General duality in vector optimization

Abstract

Vector minimization of a relation F valued in an ordered vector space under a constraint A consists in finding x ₀ ∈ A w,₀ ∈ Fx$₀ such that w,₀ is minimal in FA. To a family of vector minimization problemsminimize , one associates a Lagrange relation where ξ belongs to an arbitrary class Ξ of mappings, the main purpose being to recover solutions of the original problem from the vector minimization of the Lagrange relation for an appropriate ξ. This ξ turns out to be a solution of a dual vector maximization problem. Characterizations of exact and approximate duality in terms of vector (generalized with respect to Ξ) convexity and subdifferentiability are given. They extend the theory existing in scalar optimization. Verifiable criteria for exact penalties are also provided

Keywords:

• Efficiency
• Minimality
• Maximality
• Dual problems
• Vector optimization