Oil minmax programming problems containing n-set functions

Saddle points and Lagrangian-type duality for discrete minmax fractional subset programming problems with generalized convex functions

“…Following the introduction of the notion of convexity for set functions by Morris [18] and its extension for n-set functions by Corley [8], various generalizations of convexity for set and n-set functions were proposed in [5,[14][15][16][17]20,21,25,[29][30][31][32][33][34]. More specifically, quasiconvexity and pseudoconvexity for set functions were defined in [15], and for n-set functions in [16]; generalized ρ-convexity for n-set functions was defined in [25], (F, ρ)-convexity in [20], b-vexity in [5], (ρ, b)-vexity in [21], (F, ρ, θ)-convexity for nondifferentiable set functions in [14], (F, α, ρ, θ)-V-convexity in [29,30], and (F, b, φ, ρ, θ)-univexity in [31][32][33][34].…”

“…More specifically, quasiconvexity and pseudoconvexity for set functions were defined in [15], and for n-set functions in [16]; generalized ρ-convexity for n-set functions was defined in [25], (F, ρ)-convexity in [20], b-vexity in [5], (ρ, b)-vexity in [21], (F, ρ, θ)-convexity for nondifferentiable set functions in [14], (F, α, ρ, θ)-V-convexity in [29,30], and (F, b, φ, ρ, θ)-univexity in [31][32][33][34]. For predecessors and point-function counterparts of these convexity concepts, the reader is referred to the original papers where the extensions to set and n-set functions are discussed.…”

“…Subsequently, a Lagrangian-type dual problem was constructed for (P) in [27] via a Gordan-type theorem of the alternative, and appropriate duality theorems were proved without imposing any differentiability requirements. Later, some results of [25] were generalized in [20] by replacing the notion of ρ-convexity with (F, ρ)-convexity, and in [6] by placing generalized ρ-convexity hypotheses on different combinations of the problem functions; different derivations of the dual problem of [27] were given in [4,10]. In addition, in [10] the n-set counterpart of a Lagrangian-type dual problem originally formulated by Xu [24], was presented.…”

“…For brief surveys and lists of references pertaining to various aspects of subset programming problems, including areas of applications, optimality conditions, and duality models, the reader is referred to [8,13,18,20,26].…”

Second order duality for mathematical programming involving n-set functions

“…Mond-weir type duality theorems of a nonlinear minimax programming problem involving n-set functions were established by Preda and Zalmai [26,35]. Multi-objective programs containing n-set functions have been discussed by Kim et al [11,12,13], Chou et al [5], Lai and Lin [16], Lin [17,18,19], Preda [26,27]. Preda and Stancu-Minasian [28,30,31] studied optimality and duality results for optimization problems containing vector-valued n-set functions.…”