Inmal sets. Properties and applications

Transcription

1 Inmal sets. Properties and applications Andreas Löhne Martin-Luther-Universität Halle-Wittenberg

2 How to generalize the inmum in R to the vectorial case? A inf A

3 The inmal set (Nieuwenhuis 1980) Inf A = { y R q {y} + int R q + A + int Rq + y A + int Rq + } Inf A Inmal set of A A inf A Inmum of A

4 The inmal set is a generalization of the inmum in R Inf A = { y R q {y} + int R q + A + int Rq + y A + int Rq + } For A R: a A : inf A a inf A A + int R +

5 The inmal set is a generalization of the inmum in R Inf A = { y R q {y} + int R q + A + int Rq + y A + int Rq + } For A R: ( a A : z a) = inf A z z A + int R + = z inf A + int R + inf A + int R + A + int R +

6 Part 1 Properties of inmal sets

7 The setting I: Denition Let Y be a vector space, let τ be a topology. The pair (Y, τ) is called a topological vector space if the following two axioms are satised: (L1) (y 1, y 2 ) y 1 + y 2 is continuous on Y Y into Y, (L2) (λ, y) λy is continuous on R Y into Y. Let be a partial ordering on Y. The triple (Y, τ, ) is called a partially ordered topological vector space if additionally: (O1) (y 1 y 2, y Y ) y 1 + y y 2 + y (O2) (y 1 y 2, α R + ) αy 1 αy 2

8 A basic result: Theorem 1. Let Y be a topological vector space, B Y convex and cl A B. Then, cl A B int A = A B. Proof. Let a cl A B and assume there is some b B \ A. Consider λ := inf{λ 0 λa + (1 λ)b A}. There exists (λ n ) λ such that λ n a + (1 λ n )b A for all n N. As B is convex and λ [0, 1], we get λa + (1 λ)b cl A B int A. In particular λ > 0. On the other hand, there is a sequence (λ n ) λ such that λ n a+(1 λ n )b A for all n N. This yields the contradiction λa + (1 λ)b int A.

9 The setting II: Y := Y {± } is an extended p. o. t. v. s., ordering cone C Y = int C Y (+ ) + ( ) = +

10 A close relative of the inmal set Denition. The upper closure of a subset A Y (with respect to C) is the set Cl + A := Y if A if A = {+ } {y Y {y} + int C A \ {+ } + int C} otherwise. Proposition. It holds Cl + A := Y cl ( A \ {+ } + C ) if A if A = {+ } otherwise.

11 Denition. The set of weakly minimal vectors of a subset A Y (with respect to C) is dened by wmin A := {y A ({y} int C) A = }. Lemma 1. Let A Y be an arbitrary set and let B Y be a convex set. Let Cl + A B and B \ Cl + A. Then, it holds wmin(cl + A B). A B

14 Lemma 1. Let A Y be an arbitrary set and let B Y be a convex set. Let Cl + A B and B \ Cl + A. Then, it holds wmin(cl + A B). Proof. Assuming that wmin(cl + A B) is empty, we get This implies y Cl + A B, z Cl + A B : y {z} + int C. Cl + A B (Cl + A B) + int C (Cl + A + int C) (B + int C) Cl + A + int C int Cl + A. Theorem 1 yields Cl + A B, which contradicts B \ Cl + A.

15 Special case Corollary. For every set A Y the following is equivalent: (i) = Cl + A Y, (ii) wmin Cl + A. Proof. (i) = (ii). Follows from Lemma 1 for the choice B = Y. (ii) = (i). directly from the denition of wmin.

16 Denition of inmal sets The inmal set of A Y (with respect to C) is dened by Inf A := wmin Cl + A if = Cl + A Y { } if Cl + A = Y {+ } if Cl + A =. Inf A A

17 Another result Lemma 2. Let A Y be an arbitrary set and let B Y be an open set. Then, it holds wmin(cl + A B) = (wmin Cl + A) B.

18 A basic result on inmal sets Lemma 3. For A Y with = Cl + A Y it holds Cl + A + int C Inf A + int C. Proof. Let y Cl + A + int C, then ( {y} int C ) Cl + A. We set B := {y} int C. As B is convex and open, Lemma 1 and Lemma 2 imply that = wmin(cl + A B) = (wmin Cl + A) B. Thus, there exists some z wmin Cl + A = Inf A such that y {z} + int C, whence y Inf A + int C.

19 Another basic result on inmal sets Lemma 4. For A Y with = Cl + A Y it holds Cl + A (Inf A int C) = Y. Proof. Let y Y \ Cl + A. The set B := {y} + int C is open and convex. We have Cl + A B (otherwise the contradiction y Cl + A int C = Y ). Moreover, B \ Cl + A (otherwise B Cl + A and hence the contradiction y cl B Cl + A). Lemma 1 and Lemma 2 imply = wmin(cl + A B) = (wmin Cl + A) B. Thus there is some z wmin Cl + A = Inf A such that z {y} int C, whence y Inf A int C.

20 Properties of inmal sets: Let A, B Y, = Cl + A Y, = Cl + B Y, then: Cl + A + int C = Inf A + int C, Inf A = { y Y {y} + int C Cl + A + int C y Cl + A + int C }, int Cl + A = Cl + A + int C, Inf A = bd Cl + A, Inf A = Cl + A \ ( Cl + A + int C ), Cl + A = Cl + B Inf A = Inf B, Cl + A = Cl + B Cl + A + int C = Cl + B + int C, Inf A = Inf B Inf A + int C = Inf B + int C, Cl + A = Inf A (Inf A + int C), Inf A, (Inf A int C) and (Inf A + int C) are disjoint, Inf A (Inf A int C) (Inf A + int C) = Y.

31 Further properties of inmal sets: Let A Y, then: Inf Inf A = Inf A, Cl + Cl + A = Cl + A, Inf Cl + A = Inf A, Cl + Inf A = Cl + A, Inf(Inf A + Inf B) = Inf(A + B), α Inf A = Inf(αA) for α > 0.

35 Part 2 Applications

36 Applications The concept of inmal sets is useful in vector optimization Nieuwenhuis (1980), Tanino (1988, 1992),... : mmmmmmmmmm n inmal sets used for duality results in vector optimization Hamel, Heyde, L. and Tammer ( ): mmmmmmmmmm n set-valued approach to vector optimization: space I of self-inmal subsets of Y shown to be a complete lattice Y isomorphic to a subspace of I mn vector optimization theory with inmum/supremum possible results: solution concepts and existence, duality theory, algorithms advantage: high degree of anaology to the scalar theory

41 Applications The hyperspace I of self-inmal sets B Y is called self-inmal if Inf B = B I... the space of all self-inmal subsets of Y addition: B 1 B 2 := Inf(B 1 + B 2 ), multiplication: α B := Inf(α B) ordering: B 1 B 2 : Cl + B 1 Cl + B 2.

42 Applications addition in I ordering relation in I A 2 A 1 A 2 A 1 A 3 A 2 A 1 A 1 A 2 A 1 A 3 A 2 A 3

43 2. Applications Inmum and supremum in I Theorem 2. I is a complete lattice. For nonempty sets A I we have inf A = Inf A A A, sup A = Sup A A A. A I sup A inf A

44 2. Applications Embedding of the (VOP) Let X be a set and A X. (VOP) Minimize p : X Y with respect to C over A. p : X I, p(x) := Inf { p(x)} = { p(x)} + bd C (IVP) Minimize p : X I with respect to over A.

45 2. Applications Example: The conjugate of an I-valued function Let X, X be a dual pairing, f : X I and c int C: f : X I, f (x ) := sup x X { x, x {c} f(x)}, Note that A I A I.

46 2. Applications Example: Conjugate duality Let f : R n I, g : R m I, A R m n, c int C primal objective: p : R n I, p(x) = f(x) g(ax) dual objective: d : R m I, d(u) = f (A T u) g ( u) (P) (D) P := inf x R n p (x) D := sup u R m d(u) Duality theorem. (P) and (D) satisfy the weak duality inequality D P. If f and g are convex, 0 ri (dom g A dom f), then we have strong duality, i.e., D = P.

47 2. Applications Vector optimization with inmum and supremum What else works? solution concepts existence od solutions vectorial saddle points existence of saddle points Langrange duality Attainment of the solution of the dual problem symmetric linear theory duality interpretation of cutting plane mehtods and dual methods

48 Choice of literature: Nieuwenhuis: Supremal points and generalized duality, Math. Operationsforsch. Stat., Ser. Optimization, 1980 L.; Tammer: A new approach to duality in vector optimization, Optimization, 2007 Heyde; L.; Tammer: Set-valued duality theory for multiple objective linear programs and application to mathematical nance, Math. Meth. Operations Research, 2009 Heyde; L.: Geometric duality in multiple objective linear programming, SIAM J. Optim., 2008 Hamel: A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory, manuscript Heyde; L.: Solution concepts in vector optimization. A fresh look at an old story, submitted to Optimization. L.: Vector optimization with inmum and supremum, habilitation thesis, October 2009 (I hope so ;-)