Vector Optimization with Non-Solid or Empty-Interior Cones

Abstract

Vector optimization problems generally depend on ordering cones with a non-empty interior to ensure separation, scalarization, and robust optimality outcomes. Nevertheless, numerous significant applications, especially in infinite-dimensional spaces and Banach lattices, inherently result in non-solid or empty-interior cones, where traditional optimality theory fails. This research establishes a generic framework for vector optimization in diminished cone structures. We present revised notions of efficiency, suitable efficiency, and approximation optimality that retain significance without interiority. New separation principles, founded on weak topologies and support functionals, are formulated to identify efficient sites when conventional interior-based methods are inadequate. Optimality criteria that are both necessary and sufficient are established for cone-constrained vector problems in locally convex spaces, employing techniques from convex analysis, dual pairings, and order theory within Banach lattices. Applications demonstrate that effective solutions can still be located and estimated even in the absence of an interior in the ordering cone.