with nonempty interior, which induces a partial ordering in Rp as follows: For y1 , y2 ∈ Rp y1 ≤P y2
⇐⇒
y2 − y1 ∈ P ,
y1
Stability for Properly Quasiconvex Vector Optimization Problem C.S. Lalitha · Prashanto Chatterjee
Received: 17 August 2011 / Accepted: 30 April 2012 / Published online: 17 May 2012 © Springer Science+Business Media, LLC 2012
Abstract The aim of this paper is to study the stability aspects of various types of solution set of a vector optimization problem both in the given space and in its image space by perturbing the objective function and the feasible set. The Kuratowski– Painlevé set-convergence of the sets of minimal, weak minimal and Henig proper minimal points of the perturbed problems to the corresponding minimal set of the original problem is established assuming the objective functions to be (strictly) properly quasi cone-convex. Keywords Kuratowski–Painlevé convergence · Proper quasi cone-convexity · Efficiency · Weak efficiency · Proper efficiency
1 Introduction The analysis of the solution sets of a vector optimization problem under certain perturbations, either of the feasible set or of the objective function led to the study of stability theory. This topic has been of great interest in vector optimization. Some of the books dealing with this issue are [1–3]. Among the earlier papers dealing with stability analysis, the reader is referred to [4–6] and the references therein. By using various set-convergence notions and their epigraphical versions, Attouch and Riahi
Communicated by Marcin Studniarski. C.S. Lalitha Department of Mathematics, University of Delhi South Campus, Benito Jaurez Road, New Delhi 110021, India e-mail: [email protected] P. Chatterjee () Department of Mathematics, St. Stephen’s College, University of Delhi, Delhi 110007, India e-mail: [email protected]
J Optim Theory Appl (2012) 155:492–506
493
[7] studied semicontinuity properties, which led to further investigation in this direction by Chen and Huang [8]. Further, in [9] Huang discussed the stability results of the set of efficient solutions of vector-valued and set-valued optimization problems, where the data of the approximate (perturbed) problems converge to the data of the original problem in the sense of Kuratowski–Painlevé. Oppezzi and Rossi [10, 11] used the notion of gamma-convergence for sequences of vector-valued functions and applied it to stability theory. Lucchetti and Miglierina [12] studied stability under perturbations of both the objective function and the feasible region for convex vector optimization problem whereas Miglierina and Molho [13] presented stability analysis of minimal frontiers of sectionwise connected sets. Crespi, Papalia and Rocca [14] extended the study further by considering cone quasiconvex functions. These papers deal mainly with the Kuratowski–Painlevé convergence of the sets of minimal and efficient solutions. A well known notion closely related to stability is the concept of well-posedness. One of the first attempts made in this direction, for vector optimization problems, was by Loridan [15]. The well-posedness notion considered in [14] extended the notion of well-posedness considered in [16] to study cone quasiconvex vector optimization problems. In vector optimization, there are several solution concepts depending on the partial order induced by the cone, mainly weak efficiency and efficiency. However, it was observed by Kuhn and Tucker [17] and later by Geoffrion [18] that certain efficient points have an abnormal behaviour and may not be characterized by a scalar problem. Hence, refining the notion of efficiency, the concept of proper efficiency was introduced by these authors and in a more general framework by Henig [19]. Miglierina and Molho [20] proved that properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set. Lalitha and Chatterjee [21] established a stability result in terms of the upper semicontinuity of the perturbed Henig proper efficient solution set-valued map. In [12], it was observed that the convergence fails to hold for the set of Henig proper minimal solutions in the case of convex vector optimization problem. The aim of the present paper is to investigate stability results for various types of minimal solution for a vector optimization problem. As in [12] and [14], we perturb the objective function by generating a sequence of functions converging to it in the sense of continuous convergence and also the feasible set by generating a sequence of sets converging to it in the sense of Kuratowski–Painlevé, thereby establishing the stability results for properly quasiconvex vector optimization problem. Under (strict) proper quasiconvexity assumptions, the Kuratowski–Painlevé set-convergence results are established for the sets of minimal, weak minimal and Henig proper minimal points and for the sets of efficient, weak efficient and Henig proper efficient points in the given space. Section 2 deals with the preliminaries required in the sequel. In Sect. 3, we recall the notion of (strict) proper quasi cone-convexity and other related notions like (strict) cone-convexity, quasi cone-convexity, natural quasiconvexity and coneconvexlikeness. In Sect. 4, we establish the Kuratowski–Painlevé set-convergence of the various solution sets in the image space of the perturbed problems to the corresponding solution set of the original problem. In Sect. 5, we discuss these results in the given space for sets of efficient, weak efficient and Henig proper efficient points. In the last section, we provide some concluding remarks.
494
J Optim Theory Appl (2012) 155:492–506
2 Preliminaries Let P be a closed, convex, pointed cone in Rp with nonempty interior, which induces a partial ordering in Rp as follows: For y1 , y2 ∈ Rp y1 ≤P y2
⇐⇒
y2 − y1 ∈ P ,
y1
