Vector Optimization with Infimum and Supremum by Andreas Löhne

By Andreas Löhne

The concept of Vector Optimization is constructed through a scientific utilization of infimum and supremum. so one can get life and acceptable houses of the infimum, the picture house of the vector optimization challenge is embedded right into a better house, that is a subset of the facility set, actually, the distance of self-infimal units. in accordance with this concept we determine answer innovations, lifestyles and duality effects and algorithms for the linear case. the most benefit of this strategy is the excessive measure of analogy to corresponding result of Scalar Optimization. The ideas and effects are used to provide an explanation for and to enhance essentially correct algorithms for linear vector optimization problems.

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Proof. Let A ∈ I. 54 yields Sup A = A. It follows −A = − Sup A = Inf(−A) and hence −A ∈ I. Note that the last statement is not true for A ∈ F . Nevertheless, it is sometimes easier to work with the complete lattice F in the proofs. 52. In the following proposition we use a generalization of the Minkowski sum. For A, B ⊆ I, we set A ⊕ B := {I ∈ I| ∃A ∈ A, ∃B ∈ B : I = A ⊕ B} . 56. Let A, B ⊆ I, then (i) (ii) inf A ⊕ B = inf A ⊕ inf B, sup A ⊕ B sup A ⊕ sup B . Proof. (i) If A = ∅, we have inf A ⊕ B = inf A = {+∞} and thus inf A ⊕ B = inf A ⊕ inf B = {+∞}.

Hence we have {f (¯ ¯ for (ii) ⇒ (i). By (ii), f is constant on X. x)} = f [X] ¯ all x¯ ∈ X. 1). 13. 1). 1) for every x ¯ ∈ X. 1) if and only if {¯ x} is a unique solution to (L). Proof. (i) is equivalent to ¯ ⊆S X ∧ inf f (x) = inf f (x) ¯ x∈X x∈S ∧ ¯ = Min f [S]. 12, this is equivalent to ¯: ∀¯ x∈X x ¯∈S ∧ f (¯ x) = inf f (x) x∈S ∧ {f (¯ x)} = Min f [S], which is an alternative way to express (ii). 2 (where a complete lattice Z is obtained by extending R2 by two elements ±∞), Eff (L) is not a solution to (L); whence a solution does not exist.

Let Y be a partially ordered topological vector space with an ordering cone C such that ∅ = int C = Y and let F = FC (Y ), then Fco = {A ⊆ Y | Cl + co A = A} . Proof. We have A = Cl + co A ⇐⇒ A = co A ∧ A = Cl + A ⇐⇒ ∀λ ∈ [0, 1] : A = Cl + λA + (1 − λ)A ⇐⇒ ∀λ ∈ [0, 1] : A = λ A ⊕ (1 − λ) A. 15. 30), we can also work with the convex hull of a subset B ⊆ Y and we obtain a similar characterization of the conlinear space Ico . To this end we need the following result. 60. Let Y be an extended partially ordered topological vector space with an ordering cone C such that ∅ = int C = Y .

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