Abstract
In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.
Highlights
It is known that in vector optimization as well as in Image Space Analysis (ISA) in infinite dimensional linear spaces difficulties may arise because of the possible non-solidness of ordering cones
We studied vector optimization problems involving not necessarily pointed and not necessarily solid, convex cones in real linear spaces
We proposed a Henig-type proper efficiency solution concept based on generalized dilating convex cones which have nonempty intrinsic cores
Summary
It is known that in vector optimization (see Jahn [34]) as well as in Image Space Analysis (ISA) in infinite dimensional linear spaces (see Giannessi [19,20] and references therein) difficulties may arise because of the possible non-solidness of ordering cones (for instance in the fields of optimal control, approximation theory, duality theory). It is of increasing interest to derive optimality conditions and duality results for such vector optimization problems using generalized interiority conditions (see, e.g., Adán and Novo [1,2,3,4], Bagdasar and Popovici [6], Bao and Mordukhovich [7], Borwein and Goebel [10], Borwein and Lewis [11], Grad [23,24], Grad and Pop [25], Khazayel et al [36], Zalinescu [43,44], and Cuong et al [14]).
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