Recent Developments in Vector Optimization
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Duality in Vector Optimization with Infimum and Supremum Andreas L¨ohne
3.1 Introduction We consider the vector optimization problem (VOP) minimize f : X → Y with respect to ≤ over S ⊂ X . The extended partially ordered vector space Y , ≤ is regarded to be a subset of the complete lattice (I , ) which is defined as follows: The convex and pointed ordering cone C that induces the partial ordering ≤ on Y is assumed to satisfy 0/ = intC = Y . The infimal set of a set A ⊂ Y is defined by Inf A := wMin cl (A + C), where some additional considerations with respect to the elements ±∞ will be added later. A set A satisfying A = Inf A is called self-infimal. The family of all self-infimal sets is denoted by I . The space I is equipped with a partial ordering defined by A1 A 2
: ⇐⇒
A1 + intC ⊇ A2 + intC
whenever ±∞ do not occur. If a vector y ∈ Y is identified with its infimal set Inf {y} ∈ I and if the ordering cone C is supposed to be closed, the partially ordered set (I , ) is an extension of the partially ordered set (Y, ≤). If Y is supplemented by ±∞, and I is defined correspondingly, (I , ) is a complete lattice, that is, infimum and supremum of every subset exist. We assign to f an I -valued objective function f :X →I, f (x) := Inf { f (x)} A. L¨ohne () NWF II, Institute of Mathematics, Martin-Luther-University Halle-Wittenberg, 06099 Halle (Saale), Germany e-mail: [email protected] Q.H. Ansari and J.-C. Yao (eds.), Recent Developments in Vector Optimization, Vector Optimization 1, DOI 10.1007/978-3-642-21114-0 3, © Springer-Verlag Berlin Heidelberg 2012
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and consider the related problem minimize f : X → I with respect to over S ⊆ X.
(V )
There is a close connection between the values of f and f ; that is, for all x1 , x2 ∈ X we have f (x1 ) ≤ f (x2 ) ⇐⇒ f (x1 ) f (x2 ). Since the objective space I in Problem (V ) is a complete lattice, the latter correspondence allows us to develop the theory of vector optimization based on infimum and supremum. We consider exemplary Lagrange duality and a finite dimensional variant of conjugate duality in order to demonstrate how the complete lattice (I , ) can be used to transfer scalar duality results into a vectorial framework. Finally, we compare the results with other duality schemes from the literature and point out some advantages. We present here a selection of concepts and results from [17] in order to make the reader familiar with some important ideas of vector optimization with infimum and supremum. For a comprehensive exposition the reader is referred to [17]. This book deals additionally with: • • • • • • • • •
Solution concepts based on the attainment of infimum Continuity and semicontinuity notions for I -valued functions Existence of solutions Saddle point concepts based on infimum and supremum An infinite dimensional variant of conjugate duality Type II duality and existence of solutions to the dual problem Existence of saddle points Solution concepts and duality for linear problems Algorithms for line
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