Stability results for properly quasi convex vector optimization problems

Stability and Scalarization for a Unified Vector Optimization Problem

“…and D is a closed, convex and pointed cone with int D = ∅, then the above notion coincides with the notion of gamma-convergence considered in [8,10]. In general, the above notion is weaker, as is evident from the following example.…”

“…There are various other notions of convergence of sequences of vector-valued functions, such as continuous convergence (that is, f n (x n ) → f (x) whenever x n → x), Painlevé-Kuratowski convergence for epigraphs and Mosco covergence. We observe that for M = M n = X , the notion of Γ -convergence is weaker than the notions of continuous and Mosco convergence, but stronger than the notion of Painlevé-Kuratowski convergence for epigraphs (see [5,6,10] for more details).…”

“…However, the sequence ( f n ) n∈N does not gamma converge to f as in [8,10], for any closed, convex and pointed cone in R 2 .…”

“…Stability theory in vector optimization is usually concerned with the study of the change in the behavior of solution sets, under certain perturbations of the objective function, the feasible region or the ordering cone. The set convergence serves as a basis for investigating the stability aspects of solution sets of vector optimization; see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13] and references therein.…”

“…Lalitha and Chatterjee [8,9] discussed stability aspects for vector optimization problems involving (strict) proper quasi-convex functions. Later, Li et al [10] extended the convergence results under the weaker notion of Painlevé-Kuratowski convergence for epigraphs, whereas Li et al [11] investigated the convergence results by perturbing the feasible set, the objective function and the ordering cone.…”