Frontiers in Mathematical Sciences: 3rd conference [Vector Optimization: An Introduction and Some...

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Vector Optimization: An Introduction and SomeRecent Problems

Majid Soleimani-damaneh

University of Tehran & IPM

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

Majid Soleimani-damaneh (UT & IPM) Vector Optimization December 31, 2014 1 / 46

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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

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Outline

History

Scalarization

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Outline

History

Scalarization

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Outline

History

Scalarization

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Outline

History

Scalarization

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Outline

History

Scalarization

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Outline

History

Scalarization

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Outline

History

Scalarization

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History

Francis Y. Edgeworth (1845-1926):In 1881 at King’s College (London)and later at Oxford, economicsProfessor.

Edgeworth (1881): For themulti-utility problem within thecontext of two consumer criteria, Aand B: “It is required to find a point(x , y) such that in whatever directionwe take an infinitely small step, Aand B do not increase together butthat, while one increases, the otherdecreases.”

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History

Vilfredo Pareto (1848-1923):University of Lausanne, Switzerland.

(Pareto 1906): “The optimumallocation of the resources of asociety is not attained so long as it ispossible to make at least oneindividual better off in his ownestimation while keeping others aswell off as before in their ownestimation.”

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Multiple Objective Programming (MOP)

Multiple Objective Programming (MOP):

Minf (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.

Binary MOP, Discrete MOP..Definition..

......

x∗ ∈ E is called a Pareto (efficient) solution to MOP if ∄xo ∈ E such that

fj(xo) ≤ fj(x

∗) for each j = 1, 2, . . . , p,

fj(xo) < fj(x

∗) for some j = 1, 2, . . . , p.

.Definition..

......

x∗ ∈ E is called a weak Pareto (weak efficient) solution to MOP if∄xo ∈ E such that fj(x

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

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Binary MOP, Discrete MOP.

.Definition..

......

fj(xo) ≤ fj(x

∗) for each j = 1, 2, . . . , p,

fj(xo) < fj(x

∗) for some j = 1, 2, . . . , p.

.Definition..

......

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

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......

fj(xo) ≤ fj(x

∗) for each j = 1, 2, . . . , p,

fj(xo) < fj(x

∗) for some j = 1, 2, . . . , p.

.Definition..

......

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

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......

fj(xo) ≤ fj(x

∗) for each j = 1, 2, . . . , p,

fj(xo) < fj(x

∗) for some j = 1, 2, . . . , p.

.Definition..

......

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

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

[2] Steuer, Ralph, Multiple Criteria Optimization: Theory, Computationand Application, 1985.[3] Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of MultiobjectiveOptimization, Vol. 176, Mathematics in Science and Engineering, London,1985.

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

[3] Y. Sawaragi, H. Nakayama, and T. Tanino, Theory of MultiobjectiveOptimization, Vol. 176, Mathematics in Science and Engineering, London,1985.

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

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.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 ≤ yj for each j .¶ The set of all minimals of Y is denoted by YN ,¶ y − Rp

≧ = y : y ≤ y.

Remark: It is clear that, x0 is a Pareto solution to MOP if and only iff (x0) ∈ (f (E ))N .

Question: Under what conditions, YN is nonempty

.Theorem........If Y is compact, then YN = ∅.

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.Definition..

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∄y ∈ Y s.t. y ≤ y & y = y .

≧ = y : y ≤ y.

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.Definition..

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∄y ∈ Y s.t. y ≤ y & y = y .

≧ = y : y ≤ y.

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.Definition..

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∄y ∈ Y s.t. y ≤ y & y = y .

≧ = y : y ≤ y.

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.Definition..

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∄y ∈ Y s.t. y ≤ y & y = y .

≧ = y : y ≤ y.

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.Definition..

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∄y ∈ Y s.t. y ≤ y & y = y .

≧ = y : y ≤ y.

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.Definition..

......

We say that Y has a compact section if there exists y ∈ Y such thatY ∩ (y − Rp

≧) is compact.

.Definition..

......Y is called Rp

≧-compact if Y ∩ (y − Rp≧) is compact for each y ∈ Y .

.Theorem........If Y has a compact section, then YN = ∅.

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.Definition..

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≧) is compact.

.Definition..

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.Definition..

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≧) is compact.

.Definition..

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.Definition..

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≧) is compact.

.Definition..

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.Definition..

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Y is called Rp≧-semicompact if every open cover of Y in the form

Y ⊆ ∪i∈I (yi − Rp

≧)c ,

has a finite subcover.

.Theorem..

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If Y is Rp≧-compact, then it is Rp

≧-semicompact. The converse does not

hold necessarily.

.Theorem..

......If Y is Rp

≧-semicompact, then YN = ∅.

.Theorem..

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E is compact &fi ’s are l.s.c =⇒ Y = f (E ) is Rp≧-semicompact

=⇒ YN = ∅ =⇒ XE = ∅.

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.Definition..

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≧)c ,

.Theorem..

......

hold necessarily.

.Theorem..

......If Y is Rp

.Theorem..

......

=⇒ YN = ∅ =⇒ XE = ∅.

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.Definition..

......

≧)c ,

.Theorem..

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hold necessarily.

.Theorem..

......If Y is Rp

.Theorem..

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=⇒ YN = ∅ =⇒ XE = ∅.

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.Definition..

......

≧)c ,

.Theorem..

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hold necessarily.

.Theorem..

......If Y is Rp

.Theorem..

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=⇒ YN = ∅ =⇒ XE = ∅.

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.Definition..

......

≧)c ,

.Theorem..

......

hold necessarily.

.Theorem..

......If Y is Rp

.Theorem..

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=⇒ YN = ∅ =⇒ XE = ∅.

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.Definition..

......

≧)c ,

.Theorem..

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hold necessarily.

.Theorem..

......If Y is Rp

.Theorem..

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=⇒ YN = ∅ =⇒ XE = ∅.Majid Soleimani-damaneh (UT & IPM) Vector Optimization December 31, 2014 9 / 46

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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 Economicsand Mathematical Systems, Vol. 319, Springer-Verlag, New york,Berlin, (1989).

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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 .

.Definition..

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

y ∈ Y , y ⪯ y∗ =⇒ y = y∗.

¶ The set of all minimals of Y is denoted by YN .

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(x , y) ∈ R ⇐⇒ x ⪯ y .

.Definition..

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y ∈ Y , y ⪯ y∗ =⇒ y = y∗.

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(x , y) ∈ R ⇐⇒ x ⪯ y .

.Definition..

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y ∈ Y , y ⪯ y∗ =⇒ y = y∗.

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(x , y) ∈ R ⇐⇒ x ⪯ y .

.Definition..

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y ∈ Y , y ⪯ y∗ =⇒ y = y∗.

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(x , y) ∈ R ⇐⇒ x ⪯ y .

.Definition..

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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

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.Definition..

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

.Theorem..

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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 ∈ Dis a partial order.

Due to the above theorem, hereafter X is a real vector space which isordered by a convex pointed cone C , i.e. x ⪯ y =⇒ y − x ∈ C .

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.Definition..

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.Theorem..

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C := x ∈ X : x ⪰ 0

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.Definition..

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.Theorem..

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C := x ∈ X : x ⪰ 0

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.Definition..

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.Theorem..

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C := x ∈ X : x ⪰ 0

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.Definition..

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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 = Rn and C = x : x ≥ 0, then minimal points are calledPareto (efficient) points.

A section of Y at y ∈ Y is defined by Yy := Y ∩ (y − C ).

.Lemma..

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Let X be a real vector space partially ordered by convex pointed cone C; Ify ∈ Y ⊆ X, theni. E (Yy ,C ) ⊆ E (Y ,C ),ii. E (Y ,C ) = E (Y + C ,C ).

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.Definition..

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.Lemma..

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.Definition..

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.Lemma..

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.Definition..

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.Lemma..

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.Definition..

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.Lemma..

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.Definition..

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.Lemma..

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.Definition..

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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 Shas a lower bound (w.r.t C ) in S .

.Theorem..

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Existence: Let X be a real vector space partially ordered by convexpointed cone C; and Y ⊆ X be nonempty. ThenE (Y ,C ) = ∅ if and only if Y has an inductively ordered section.

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.Definition..

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.Definition..

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.Theorem..

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.Definition..

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.Definition..

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.Theorem..

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Let X be a TVS.

.Definition..

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The ordering cone C has the Daniell property if the infimum of eachdecreasing net (w.r.t C ) in C exists and is also the topological limit of thenet.

.Definition..

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The ordering cone C is boundedly order complete if each decreasingbounded 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 = ( 1n ,

1n )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), whichmakes 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.

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Let X be a TVS.

.Definition..

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.Definition..

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Let X be a TVS.

.Definition..

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.Definition..

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Let X be a TVS.

.Definition..

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.Definition..

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Let X be a TVS.

.Definition..

......

.Definition..

......

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 = ( 1n ,

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Let X be a TVS.

.Definition..

......

.Definition..

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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 = ( 1n ,1n )n∈N.It is decreasing w.r.t C with (0, 0) as a lower

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Let X be a TVS.

.Definition..

......

.Definition..

......

1n )n∈N.

It is decreasing w.r.t C with (0, 0) as a lowerbound.Contradiction: (a, b) = inf xn.If a, b > 0, then ∃n; xn ⪯ (a, b), whichmakes 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.

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Let X be a TVS.

.Definition..

......

.Definition..

......

bound.Contradiction: (a, b) = inf xn.

If a, b > 0, then ∃n; xn ⪯ (a, b), whichmakes 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.

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Let X be a TVS.

.Definition..

......

.Definition..

......

bound.Contradiction: (a, b) = inf xn.If a, b > 0, then ∃n; xn ⪯ (a, b), whichmakes 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.

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Let X be a TVS.

.Definition..

......

.Definition..

......

bound.Contradiction: (a, b) = inf xn.If a, b > 0, then ∃n; xn ⪯ (a, b), whichmakes 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.

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Let X be a TVS.

.Definition..

......

.Definition..

......

bound.Contradiction: (a, b) = inf xn.If a, b > 0, then ∃n; xn ⪯ (a, b), whichmakes 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.

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Let X be a TVS.

.Definition..

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.Definition..

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.Theorem..

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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 aclosed 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.

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.Theorem..

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.Theorem..

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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 aclosed and bounded section (boundedness is wrt to the topology).iii. Y has a compact section.

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.Theorem..

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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 aclosed and bounded section (boundedness is wrt to the topology).

iii. Y has a compact section.

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.Theorem..

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.Theorem..

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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 −Minf (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:Minf (x) : x ∈ E, (MOP)

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

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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 −Minf (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:Minf (x) : x ∈ E, (MOP)

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

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Vector optimization/Multiple Objective Programming An Application: Performance Analysis

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

The minimal set satisfying axioms 1-4 is(X ,Y ) :

∑nj=1 µjXj ≤ X ,

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.

Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.

φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.

Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

Performance AnalysisAssume 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 beproduced by X.Let DMUo(Xo ,Yo) be under evaluation.φo = maxφ : (Xo , φYo) ∈ PPS.Axioms:1. Observations: (Xj ,Yj) ∈ PPS , ∀j = 1, 2, . . . , n.2. Possibility:(X ,Y ) ∈ PPS , X ≥ X , 0 ≤ Y ≤ Y =⇒ (X , Y ) ∈ PPS .3. Unbounded ray: (X ,Y ) ∈ PPS , λ ≥ 0 =⇒ (λX , λY ) ∈ PPS .4. Convexity: PPS is a convex set.

.Theorem..

......

∑nj=1 µjYj ≥ Y ≥ 0, µ ≥ 0.

. . . . . .

φo = max φs.t.

∑nj=1 µjXj ≤ Xo ,∑nj=1 µjYj ≥ φYo ,

µj ≥ 0, j = 1, 2, . . . , n.

Question 1. If the efficiency index φo remains unchanged, but the inputsincrease, how much should the outputs of DMUo increase

. . . . . .

φo = max φs.t.

µj ≥ 0, j = 1, 2, . . . , n.

Question 1. If the efficiency index φo remains unchanged, but the inputsincrease, how much should the outputs of DMUo increase

. . . . . .

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 β∗o provided that the efficiency

index of DMUo is still φo . In fact,

β∗o = (β∗

1o , β∗2o , ..., β

∗mo)

t = Yo +∆Yo ; ∆Yo ≥ 0.

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

φn+1 = max φs.t.

∑nj=1 µjXj + µn+1αo ≤ αo ,∑nj=1 µjYj + µn+1β

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

. . . . . .

Output estimation

αo = Xo +∆Xo ; ∆Xo ≥ 0, ∆Xo = 0.

β∗o = (β∗

1o , β∗2o , ..., β

∗mo)

t = Yo +∆Yo ; ∆Yo ≥ 0.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

. . . . . .

Output estimation

αo = Xo +∆Xo ; ∆Xo ≥ 0, ∆Xo = 0.

β∗o = (β∗

1o , β∗2o , ..., β

∗mo)

t = Yo +∆Yo ; ∆Yo ≥ 0.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

. . . . . .

Output estimation

αo = Xo +∆Xo ; ∆Xo ≥ 0, ∆Xo = 0.

β∗o = (β∗

1o , β∗2o , ..., β

∗mo)

t = Yo +∆Yo ; ∆Yo ≥ 0.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

. . . . . .

Output estimation

αo = Xo +∆Xo ; ∆Xo ≥ 0, ∆Xo = 0.

β∗o = (β∗

1o , β∗2o , ..., β

∗mo)

t = Yo +∆Yo ; ∆Yo ≥ 0.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

. . . . . .

Output estimation

φo = max φs.t.

µj ≥ 0, j = 1, 2, . . . , n.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

Aim: φo = φn+1.

MOLP (I ) :

max (β1o , β2o , ..., βso)s.t.

∑nj=1 µjXj ≤ αo ,∑nj=1 µjYj ≥ φoβo ,

βo ≥ Yo ,µj ≥ 0, j = 1, 2, . . . , n.

. . . . . .

Output estimation

φo = max φs.t.

µj ≥ 0, j = 1, 2, . . . , n.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

Aim: φo = φn+1.

MOLP (I ) :

βo ≥ Yo ,µj ≥ 0, j = 1, 2, . . . , n.

. . . . . .

Output estimation

φo = max φs.t.

µj ≥ 0, j = 1, 2, . . . , n.

φn+1 = max φs.t.

∗o ≥ φβ∗

µj ≥ 0, j = 1, 2, . . . , n + 1.

Aim: φo = φn+1.

MOLP (I ) :

βo ≥ Yo ,µj ≥ 0, j = 1, 2, . . . , n.

. . . . . .

Output estimation

.Theorem..

......

Suppose that φo > 1 and the inputs for DMUo are going to increase fromXo to αo = Xo +∆Xo , where ∆Xo ≥ 0 and ∆Xo = 0.

(i) Let (µ∗, β∗o) be a Weak Pareto solution of MOLP (I). Then, when the

outputs of DMUo are increased to β∗o we have φ(αo , β

∗o) = φ(Xo ,Yo).

(ii) Conversely, let (µ∗, β∗o) be a feasible solution of MOLP (I). If

φ(αo , β∗o) = φ(Xo ,Yo), then (µ∗, β∗

o) is a Weak Pareto solution toMOLP (I).

. . . . . .

References

A. Hadi-Vencheh, A.A. Foroughi, M. Soleimani-damaneh, A DEAmodel for resource allocation. Economic Modelling 25 (2008), pp.983-993.

M. Soleimani-damaneh, P.J. Korhonen, J. Wallenius, On valueefficiency, Optimization 63 (2014) 617-631.

Q.L. Wei, J. Zhang, X. Zhang, An inverse DEA model forinput/output estimate. European Journal of Operational Research 121(1) (2000), pp. 151-163.

. . . . . .

Scalarization

Weight Sum

ϵ-constraint

Pascoletti-Serafini

NBI method

Nonlinear scalarization (Gerth function)

. . . . . .

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..

......

λj fj(x) over E for some λ1, . . . , λp ≥ 0, (not allzero) then x∗ is a weak Pareto solution of MOP.

.Theorem..

......

Let f be a convex function and E be a convex set. If x∗ is a weak Paretosolution of MOP, then x∗ is a minimizer of

∑λj fj(x) over E for some

λ1, . . . , λp ≥ 0 (not all zero).

. . . . . .

Scalarization

.Theorem..

......

.Theorem..

......

.Theorem..

......

λ1, . . . , λp ≥ 0 (not all zero).

. . . . . .

Scalarization

.Theorem..

......

.Theorem..

......

.Theorem..

......

λ1, . . . , λp ≥ 0 (not all zero).

. . . . . .

Some recent issues Proper efficiency

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

Properly efficient solutions are efficient solutions in which, given anyobjective, the trade-off between that objective and some otherobjective is bounded.

This notion was dealt with initially by Kuhn and Tucker (1951) andwas precised thereafter by Geoffrion (1968) for multiple-objectiveoptimization problems (MOPs) in finite dimensional Euclidean spaceswith natural ordering cone.

For MOPs with unnatural ordering cones, the definition of properefficiency has been extended by Benson (1979), Borwein (1977), andHenig (1982).

. . . . . .

.Definition..

......

A feasible solution x ∈ X is called properly efficient solution of MOP inGeoffrion’s sense, if it is efficient and there is a real number M > 0 suchthat for all i ∈ 1, 2, ..., p and x ∈ X satisfying fi (x) < fi (x) there existsan index j ∈ 1, 2, ..., p such that fj(x) > fj(x) and

fi (x)− fi (x)

fj(x)− fj(x)≤ M.

.Definition..

......

A feasible solution x ∈ X is called properly efficient solution of MOP inHenig’s sense, if (f (x)− C ) ∩ f (X ) = f (x), for some convex pointedcone C with Rp

≧\0 ⊆ int(C ).

.Theorem........The above two definitions are equivalent for MOP.

Benson definition, Borwein definition, etc.

. . . . . .

.Definition..

......

fi (x)− fi (x)

.Definition..

......

≧\0 ⊆ int(C ).

. . . . . .

.Definition..

......

fi (x)− fi (x)

.Definition..

......

≧\0 ⊆ int(C ).

. . . . . .

.Definition..

......

fi (x)− fi (x)

.Definition..

......

≧\0 ⊆ int(C ).

Benson definition, Borwein definition, etc.Majid Soleimani-damaneh (UT & IPM) Vector Optimization December 31, 2014 27 / 46

. . . . . .

.Theorem..

......

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 Paretosolution of MOP, then x∗ is a minimizer of

λ1, . . . , λp > 0.

. . . . . .

.Theorem..

......

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 Paretosolution of MOP, then x∗ is a minimizer of

λ1, . . . , λp > 0.

. . . . . .

.Definition..

......

A feasible solution x ∈ X is called strongly proper efficient for MOP if it isefficient and there is a real number M > 0 such that for all i ∈ 1, 2, ..., pand x ∈ X satisfying fi (x) < fi (x), we have fi (x)−fi (x)

fj (x)−fj (x)≤ M, for all

j ∈ 1, 2, ..., p with fj(x) > fj(x).

Questions:

Obtaining strongly proper efficient solutions,

Defining and characterizing strongly proper efficiency for infinitedimensional case.

. . . . . .

.Definition..

......

A feasible solution x ∈ X is called strongly proper efficient for MOP if it isefficient and there is a real number M > 0 such that for all i ∈ 1, 2, ..., pand x ∈ X satisfying fi (x) < fi (x), we have fi (x)−fi (x)

fj (x)−fj (x)≤ M, for all

j ∈ 1, 2, ..., p with fj(x) > fj(x).

Questions:

Obtaining strongly proper efficient solutions,

Defining and characterizing strongly proper efficiency for infinitedimensional case.

. . . . . .

K. Khaledian, E. Khorram, M. Soleimani-damaneh, Strongly properefficient solutions: efficient solutions with bounded trade-offs. Journalof Optimization Theory and Applications. (Accepted).

I. Kaliszewski, Quantitative Pareto Analysis by Cone SeparationTechnique. Kluwer Academic Publishers, Dordrecht, 1994.

M. Ehrgott, Multicriteria Optimization, Springer, Berlin, 2005.

. . . . . .

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 )) = ∅.

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.When cor(A) = ∅ we say that A is solid. If cor(K ) = ∅, thencor(K ) ∪ 0 is a convex cone, cor(K ) + K = cor(K ) andcor(cor(K )) = cor(K ).

vcl(A) = b ∈ X : ∃x ∈ X ; ∀λ′> 0 , ∃λ ∈ [0, λ

′] ; b + λx ∈ A.

. . . . . .

......

(f (E )− f (x0)) ∩ (−C\0) = ∅.

(f (Ω)− f (x0)) ∩ (−int(C )) = ∅.

cor(A) = x ∈ A : ∀x ′ ∈ X , ∃λ′> 0; ∀λ ∈ [0, λ

′], x + λx

vcl(A) = b ∈ X : ∃x ∈ X ; ∀λ′> 0 , ∃λ ∈ [0, λ

′] ; b + λx ∈ A.

. . . . . .

......

(f (E )− f (x0)) ∩ (−C\0) = ∅.

(f (Ω)− f (x0)) ∩ (−int(C )) = ∅.

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 ) = ∅, thencor(K ) ∪ 0 is a convex cone, cor(K ) + K = cor(K ) andcor(cor(K )) = cor(K ).

vcl(A) = b ∈ X : ∃x ∈ X ; ∀λ′> 0 , ∃λ ∈ [0, λ

′] ; b + λx ∈ A.

. . . . . .

......

(f (E )− f (x0)) ∩ (−C\0) = ∅.

(f (Ω)− f (x0)) ∩ (−int(C )) = ∅.

cor(A) = x ∈ A : ∀x ′ ∈ X , ∃λ′> 0; ∀λ ∈ [0, λ

′], x + λx

vcl(A) = b ∈ X : ∃x ∈ X ; ∀λ′> 0 , ∃λ ∈ [0, λ

′] ; b + λx ∈ A.

. . . . . .

......

(f (E )− f (x0)) ∩ (−C\0) = ∅.

(f (Ω)− f (x0)) ∩ (−int(C )) = ∅.

cor(A) = x ∈ A : ∀x ′ ∈ X , ∃λ′> 0; ∀λ ∈ [0, λ

′], x + λx

vcl(A) = b ∈ X : ∃x ∈ X ; ∀λ′> 0 , ∃λ ∈ [0, λ

′] ; b + λx ∈ A.

. . . . . .

.Theorem..

......

Let M,K be solid nontrivial convex cones in X . If M ∩ cor(K ) = ∅, thenthere 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 allm ∈ cor(M).

.Theorem..

......

Let M,K be two convex, nontrivial, and vectorially closed cones in X suchthat M,K are relatively solid and K+ is solid. If M ∩ K = 0, then thereexists a functional l ∈ X ′\0 such that,

⟨l , k⟩ ≥ 0 ≥ ⟨l ,m⟩ ∀(k ∈ K ,m ∈ M),

and furthermore,⟨l , k⟩ > 0 ∀k ∈ K\0.

. . . . . .

.Theorem..

......

Let M,K be solid nontrivial convex cones in X . If M ∩ cor(K ) = ∅, thenthere 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 allm ∈ cor(M).

.Theorem..

......

Let M,K be two convex, nontrivial, and vectorially closed cones in X suchthat M,K are relatively solid and K+ is solid. If M ∩ K = 0, then thereexists a functional l ∈ X ′\0 such that,

⟨l , k⟩ ≥ 0 ≥ ⟨l ,m⟩ ∀(k ∈ K ,m ∈ M),

and furthermore,⟨l , k⟩ > 0 ∀k ∈ K\0.

. . . . . .

.Theorem..

......

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

M. Adan, V. Novo, Optimality conditions for vector optimizationproblems with generalized convexity in real linear spaces,Optimization, 51 (2002) 73-91.T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers formultiobjective problems: existence and optimality conditions, Math.Prog. 122 (2010) 301-347.E. Kiyani, M. Soleimani-damaneh, Approximate proper efficiency onreal linear vector spaces. Pacific Journal of Optimization. (Accepted)E. Kiyani, M. Soleimani-damaneh, Algebraic interior and separationon linear vector spaces: Some comments. Journal of OptimizationTheory and Applications 161 (2014) 994-998.

. . . . . .

.Theorem..

......

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

M. Adan, V. Novo, Optimality conditions for vector optimizationproblems with generalized convexity in real linear spaces,Optimization, 51 (2002) 73-91.T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers formultiobjective problems: existence and optimality conditions, Math.Prog. 122 (2010) 301-347.E. Kiyani, M. Soleimani-damaneh, Approximate proper efficiency onreal linear vector spaces. Pacific Journal of Optimization. (Accepted)E. Kiyani, M. Soleimani-damaneh, Algebraic interior and separationon linear vector spaces: Some comments. Journal of OptimizationTheory and Applications 161 (2014) 994-998.

. . . . . .

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.

A function h : A ⊆ X −→ R is said to be Lipschitz on A if thereexists 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 ofx . Also, h is locally Lipschitz on A if it is Lipschitz near x for everyx ∈ A.

. . . . . .

|h(x)− h(y)| ≤ k∥x − y∥ ∀x , y ∈ A.

. . . . . .

|h(x)− h(y)| ≤ k∥x − y∥ ∀x , y ∈ A.

. . . . . .

The notion of generalized diff. plays a fundamental role in modernvariational 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 thedirection d , denoted by h(x ; d), is defined as

h(x ; d) = lim supx −→ xt ↓ 0

(1/t)[h(x + td)− h(x)].

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 withweak∗-topology, and ⟨., .⟩ exhibits the duality pairing.

. . . . . .

Consider h as a locally Lipschitz function from X into R.

The Clarke’s generalized directional derivative of h at x in thedirection d , denoted by h(x ; d), is defined as

(1/t)[h(x + td)− h(x)].

∂h(x) = ξ∗ ∈ X ∗ : h(x ; d) ≥ ⟨ξ∗, d⟩, ∀d ∈ X,

. . . . . .

(1/t)[h(x + td)− h(x)].

∂h(x) = ξ∗ ∈ X ∗ : h(x ; d) ≥ ⟨ξ∗, d⟩, ∀d ∈ X,

. . . . . .

(1/t)[h(x + td)− h(x)].

∂h(x) = ξ∗ ∈ X ∗ : h(x ; d) ≥ ⟨ξ∗, d⟩, ∀d ∈ X,

. . . . . .

.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 ξ∗i be sequences in X and X ∗ such that ξ∗i ∈ ∂h(xi ).Suppose that xi converges to x, and that ξ∗ is a cluster point of ξ∗i in theweak∗−topology. Then ξ∗ ∈ ∂h(x).

.Theorem..

......

Let x , y ∈ X , and suppose that h is Lipschitz on an open set containingthe line segment [x , y ]. Then there exists a point u ∈ (x , y) such that

h(y)− h(x) ∈ ⟨∂h(u), y − x⟩.

. . . . . .

.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 ξ∗i be sequences in X and X ∗ such that ξ∗i ∈ ∂h(xi ).Suppose that xi converges to x, and that ξ∗ is a cluster point of ξ∗i in theweak∗−topology. Then ξ∗ ∈ ∂h(x).

.Theorem..

......

Let x , y ∈ X , and suppose that h is Lipschitz on an open set containingthe line segment [x , y ]. Then there exists a point u ∈ (x , y) such that

h(y)− h(x) ∈ ⟨∂h(u), y − x⟩.

. . . . . .

.Theorem..

......

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

0 ∈m∑i=1

vi∂fi (x) +

p∑j=1

uj∂gj(x)

andujgj(x) = 0; j = 1, 2, . . . , p.

. . . . . .

[1] F.H. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmoothanalysis and control theory, Springer Verlag, New York, 1998.

[2, 3] B.S. Mordukhovich, Variations analysis and generalizeddifferentiation, I: Basic theory & II: Applications. Springer, 2006.

[4] M. Soleimani-damaneh, Nonsmooth Optimization Using MordukhovichsSubdifferential. SIAM J. Control Optim. 48 (2010) 3403-3432.

[5] Soleimani-damaneh, J.J. Nieto, Nonsmooth multiple-objectiveoptimization in separable Hilbert spaces. Nonlinear Analysis 71 (2009)4553-4558.

. . . . . .

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 vectoroptimization, Transactions of the American Mathematical Society,338 (1993) 105-122.[∗] M. Ehrgott, 2005.

. . . . . .

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 vectoroptimization, Transactions of the American Mathematical Society,338 (1993) 105-122.[∗] M. Ehrgott, 2005.

. . . . . .

Applications in financial mathematics:

[∗] S. Utz, M. Wimmer, M. Hirschberger, and R. E. Steuer,Tri-Criterion Inverse Portfolio Optimization with Application toSocially Responsible Mutual Funds, European Journal of OperationalResearch, Vol. 234 (2014) 491-498.[∗] M. Hirschberger, R.E. Steuer, S. Utz, M. Wimmer and Y. Qi,Computing the Nondominated Surface in Tri-Criterion PortfolioSelection, 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-objectiveProgramming Utilizing Scalarization Approaches. Optimization. 62(2013) 703-720.

. . . . . .

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

. . . . . .

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

. . . . . .

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, Aparallel multistart algorithm for the closest string problem. Computersand Operations Research 35 (2008) 3636-3643.

[∗] M. Soleimani-damaneh, An optimization modelling for stringselection in molecular biology using Pareto optimality, AppliedMathematical Modelling, 35 (2011) 3887-3892.

. . . . . .

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, Aparallel multistart algorithm for the closest string problem. Computersand Operations Research 35 (2008) 3636-3643.

[∗] M. Soleimani-damaneh, An optimization modelling for stringselection in molecular biology using Pareto optimality, AppliedMathematical Modelling, 35 (2011) 3887-3892.

. . . . . .

ORO2013

. . . . . .

ORO2013

. . . . . .

ORO2011

. . . . . .

MCDM society

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

GCM society

. . . And hopefully ORO2016 conference in Tehran.

. . . . . .

Thanks

MANY THANKS for your ATTENTION

Frontiers in Mathematical Sciences: 3rd conference [Vector Optimization: An Introduction and Some...

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