Vector Optimization: An Introduction and Some

Vector Optimization: An Introduction and Some Recent Problems Majid Soleimani-damaneh University of Tehran & IPM

Frontiers in Mathematical Sciences: 3rd conference IPM, Tehran, December 31, 2014

.

Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

.

.

.

.

December 31, 2014

.

1 / 46

Outline

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming Minimals: Infinite dimensional spaces

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming Minimals: Infinite dimensional spaces Minimals: Finite dimensional spaces

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming Minimals: Infinite dimensional spaces Minimals: Finite dimensional spaces Vector optimization/Multiple Objective Programming; * An application: Performance Analysis

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming Minimals: Infinite dimensional spaces Minimals: Finite dimensional spaces Vector optimization/Multiple Objective Programming; * An application: Performance Analysis Scalarization

.


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

Outline History Multiple Objective Programming Minimals: Infinite dimensional spaces Minimals: Finite dimensional spaces Vector optimization/Multiple Objective Programming; * An application: Performance Analysis Scalarization Some recent issues: * Proper efficiency, * VOP without topology, * Nonsmooth Optimization, * More problems .


Vector Optimization

.

.

.

.

December 31, 2014

.

2 / 46

History

Edgeworth (1881): For the multi-utility problem within the context of two consumer criteria, A and B: “It is required to find a point Francis Y. Edgeworth (1845-1926): (x, y ) such that in whatever direction In 1881 at King’s College (London) we take an infinitely small step, A and later at Oxford, economics and B do not increase together but Professor. that, while one increases, the other decreases.”. . . . . . Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

December 31, 2014

3 / 46

History

(Pareto 1906): “The optimum allocation of the resources of a society is not attained so long as it is possible to make at least one individual better off in his own estimation while keeping others as well off as before in their own estimation.” Vilfredo Pareto (1848-1923): University of Lausanne, Switzerland.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

4 / 46

Multiple Objective Programming (MOP)

Multiple Objective Programming (MOP): Min{f (x) : x ∈ E },

(MOP)

where f (x) = (f1 (x), f2 (x), . . . , fp (x))T , where E ⊆ Rn is a nonempty set; and f : Rn −→ Rp .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

5 / 46



(MOP)

where f (x) = (f1 (x), f2 (x), . . . , fp (x))T , where E ⊆ Rn is a nonempty set; and f : Rn −→ Rp . Binary MOP, Discrete MOP.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

5 / 46



(MOP)

where f (x) = (f1 (x), f2 (x), . . . , fp (x))T , where E ⊆ Rn is a nonempty set; and f : Rn −→ Rp . Binary MOP, Discrete MOP. . Definition . x ∗ ∈ E is called a Pareto (efficient) solution to MOP if ∄x o ∈ E such that fj (x o ) ≤ fj (x ∗ ) for each j = 1, 2, . . . , p, .

fj (x o ) < fj (x ∗ ) for some j = 1, 2, . . . , p.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

5 / 46



(MOP)

where f (x) = (f1 (x), f2 (x), . . . , fp (x))T , where E ⊆ Rn is a nonempty set; and f : Rn −→ Rp . Binary MOP, Discrete MOP. . Definition . x ∗ ∈ E is called a Pareto (efficient) solution to MOP if ∄x o ∈ E such that fj (x o ) ≤ fj (x ∗ ) for each j = 1, 2, . . . , p, fj (x o ) < fj (x ∗ ) for some j = 1, 2, . . . , p. . . Definition . x ∗ ∈ E is called a weak Pareto (weak efficient) solution to MOP if o ∈ E such that f (x o ) < f (x ∗ ) for each j = 1, 2, . . . , p. ∄x j j . .


Vector Optimization

.

.

.

.

December 31, 2014

.

5 / 46


[1] Stadler, W., A Survey of Multicriteria Optimization, or the Vector Maximum Problem, Journal of Optimization Theory and Applications, Vol. 29, pp. 1-52, 1979.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

6 / 46


[1] Stadler, W., A Survey of Multicriteria Optimization, or the Vector Maximum Problem, Journal of Optimization Theory and Applications, Vol. 29, pp. 1-52, 1979. [2] Steuer, Ralph, Multiple Criteria Optimization: Theory, Computation and Application, 1985.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

6 / 46


[1] Stadler, W., A Survey of Multicriteria Optimization, or the Vector Maximum Problem, Journal of Optimization Theory and Applications, Vol. 29, pp. 1-52, 1979. [2] Steuer, Ralph, Multiple Criteria Optimization: Theory, Computation and Application, 1985. [3] Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of Multiobjective Optimization, Vol. 176, Mathematics in Science and Engineering, London, 1985.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

6 / 46

Minimals: Finite dimensional spaces

.


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


. Definition . Let Y ⊂ Rp be nonempty. y¯ ∈ Y is called a minimal (Pareto (efficient) point) of Y if ∄y ∈ Y s.t. y ≤ y¯ & y ̸= y¯ . .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


. Definition . Let Y ⊂ Rp be nonempty. y¯ ∈ Y is called a minimal (Pareto (efficient) point) of Y if ∄y ∈ Y s.t. y ≤ y¯ & y ̸= y¯ . . ¶y ≤ y¯ means yj ≤ y¯j for each j. ¶ The set of all minimals of Y is denoted by YN , ¶ y¯ − R≧p = {y : y ≤ y¯ }.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


. Definition . Let Y ⊂ Rp be nonempty. y¯ ∈ Y is called a minimal (Pareto (efficient) point) of Y if ∄y ∈ Y s.t. y ≤ y¯ & y ̸= y¯ . . ¶y ≤ y¯ means yj ≤ y¯j for each j. ¶ The set of all minimals of Y is denoted by YN , ¶ y¯ − R≧p = {y : y ≤ y¯ }. Remark: It is clear that, x 0 is a Pareto solution to MOP if and only if f (x 0 ) ∈ (f (E ))N .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


. Definition . Let Y ⊂ Rp be nonempty. y¯ ∈ Y is called a minimal (Pareto (efficient) point) of Y if ∄y ∈ Y s.t. y ≤ y¯ & y ̸= y¯ . . ¶y ≤ y¯ means yj ≤ y¯j for each j. ¶ The set of all minimals of Y is denoted by YN , ¶ y¯ − R≧p = {y : y ≤ y¯ }. Remark: It is clear that, x 0 is a Pareto solution to MOP if and only if f (x 0 ) ∈ (f (E ))N . Question: Under what conditions, YN is nonempty

.


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


. Definition . Let Y ⊂ Rp be nonempty. y¯ ∈ Y is called a minimal (Pareto (efficient) point) of Y if ∄y ∈ Y s.t. y ≤ y¯ & y ̸= y¯ . . ¶y ≤ y¯ means yj ≤ y¯j for each j. ¶ The set of all minimals of Y is denoted by YN , ¶ y¯ − R≧p = {y : y ≤ y¯ }. Remark: It is clear that, x 0 is a Pareto solution to MOP if and only if f (x 0 ) ∈ (f (E ))N . Question: Under what conditions, YN is nonempty . Theorem . If . Y is compact, then YN ̸= ∅. .


Vector Optimization

.

.

.

.

December 31, 2014

.

7 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

8 / 46


. Definition . We say that Y has a compact section if there exists y¯ ∈ Y such that Y ∩ (¯ y − R≧p ) is compact. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

8 / 46


. Definition . We say that Y has a compact section if there exists y¯ ∈ Y such that Y ∩ (¯ y − R≧p ) is compact. . . Definition . Y is called R≧p -compact if Y ∩ (¯ y − R≧p ) is compact for each y¯ ∈ Y . .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

8 / 46


. Definition . We say that Y has a compact section if there exists y¯ ∈ Y such that Y ∩ (¯ y − R≧p ) is compact. . . Definition . Y is called R≧p -compact if Y ∩ (¯ y − R≧p ) is compact for each y¯ ∈ Y . . . Theorem . If . Y has a compact section, then YN ̸= ∅.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

8 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

9 / 46


. Definition . Y is called R≧p -semicompact if every open cover of Y in the form Y ⊆ ∪i∈I (y i − R≧p )c , has . a finite subcover.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

9 / 46


. Definition . Y is called R≧p -semicompact if every open cover of Y in the form Y ⊆ ∪i∈I (y i − R≧p )c , has . a finite subcover. . Theorem . If Y is R≧p -compact, then it is R≧p -semicompact. The converse does not hold necessarily. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

9 / 46


. Definition . Y is called R≧p -semicompact if every open cover of Y in the form Y ⊆ ∪i∈I (y i − R≧p )c , has . a finite subcover. . Theorem . If Y is R≧p -compact, then it is R≧p -semicompact. The converse does not hold necessarily. . . Theorem . If Y is R≧p -semicompact, then YN ̸= ∅. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

9 / 46


. Definition . Y is called R≧p -semicompact if every open cover of Y in the form Y ⊆ ∪i∈I (y i − R≧p )c , has . a finite subcover. . Theorem . If Y is R≧p -compact, then it is R≧p -semicompact. The converse does not hold necessarily. . . Theorem . If Y is R≧p -semicompact, then YN ̸= ∅. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

9 / 46


. Definition . Y is called R≧p -semicompact if every open cover of Y in the form Y ⊆ ∪i∈I (y i − R≧p )c , has . a finite subcover. . Theorem . If Y is R≧p -compact, then it is R≧p -semicompact. The converse does not hold necessarily. . . Theorem . If Y is R≧p -semicompact, then YN ̸= ∅. . . Theorem . E is compact &fi ’s are l.s.c =⇒ Y = f (E ) is R≧p -semicompact =⇒ YN ̸= ∅ =⇒ XE ̸= ∅. . . . . . Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

.

December 31, 2014

.

9 / 46


M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005. R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York (1986). D.T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 319, Springer-Verlag, New york, Berlin, (1989).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

10 / 46

Minimals: Infinite dimensional spaces

Let X be a nonempty set.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

11 / 46


Let X be a nonempty set. R ⊆ X × X is called a partial order if it is □ Reflexive: (x, x) ∈ R, ∀x ∈ X □ Antisymmetric: (x, y ) ∈ R, (y , x) ∈ R =⇒ x = y , □ Transitive: (x, y ) ∈ R, (y , z) ∈ R =⇒ (x, z) ∈ R.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

11 / 46


Let X be a nonempty set. R ⊆ X × X is called a partial order if it is □ Reflexive: (x, x) ∈ R, ∀x ∈ X □ Antisymmetric: (x, y ) ∈ R, (y , x) ∈ R =⇒ x = y , □ Transitive: (x, y ) ∈ R, (y , z) ∈ R =⇒ (x, z) ∈ R. Notation: ⪯ is used for denoting a partial order, i.e. (x, y ) ∈ R ⇐⇒ x ⪯ y .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

11 / 46



. Definition . Let (X , ⪯) be a partially ordered set and Y ⊆ X . y ∗ ∈ Y is called a minimal of Y if y ∈ Y , y ⪯ y ∗ =⇒ y = y ∗ . . .


Vector Optimization

.

.

.

.

December 31, 2014

.

11 / 46



. Definition . Let (X , ⪯) be a partially ordered set and Y ⊆ X . y ∗ ∈ Y is called a minimal of Y if y ∈ Y , y ⪯ y ∗ =⇒ y = y ∗ . . ¶ The set of all minimals of Y is denoted by YN.. Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

.

.

.

.

December 31, 2014

.

11 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

12 / 46


. Definition . Let (X , ⪯) be a partially ordered real vector space. C ⊆ X is called a cone if λC ⊆ C for each λ ≥ 0. A cone C is convex if C + C ⊆ C . Moreover, a cone C is called pointed if C . ∩ (−C ) = {0}.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

12 / 46


. Definition . Let (X , ⪯) be a partially ordered real vector space. C ⊆ X is called a cone if λC ⊆ C for each λ ≥ 0. A cone C is convex if C + C ⊆ C . Moreover, a cone C is called pointed if C . ∩ (−C ) = {0}. . Theorem . Let (X , ⪯) be a partially ordered real vector space; then C := {x ∈ X : x ⪰ 0} is a convex pointed cone. Conversely, if D ⊆ X is a pointed convex cone, then ⪯:= {(x, y ) ∈ X × X : y − x ∈ D} is . a partial order.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

12 / 46


. Definition . Let (X , ⪯) be a partially ordered real vector space. C ⊆ X is called a cone if λC ⊆ C for each λ ≥ 0. A cone C is convex if C + C ⊆ C . Moreover, a cone C is called pointed if C . ∩ (−C ) = {0}. . Theorem . Let (X , ⪯) be a partially ordered real vector space; then C := {x ∈ X : x ⪰ 0} is a convex pointed cone. Conversely, if D ⊆ X is a pointed convex cone, then ⪯:= {(x, y ) ∈ X × X : y − x ∈ D} is . a partial order. Due to the above theorem, hereafter X is a real vector space which is ordered by a convex pointed cone C , i.e. x ⪯ y =⇒ y. − x. ∈ C. . . . Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

December 31, 2014

.

12 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C . y. ∗ ∈ Y ⊆ X is called a minimal of Y w.r.t C if (y ∗ − C ) ∩ (Y ) = {y ∗ }.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C . y. ∗ ∈ Y ⊆ X is called a minimal of Y w.r.t C if (y ∗ − C ) ∩ (Y ) = {y ∗ }. The set of minimals of Y w.r.t C is denoted by E (Y , C ).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C . y. ∗ ∈ Y ⊆ X is called a minimal of Y w.r.t C if (y ∗ − C ) ∩ (Y ) = {y ∗ }. The set of minimals of Y w.r.t C is denoted by E (Y , C ). If X = R n and C = {x : x ≥ 0}, then minimal points are called Pareto (efficient) points.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C . y. ∗ ∈ Y ⊆ X is called a minimal of Y w.r.t C if (y ∗ − C ) ∩ (Y ) = {y ∗ }. The set of minimals of Y w.r.t C is denoted by E (Y , C ). If X = R n and C = {x : x ≥ 0}, then minimal points are called Pareto (efficient) points. A section of Y at y ∈ Y is defined by Yy := Y ∩ (y − C ).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C . y. ∗ ∈ Y ⊆ X is called a minimal of Y w.r.t C if (y ∗ − C ) ∩ (Y ) = {y ∗ }. The set of minimals of Y w.r.t C is denoted by E (Y , C ). If X = R n and C = {x : x ≥ 0}, then minimal points are called Pareto (efficient) points. A section of Y at y ∈ Y is defined by Yy := Y ∩ (y − C ). . Lemma . Let X be a real vector space partially ordered by convex pointed cone C ; If y ∈ Y ⊆ X , then i. E (Yy , C ) ⊆ E (Y , C ), ii. . E (Y , C ) = E (Y + C , C ). .


Vector Optimization

.

.

.

.

December 31, 2014

.

13 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C ; and S ⊆ X . y is called a lower bound of S (w.r.t C ) if S ⊆ y + C . .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

14 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C ; and S ⊆ X . y is called a lower bound of S (w.r.t C ) if S ⊆ y + C . . . Definition . Let X be a real vector space partially ordered by convex pointed cone C . S ⊆ X is called inductively ordered if each decreasing net (w.r.t C ) in S .has a lower bound (w.r.t C ) in S.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

14 / 46


. Definition . Let X be a real vector space partially ordered by convex pointed cone C ; and S ⊆ X . y is called a lower bound of S (w.r.t C ) if S ⊆ y + C . . . Definition . Let X be a real vector space partially ordered by convex pointed cone C . S ⊆ X is called inductively ordered if each decreasing net (w.r.t C ) in S .has a lower bound (w.r.t C ) in S. . Theorem . Existence: Let X be a real vector space partially ordered by convex pointed cone C ; and Y ⊆ X be nonempty. Then .E (Y , C ) ̸= ∅ if and only if Y has an inductively ordered section.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

14 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .It is decreasing w.r.t C with (0, 0) as a lower bound. Contradiction: (a, b) = inf xn .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .It is decreasing w.r.t C with (0, 0) as a lower bound. Contradiction: (a, b) = inf xn .If a, b > 0, then ∃n; xn ⪯ (a, b), which makes a contradiction. .


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .It is decreasing w.r.t C with (0, 0) as a lower bound. Contradiction: (a, b) = inf xn .If a, b > 0, then ∃n; xn ⪯ (a, b), which makes a contradiction.(0, 0) and (−1, 0) are two lowerbounds for xn . .


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .It is decreasing w.r.t C with (0, 0) as a lower bound. Contradiction: (a, b) = inf xn .If a, b > 0, then ∃n; xn ⪯ (a, b), which makes a contradiction.(0, 0) and (−1, 0) are two lowerbounds for xn . So, (0, 0) ⪯ (a, b) =⇒ (a, b) = (0, 0). .


Vector Optimization

.

.

.

.

December 31, 2014

.

15 / 46


Let X be a TVS. . Definition . The ordering cone C has the Daniell property if the infimum of each decreasing net (w.r.t C ) in C exists and is also the topological limit of the .net. . Definition . The ordering cone C is boundedly order complete if each decreasing bounded net (w.r.t topology) has infimum (w.r.t C ) in X . . Remark. Let X = R2 and C = {x ∈ R2 : x > 0} ∪ {0}. Hence, (x1 , x2 ) ⪯ (y1 , y2 ) ⇐⇒ (x1 , x2 ) = (y1 , y2 ) or (x1 , x2 ) < (y1 , y2 ). Consider xn = {( n1 , n1 )}n∈N .It is decreasing w.r.t C with (0, 0) as a lower bound. Contradiction: (a, b) = inf xn .If a, b > 0, then ∃n; xn ⪯ (a, b), which makes a contradiction.(0, 0) and (−1, 0) are two lowerbounds for xn . So, (0, 0) ⪯ (a, b) =⇒ (a, b) = (0, 0).Hence (−1, 0) ⪯ (a, b) =⇒ (−1, 0) ⪯ (0, 0) which is a contradiction. . . . . . . Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

December 31, 2014

15 / 46


.


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46


. Theorem . Let X be a real vector space partially ordered by convex pointed cone C , and Y ⊆ X . Then under each of the following assumptions, E (Y , C ) ̸= ∅.

.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46


. Theorem . Let X be a real vector space partially ordered by convex pointed cone C , and Y ⊆ X . Then under each of the following assumptions, E (Y , C ) ̸= ∅. i. C has Daniell property and Y has a closed and bounded section (boundedness is wrt to the order induced by C ).

.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46


. Theorem . Let X be a real vector space partially ordered by convex pointed cone C , and Y ⊆ X . Then under each of the following assumptions, E (Y , C ) ̸= ∅. i. C has Daniell property and Y has a closed and bounded section (boundedness is wrt to the order induced by C ). ii. C has Daniell property, C is boundedly order complete, and Y has a closed and bounded section (boundedness is wrt to the topology). .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46


. Theorem . Let X be a real vector space partially ordered by convex pointed cone C , and Y ⊆ X . Then under each of the following assumptions, E (Y , C ) ̸= ∅. i. C has Daniell property and Y has a closed and bounded section (boundedness is wrt to the order induced by C ). ii. C has Daniell property, C is boundedly order complete, and Y has a closed and bounded section (boundedness is wrt to the topology). iii. Y has a compact section. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46


. Theorem . Let X be a real vector space partially ordered by convex pointed cone C , and Y ⊆ X . Then under each of the following assumptions, E (Y , C ) ̸= ∅. i. C has Daniell property and Y has a closed and bounded section (boundedness is wrt to the order induced by C ). ii. C has Daniell property, C is boundedly order complete, and Y has a closed and bounded section (boundedness is wrt to the topology). iii. Y has a compact section. . J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993) 105-122. .


Vector Optimization

.

.

.

.

December 31, 2014

.

16 / 46

Vector optimization/Multiple Objective Programming

General problem: X , Z are two real linear vector spaces. Z is partially ordered by nontrivial ordering convex cone C . The VOP: C − Min{f (x) : x ∈ E },

(VOP)

where E ⊆ X is a nonempty set; and f : X −→ Z .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

17 / 46


General problem: X , Z are two real linear vector spaces. Z is partially ordered by nontrivial ordering convex cone C . The VOP: C − Min{f (x) : x ∈ E },

(VOP)

where E ⊆ X is a nonempty set; and f : X −→ Z . Special case: Multiple Objective Programming (MOP): X = Rn , Z = Rm , and C = Rm ≧. The MOP: Min{f (x) : x ∈ E },

(MOP)

where f (x) = (f1 (x), f2 (x), . . . , fm (x))T .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

17 / 46


An Application: Performance Analysis

Performance Analysis Assume that there are n peer Decision Making units (DMUs).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0. Production Possibility Set (PPS): {(X , Y ) ∈ Rm × Rs : Y can be produced by X }.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0. Production Possibility Set (PPS): {(X , Y ) ∈ Rm × Rs : Y can be produced by X }. Let DMUo (Xo , Yo ) be under evaluation.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0. Production Possibility Set (PPS): {(X , Y ) ∈ Rm × Rs : Y can be produced by X }. Let DMUo (Xo , Yo ) be under evaluation. φo = max{φ : (Xo , φYo ) ∈ PPS}.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0. Production Possibility Set (PPS): {(X , Y ) ∈ Rm × Rs : Y can be produced by X }. Let DMUo (Xo , Yo ) be under evaluation. φo = max{φ : (Xo , φYo ) ∈ PPS}. Axioms: 1. Observations: (Xj , Yj ) ∈ PPS, ∀j = 1, 2, . . . , n. 2. Possibility: ¯ ≥ X , 0 ≤ Y¯ ≤ Y =⇒ (X ¯ , Y¯ ) ∈ PPS. (X , Y ) ∈ PPS, X 3. Unbounded ray: (X , Y ) ∈ PPS, λ ≥ 0 =⇒ (λX , λY ) ∈ PPS. 4. Convexity: PPS is a convex set.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



Performance Analysis Assume that there are n peer Decision Making units (DMUs). DMUj : Xj 99K Yj ; Xj ∈ Rm , Yj ∈ Rs , Xj > 0, Yj > 0. Production Possibility Set (PPS): {(X , Y ) ∈ Rm × Rs : Y can be produced by X }. Let DMUo (Xo , Yo ) be under evaluation. φo = max{φ : (Xo , φYo ) ∈ PPS}. Axioms: 1. Observations: (Xj , Yj ) ∈ PPS, ∀j = 1, 2, . . . , n. 2. Possibility: ¯ ≥ X , 0 ≤ Y¯ ≤ Y =⇒ (X ¯ , Y¯ ) ∈ PPS. (X , Y ) ∈ PPS, X 3. Unbounded ray: (X , Y ) ∈ PPS, λ ≥ 0 =⇒ (λX , λY ) ∈ PPS. 4. Convexity: PPS is a convex set. . Theorem . The minimal∑set satisfying axioms ∑n 1-4 is n {(X , Y ) : µ X ≤ X , j=1 j j j=1 µj Yj ≥ Y ≥ 0, µ ≥ 0}. . .


Vector Optimization

.

.

.

.

December 31, 2014

.

18 / 46



φo = max φ ∑n s.t. µj Xj ≤ Xo , ∑j=1 n j=1 µj Yj ≥ φYo , µj ≥ 0, j = 1, 2, . . . , n.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

19 / 46



φo = max φ ∑n s.t. µj Xj ≤ Xo , ∑j=1 n j=1 µj Yj ≥ φYo , µj ≥ 0, j = 1, 2, . . . , n. Question 1. If the efficiency index φo remains unchanged, but the inputs increase, how much should the outputs of DMUo increase

.


Vector Optimization

.

.

.

.

December 31, 2014

.

19 / 46



Output estimation

.


Vector Optimization

.

.

.

.

December 31, 2014

.

20 / 46



Output estimation Suppose that the inputs of DMUo are increased from Xo to αo = Xo + ∆Xo ; ∆Xo ≥ 0, ∆Xo ̸= 0.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

20 / 46



Output estimation Suppose that the inputs of DMUo are increased from Xo to αo = Xo + ∆Xo ; ∆Xo ≥ 0, ∆Xo ̸= 0. The aim: estimating the output vector βo∗ provided that the efficiency index of DMUo is still φo . In fact, ∗ ∗ ∗ t βo∗ = (β1o , β2o , ..., βmo ) = Yo + ∆Yo ; ∆Yo ≥ 0.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

20 / 46




Suppose DMUn+1 represents DMUo after changing the inputs and outputs.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

20 / 46




Suppose DMUn+1 represents DMUo after changing the inputs and outputs. φn+1 = max φ ∑n s.t. µ X + µn+1 αo ≤ αo , ∑nj=1 j j ∗ ∗ µ j=1 j Yj + µn+1 βo ≥ φβo , µj ≥ 0, j = 1, 2, . . . , n + 1. .


Vector Optimization

.

.

.

.

December 31, 2014

.

20 / 46



Output estimation φo = max φ ∑n s.t. µj Xj ≤ Xo , ∑j=1 n j=1 µj Yj ≥ φYo , µj ≥ 0, j = 1, 2, . . . , n. φn+1 = max φ ∑n s.t. µ X + µn+1 αo ≤ αo , ∑nj=1 j j ∗ ∗ j=1 µj Yj + µn+1 βo ≥ φβo , µj ≥ 0, j = 1, 2, . . . , n + 1.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

21 / 46



Output estimation φo = max φ ∑n s.t. µj Xj ≤ Xo , ∑j=1 n j=1 µj Yj ≥ φYo , µj ≥ 0, j = 1, 2, . . . , n. φn+1 = max φ ∑n s.t. µ X + µn+1 αo ≤ αo , ∑nj=1 j j ∗ ∗ j=1 µj Yj + µn+1 βo ≥ φβo , µj ≥ 0, j = 1, 2, . . . , n + 1. Aim: φo = φn+1 .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

21 / 46



Output estimation φo = max φ ∑n s.t. µj Xj ≤ Xo , ∑j=1 n j=1 µj Yj ≥ φYo , µj ≥ 0, j = 1, 2, . . . , n. φn+1 = max φ ∑n s.t. µ X + µn+1 αo ≤ αo , ∑nj=1 j j ∗ ∗ j=1 µj Yj + µn+1 βo ≥ φβo , µj ≥ 0, j = 1, 2, . . . , n + 1. Aim: φo = φn+1 . MOLP (I ) :

max (β , β2o , ..., βso ) ∑1o n s.t. µ X ≤ αo , ∑nj=1 j j µ j=1 j Yj ≥ φo βo , βo ≥ Yo , µj ≥ 0, j = 1, 2, . . . , n. .


Vector Optimization

.

.

.

.

December 31, 2014

.

21 / 46



Output estimation

. Theorem . Suppose that φo > 1 and the inputs for DMUo are going to increase from Xo to αo = Xo + ∆Xo , where ∆Xo ≥ 0 and ∆Xo ̸= 0. (i) Let (µ∗ , βo∗ ) be a Weak Pareto solution of MOLP (I). Then, when the outputs of DMUo are increased to βo∗ we have φ(αo , βo∗ ) = φ(Xo , Yo ). (ii) Conversely, let (µ∗ , βo∗ ) be a feasible solution of MOLP (I). If φ(αo , βo∗ ) = φ(Xo , Yo ), then (µ∗ , βo∗ ) is a Weak Pareto solution to MOLP (I). .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

22 / 46



References

A. Hadi-Vencheh, A.A. Foroughi, M. Soleimani-damaneh, A DEA model for resource allocation. Economic Modelling 25 (2008), pp. 983-993. M. Soleimani-damaneh, P.J. Korhonen, J. Wallenius, On value efficiency, Optimization 63 (2014) 617-631. Q.L. Wei, J. Zhang, X. Zhang, An inverse DEA model for input/output estimate. European Journal of Operational Research 121 (1) (2000), pp. 151-163.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

23 / 46

Scalarization

Weight Sum ϵ-constraint Pascoletti-Serafini NBI method Nonlinear scalarization (Gerth function)

.


Vector Optimization

.

.

.

.

December 31, 2014

.

24 / 46

Scalarization

. Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp > 0, then x ∗ is . a Pareto solution of MOP.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

25 / 46

Scalarization

. Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp > 0, then x ∗ is . a Pareto solution of MOP. . Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp ≥ 0, (not all ∗ is a weak Pareto solution of MOP. zero) then x .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

25 / 46

Scalarization

. Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp > 0, then x ∗ is . a Pareto solution of MOP. . Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp ≥ 0, (not all ∗ is a weak Pareto solution of MOP. zero) then x . . Theorem . ∗ Let f be a convex function and E be a convex ∑ set. If x is a weak Pareto ∗ solution of MOP, then x is a minimizer of λj fj (x) over E for some λ , . . . , λ ≥ 0 (not all zero). p .1

.


Vector Optimization

.

.

.

.

December 31, 2014

.

25 / 46

Some recent issues

Proper efficiency

Proper efficiency is one of the most important solution concepts in multiple-objective programming.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

26 / 46

Some recent issues

Proper efficiency

Proper efficiency is one of the most important solution concepts in multiple-objective programming. Properly efficient solutions are efficient solutions in which, given any objective, the trade-off between that objective and some other objective is bounded.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

26 / 46

Some recent issues

Proper efficiency

Proper efficiency is one of the most important solution concepts in multiple-objective programming. Properly efficient solutions are efficient solutions in which, given any objective, the trade-off between that objective and some other objective is bounded. This notion was dealt with initially by Kuhn and Tucker (1951) and was precised thereafter by Geoffrion (1968) for multiple-objective optimization problems (MOPs) in finite dimensional Euclidean spaces with natural ordering cone.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

26 / 46

Some recent issues

Proper efficiency

Proper efficiency is one of the most important solution concepts in multiple-objective programming. Properly efficient solutions are efficient solutions in which, given any objective, the trade-off between that objective and some other objective is bounded. This notion was dealt with initially by Kuhn and Tucker (1951) and was precised thereafter by Geoffrion (1968) for multiple-objective optimization problems (MOPs) in finite dimensional Euclidean spaces with natural ordering cone. For MOPs with unnatural ordering cones, the definition of proper efficiency has been extended by Benson (1979), Borwein (1977), and Henig (1982).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

26 / 46

Some recent issues

Proper efficiency

. Definition . A feasible solution xˆ ∈ X is called properly efficient solution of MOP in Geoffrion’s sense, if it is efficient and there is a real number M > 0 such that for all i ∈ {1, 2, ..., p} and x ∈ X satisfying fi (x) < fi (ˆ x ) there exists an index j ∈ {1, 2, ..., p} such that fj (x) > fj (ˆ x ) and

.

fi (ˆ x ) − fi (x) ≤ M. fj (x) − fj (ˆ x)

.


Vector Optimization

.

.

.

.

December 31, 2014

.

27 / 46

Some recent issues

Proper efficiency

. Definition . A feasible solution xˆ ∈ X is called properly efficient solution of MOP in Geoffrion’s sense, if it is efficient and there is a real number M > 0 such that for all i ∈ {1, 2, ..., p} and x ∈ X satisfying fi (x) < fi (ˆ x ) there exists an index j ∈ {1, 2, ..., p} such that fj (x) > fj (ˆ x ) and fi (ˆ x ) − fi (x) ≤ M. fj (x) − fj (ˆ x)

. . Definition . A feasible solution xˆ ∈ X is called properly efficient solution of MOP in Henig’s sense, if (f (ˆ x ) − C ) ∩ f (X ) = {f (ˆ x )}, for some convex pointed cone C with Rp≧ \{0} ⊆ int(C ). .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

27 / 46

Some recent issues

Proper efficiency


. . Definition . A feasible solution xˆ ∈ X is called properly efficient solution of MOP in Henig’s sense, if (f (ˆ x ) − C ) ∩ f (X ) = {f (ˆ x )}, for some convex pointed cone C with Rp≧ \{0} ⊆ int(C ). . . Theorem . The above two definitions are equivalent for MOP. . .


Vector Optimization

.

.

.

.

December 31, 2014

.

27 / 46

Some recent issues

Proper efficiency


. . Definition . A feasible solution xˆ ∈ X is called properly efficient solution of MOP in Henig’s sense, if (f (ˆ x ) − C ) ∩ f (X ) = {f (ˆ x )}, for some convex pointed cone C with Rp≧ \{0} ⊆ int(C ). . . Theorem . The above two definitions are equivalent for MOP. . Benson definition, Borwein definition, etc. Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

.

.

.

.

.

December 31, 2014

.

27 / 46

Some recent issues

Proper efficiency

. Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp > 0, then x ∗ is . a proper efficient solution of MOP.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

28 / 46

Some recent issues

Proper efficiency

. Theorem . ∑ If x ∗ is a minimizer of λj fj (x) over E for some λ1 , . . . , λp > 0, then x ∗ is . a proper efficient solution of MOP. . Theorem . ∗ Let f be a convex function and E be a convex ∑ set. If x is a proper Pareto ∗ solution of MOP, then x is a minimizer of λj fj (x) over E for some .λ1 , . . . , λp > 0.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

28 / 46

Some recent issues

Proper efficiency

. Definition . A feasible solution xˆ ∈ X is called strongly proper efficient for MOP if it is efficient and there is a real number M > 0 such that for all i ∈ {1, 2, ..., p} x )−fi (x) and x ∈ X satisfying fi (x) < fi (ˆ x ), we have ffji (ˆ (x)−fj (ˆ x ) ≤ M, for all x ). .j ∈ {1, 2, ..., p} with fj (x) > fj (ˆ

.


Vector Optimization

.

.

.

.

December 31, 2014

.

29 / 46

Some recent issues

Proper efficiency

. Definition . A feasible solution xˆ ∈ X is called strongly proper efficient for MOP if it is efficient and there is a real number M > 0 such that for all i ∈ {1, 2, ..., p} x )−fi (x) and x ∈ X satisfying fi (x) < fi (ˆ x ), we have ffji (ˆ (x)−fj (ˆ x ) ≤ M, for all x ). .j ∈ {1, 2, ..., p} with fj (x) > fj (ˆ Questions: Obtaining strongly proper efficient solutions, Defining and characterizing strongly proper efficiency for infinite dimensional case.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

29 / 46

Some recent issues

Proper efficiency

K. Khaledian, E. Khorram, M. Soleimani-damaneh, Strongly proper efficient solutions: efficient solutions with bounded trade-offs. Journal of Optimization Theory and Applications. (Accepted). I. Kaliszewski, Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Dordrecht, 1994. M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

30 / 46

Some recent issues

VOP without topology

When X is a TVS: . Definition . x0 ∈ E is called an efficient solution of (VOP) if (f (E ) − f (x0 )) ∩ (−C \{0}) = ∅. Furthermore, when C is solid, x0 ∈ Ω is called a weak efficient solution of (VOP) if (f (Ω) − f (x0 )) ∩ (−int(C )) = ∅. .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

31 / 46

Some recent issues


When X is a TVS: . Definition . x0 ∈ E is called an efficient solution of (VOP) if (f (E ) − f (x0 )) ∩ (−C \{0}) = ∅. Furthermore, when C is solid, x0 ∈ Ω is called a weak efficient solution of (VOP) if (f (Ω) − f (x0 )) ∩ (−int(C )) = ∅. . Let X be a real vector space, without any particular topology.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

31 / 46

Some recent issues


When X is a TVS: . Definition . x0 ∈ E is called an efficient solution of (VOP) if (f (E ) − f (x0 )) ∩ (−C \{0}) = ∅. Furthermore, when C is solid, x0 ∈ Ω is called a weak efficient solution of (VOP) if (f (Ω) − f (x0 )) ∩ (−int(C )) = ∅. . Let X be a real vector space, without any particular topology. ′

′

′

′

cor (A) = {x ∈ A : ∀x ∈ X , ∃λ > 0; ∀λ ∈ [0, λ ], x + λx ∈ A}.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

31 / 46

Some recent issues



′

′

′

cor (A) = {x ∈ A : ∀x ∈ X , ∃λ > 0; ∀λ ∈ [0, λ ], x + λx ∈ A}. When cor (A) ̸= ∅ we say that A is solid. If cor (K ) ̸= ∅, then cor (K ) ∪ {0} is a convex cone, cor (K ) + K = cor (K ) and cor (cor (K )) = cor (K ).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

31 / 46

Some recent issues



′

′

′

cor (A) = {x ∈ A : ∀x ∈ X , ∃λ > 0; ∀λ ∈ [0, λ ], x + λx ∈ A}. When cor (A) ̸= ∅ we say that A is solid. If cor (K ) ̸= ∅, then cor (K ) ∪ {0} is a convex cone, cor (K ) + K = cor (K ) and cor (cor (K )) = cor (K ). ′

′

vcl(A) = {b ∈ X : ∃x ∈ X ; ∀λ > 0 , ∃λ ∈ [0, λ ] ; b + λx ∈ A}. .


Vector Optimization

.

.

.

.

December 31, 2014

.

31 / 46

Some recent issues


. Theorem . Let M, K be solid nontrivial convex cones in X . If M ∩ cor (K ) = ∅, then ′ there exists a functional l ∈ X \{0} such that, ⟨l, m⟩ ≤ 0 ≤ ⟨l, k⟩

∀(k ∈ K , m ∈ M),

and furthermore, ⟨l, k⟩ > 0 for all k ∈ cor (K ), and ⟨l, m⟩ < 0 for all .m ∈ cor (M).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

32 / 46

Some recent issues


. Theorem . Let M, K be solid nontrivial convex cones in X . If M ∩ cor (K ) = ∅, then ′ there exists a functional l ∈ X \{0} such that, ⟨l, m⟩ ≤ 0 ≤ ⟨l, k⟩

∀(k ∈ K , m ∈ M),

and furthermore, ⟨l, k⟩ > 0 for all k ∈ cor (K ), and ⟨l, m⟩ < 0 for all .m ∈ cor (M). . Theorem . Let M, K be two convex, nontrivial, and vectorially closed cones in X such that M, K are relatively solid and K + is solid. If M ∩ K = {0}, then there exists a functional l ∈ X ′ \{0} such that, ⟨l, k⟩ ≥ 0 ≥ ⟨l, m⟩

∀(k ∈ K , m ∈ M),

and furthermore, .

⟨l, k⟩ > 0

∀k ∈ K \{0}. .


Vector Optimization

.

.

.

.

December 31, 2014

.

32 / 46

Some recent issues


. Theorem . (Alternative theorem) Let K be a nontrivial solid pointed convex cone and let A be a nonempty subset of X . If vcl(cone(A) + K ) is convex, then one and only one of the following alternatives is valid: (i) A ∩ (−cor (K )) ̸= ∅ + + .(ii) A ∩ K ̸= {0}.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

33 / 46

Some recent issues


. Theorem . (Alternative theorem) Let K be a nontrivial solid pointed convex cone and let A be a nonempty subset of X . If vcl(cone(A) + K ) is convex, then one and only one of the following alternatives is valid: (i) A ∩ (−cor (K )) ̸= ∅ + + .(ii) A ∩ K ̸= {0}. M. Adan, V. Novo, Optimality conditions for vector optimization problems with generalized convexity in real linear spaces, Optimization, 51 (2002) 73-91. T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Prog. 122 (2010) 301-347. E. Kiyani, M. Soleimani-damaneh, Approximate proper efficiency on real linear vector spaces. Pacific Journal of Optimization. (Accepted) E. Kiyani, M. Soleimani-damaneh, Algebraic interior and separation on linear vector spaces: Some comments. Journal of Optimization Theory and Applications 161 (2014) 994-998. . . . . . . Majid Soleimani-damaneh (UT & IPM)

Vector Optimization

December 31, 2014

33 / 46

Some recent issues

Nonsmooth Optimization

Consider the following MOP: min{(f1 (x), . . . , fp (x)) : x ∈ E , gj (x) ≤ 0; for all j = 1, 2, . . . , m}, where E is a nonempty open set in X , a real Banach space. X ∗ denotes the topological dual of X equipped with weak∗ topology. fi , gj : X −→ R are real-valued functions.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

34 / 46

Some recent issues


Consider the following MOP: min{(f1 (x), . . . , fp (x)) : x ∈ E , gj (x) ≤ 0; for all j = 1, 2, . . . , m}, where E is a nonempty open set in X , a real Banach space. X ∗ denotes the topological dual of X equipped with weak∗ topology. fi , gj : X −→ R are real-valued functions. A function h : A ⊆ X −→ R is said to be Lipschitz on A if there exists a k ∈ R such that |h(x) − h(y )| ≤ k∥x − y ∥ ∀x, y ∈ A.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

34 / 46

Some recent issues


Consider the following MOP: min{(f1 (x), . . . , fp (x)) : x ∈ E , gj (x) ≤ 0; for all j = 1, 2, . . . , m}, where E is a nonempty open set in X , a real Banach space. X ∗ denotes the topological dual of X equipped with weak∗ topology. fi , gj : X −→ R are real-valued functions. A function h : A ⊆ X −→ R is said to be Lipschitz on A if there exists a k ∈ R such that |h(x) − h(y )| ≤ k∥x − y ∥ ∀x, y ∈ A.

h is said to be Lipschitz near x if it is Lipschitz on a neighborhood of x. Also, h is locally Lipschitz on A if it is Lipschitz near x for every x ∈ A. .


Vector Optimization

.

.

.

.

December 31, 2014

.

34 / 46

Some recent issues


The notion of generalized diff. plays a fundamental role in modern variational analysis and optimization: Clarkes gradients, Mordukhovich’s subdif., Limiting subdif. in Hilbert spaces.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

35 / 46

Some recent issues


The notion of generalized diff. plays a fundamental role in modern variational analysis and optimization: Clarkes gradients, Mordukhovich’s subdif., Limiting subdif. in Hilbert spaces. Consider h as a locally Lipschitz function from X into R.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

35 / 46

Some recent issues


The notion of generalized diff. plays a fundamental role in modern variational analysis and optimization: Clarkes gradients, Mordukhovich’s subdif., Limiting subdif. in Hilbert spaces. Consider h as a locally Lipschitz function from X into R. The Clarke’s generalized directional derivative of h at x¯ in the direction d, denoted by h◦ (¯ x ; d), is defined as h◦ (¯ x ; d) = lim sup (1/t)[h(x + td) − h(x)]. x −→ x¯ t↓0

.


Vector Optimization

.

.

.

.

December 31, 2014

.

35 / 46

Some recent issues


The notion of generalized diff. plays a fundamental role in modern variational analysis and optimization: Clarkes gradients, Mordukhovich’s subdif., Limiting subdif. in Hilbert spaces. Consider h as a locally Lipschitz function from X into R. The Clarke’s generalized directional derivative of h at x¯ in the direction d, denoted by h◦ (¯ x ; d), is defined as h◦ (¯ x ; d) = lim sup (1/t)[h(x + td) − h(x)]. x −→ x¯ t↓0 The Clarke’s generalized gradient of h at x¯ is given by ∂h(¯ x ) = {ξ ∗ ∈ X ∗ : h◦ (¯ x ; d) ≥ ⟨ξ ∗ , d⟩, ∀d ∈ X }, in which X ∗ is the topological dual of X equipped with weak∗ -topology, and ⟨., .⟩ exhibits the duality pairing. .


Vector Optimization

.

.

.

.

December 31, 2014

.

35 / 46

Some recent issues


. Theorem . Let h be Lipschitz near x with Lipschitz constant K , then (i) ∂h(x) is a nonempty, convex, and weak∗ -compact set. (ii) ∥ξ ∗ ∥∗ ≤ K for every ξ ∗ ∈ ∂h(x), where ∥ξ ∗ ∥∗ = sup{⟨ξ ∗ , v ⟩ : v ∈ X , ∥v ∥ ≤ 1}. (iii) Let xi and ξi∗ be sequences in X and X ∗ such that ξi∗ ∈ ∂h(xi ). Suppose that xi converges to x, and that ξ ∗ is a cluster point of ξi∗ in the ∗ ∗ .weak −topology. Then ξ ∈ ∂h(x).

.


Vector Optimization

.

.

.

.

December 31, 2014

.

36 / 46

Some recent issues


. Theorem . Let h be Lipschitz near x with Lipschitz constant K , then (i) ∂h(x) is a nonempty, convex, and weak∗ -compact set. (ii) ∥ξ ∗ ∥∗ ≤ K for every ξ ∗ ∈ ∂h(x), where ∥ξ ∗ ∥∗ = sup{⟨ξ ∗ , v ⟩ : v ∈ X , ∥v ∥ ≤ 1}. (iii) Let xi and ξi∗ be sequences in X and X ∗ such that ξi∗ ∈ ∂h(xi ). Suppose that xi converges to x, and that ξ ∗ is a cluster point of ξi∗ in the ∗ ∗ .weak −topology. Then ξ ∈ ∂h(x). . Theorem . Let x, y ∈ X , and suppose that h is Lipschitz on an open set containing the line segment [x, y ]. Then there exists a point u ∈ (x, y ) such that .

h(y ) − h(x) ∈ ⟨∂h(u), y − x⟩. .


Vector Optimization

.

.

.

.

December 31, 2014

.

36 / 46

Some recent issues


. Theorem . (necessary condition) Let x¯ ∈ S be a feasible solution for (MOP) and I (¯ x ) = {j : gj (¯ x ) = 0}. Suppose that fi for i = 1, 2, . . . , m and gj for j ∈ I (¯ x ) are Lipschitz near x¯ and gj for j ∈ / I (¯ x ) is continuous at x¯. If x¯ is a Weak Pareto solution of (MOP), then there exists a u = (v1 , . . . , vm , u1 , . . . , up ) ⩾ 0 such that 0∈

m ∑ i=1

vi ∂fi (¯ x) +

p ∑

uj ∂gj (¯ x)

j=1

and .

uj gj (¯ x ) = 0; j = 1, 2, . . . , p.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

37 / 46

Some recent issues


[1] F.H. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth analysis and control theory, Springer Verlag, New York, 1998. [2, 3] B.S. Mordukhovich, Variations analysis and generalized differentiation, I: Basic theory & II: Applications. Springer, 2006. [4] M. Soleimani-damaneh, Nonsmooth Optimization Using Mordukhovichs Subdifferential. SIAM J. Control Optim. 48 (2010) 3403-3432. [5] Soleimani-damaneh, J.J. Nieto, Nonsmooth multiple-objective optimization in separable Hilbert spaces. Nonlinear Analysis 71 (2009) 4553-4558.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

38 / 46

Some recent issues

More problem

Under what conditions YN is connected

.


Vector Optimization

.

.

.

.

December 31, 2014

.

39 / 46

Some recent issues

More problem

Under what conditions YN is connected [∗] M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

39 / 46

Some recent issues

More problem

Under what conditions YN is connected [∗] M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005. Under what conditions YPN is dense in YN .

.


Vector Optimization

.

.

.

.

December 31, 2014

.

39 / 46

Some recent issues

More problem

Under what conditions YN is connected [∗] M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005. Under what conditions YPN is dense in YN . [∗] J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993) 105-122. [∗] M. Ehrgott, 2005.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

39 / 46

Some recent issues

More problem

Applications in financial mathematics:

.


Vector Optimization

.

.

.

.

December 31, 2014

.

40 / 46

Some recent issues

More problem

Applications in financial mathematics: [∗] S. Utz, M. Wimmer, M. Hirschberger, and R. E. Steuer, Tri-Criterion Inverse Portfolio Optimization with Application to Socially Responsible Mutual Funds, European Journal of Operational Research, Vol. 234 (2014) 491-498. [∗] M. Hirschberger, R.E. Steuer, S. Utz, M. Wimmer and Y. Qi, Computing the Nondominated Surface in Tri-Criterion Portfolio Selection, Operations Research, 61 (2013) 169-183.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

40 / 46

Some recent issues

More problem

Applications in financial mathematics: [∗] S. Utz, M. Wimmer, M. Hirschberger, and R. E. Steuer, Tri-Criterion Inverse Portfolio Optimization with Application to Socially Responsible Mutual Funds, European Journal of Operational Research, Vol. 234 (2014) 491-498. [∗] M. Hirschberger, R.E. Steuer, S. Utz, M. Wimmer and Y. Qi, Computing the Nondominated Surface in Tri-Criterion Portfolio Selection, Operations Research, 61 (2013) 169-183. Approximate weak/proper efficiency: [∗] B.A. Ghaznavai-Ghosoni, E. Khorram, M. Soleimani-damaneh, Approximate Weakly/Properly Efficient Solutions in Multi-objective Programming Utilizing Scalarization Approaches. Optimization. 62 (2013) 703-720.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

40 / 46

Some recent issues

More problem

Applications in Molecular Biology:

.


Vector Optimization

.

.

.

.

December 31, 2014

.

41 / 46

Some recent issues

More problem

Applications in Molecular Biology: [∗] Y. Cherruault, Global Optimization in Biology and Medicine. Math. Comput. Modelling 20 (1994) 119-132. [∗] J.G. Ecker, M. Kupferschmid, C.E. Lawrence, A.A. Reilly, A.C.H. Scott, An application of nonlinear optimization in molecular biology. European Journal of Operational Research 138 (2002) 452-458. [∗] P. Festa, On some optimization problems in molecular biology. Mathematical Biosciences, 207 (2007) 219-234. [∗] F.C. Gomes, C.N. Meneses, P.M. Pardalos, G.V.R. Viana, A parallel multistart algorithm for the closest string problem. Computers and Operations Research 35 (2008) 3636-3643. [∗] M. Soleimani-damaneh, An optimization modelling for string selection in molecular biology using Pareto optimality, Applied Mathematical Modelling, 35 (2011) 3887-3892. .


Vector Optimization

.

.

.

.

December 31, 2014

.

41 / 46

ORO2013

.


Vector Optimization

.

.

.

.

December 31, 2014

.

42 / 46

ORO2013

.


Vector Optimization

.

.

.

.

December 31, 2014

.

43 / 46

ORO2011

.


Vector Optimization

.

.

.

.

December 31, 2014

.

44 / 46

...

MCDM society ▶

23rd International Conference on Multiple Criteria Decision Making MCDM 2015, August 2nd7th, 2015, Hamburg, Germany.

GCM society . . . And hopefully ORO2016 conference in Tehran.

.


Vector Optimization

.

.

.

.

December 31, 2014

.

45 / 46

Thanks

MANY THANKS for your ATTENTION

.


Vector Optimization

.

.

.

.

December 31, 2014

.

46 / 46

Vector Optimization: An Introduction and Some

Vector Optimization: An Introduction and Some

Suggest Documents

OPTIMIZATION An introduction

An Introduction to Optimization Chong and Zak.pdf

Program Composition and Optimization: An Introduction - DROPS

An Introduction to Support Vector Machines - CiteSeerX

Some Vector Sequence Spaces-II 1 Introduction - m-hikari

Bit-Vector Optimization

[PDF] An Introduction to Support Vector Machines and ... - Google Sites

Introduction to J1939 - Vector

Introduction to J1939 - Vector

an introduction to multiobjective optimization techniques

Java Optimization for Superscalar and Vector Architectures

Java Optimization for Superscalar and Vector Architectures

C1, vector optimization problems and Riemann derivatives

vector ant colony optimization and travelling

From scalar to vector optimization

Duality in vector optimization

Cyclic Games: An Introduction and Some Examples - CiteSeerX

1 Understanding childhood: an introduction to some key themes and ...

An Introduction to Optimality Criteria and Some Results on Optimal ...

An Introduction to the Fundamental Group and some Applications

An Introduction to Optimization Models and Methodswww.researchgate.net › publication › fulltext › 31426680

Introduction to vector and tensor analysis

VECTOR MEASURES AND NUCLEAR OPERATORS 1. Introduction

Introduction . Introduction 3. Some Resul 3. Some ...