new versions of the ekeland variational principle in ... · new versions of the ekeland variational...

New versions of the Ekeland variationalprinciple in vector optimization

C. Gutiérrez

Departamento de Matemática AplicadaUniversidad de Valladolid

Second ALEL Meeting on Optimization and Applications

Universidad de AlicanteDepartamento de Estadística e Investigación Operativa

June 26-27, 2009

Coauthors: B. Jiménez and V. Novo

Departamento de Matemática Aplicada IUniversidad Nacional de Educación a Distancia

Introduction. Motivation

(X ,d) complete metric space, g : X → R ∪ {+∞} proper

Min{g(x) : x ∈ X} (SOP)

DefinitionConsider ε > 0. x0 ∈ X is an ε-solution of problem (SOP), denotedx0 ∈ AMin(g, ε), if g(x0)− ε ≤ g(x) for all x ∈ X

Theorem (Ekeland, 1974)

Suppose that g is lower semicontinuous (l.s.) and lower bounded(l.b.). For each x0 ∈ AMin(g, ε) and α > 0 there exists x ∈ X suchthat (a) g(x) ≤ g(x0), (b) d(x , x0) ≤ α and

(c) g(x) < g(x) + (ε/α)d(x , x) ∀ x ∈ X\{x}

(c) x is a strict solution of the perturbed optimization problem

Min{g(x) + (ε/α)d(x , x) : x ∈ X},

denoted x ∈ SMin(g + (ε/α)d(·, x))

Introduction. Motivation

Ekeland Variational Principle (EVP) has become an essential tool inseveral areas:

1 Nonlinear Analysis

Fixed point theoremsGeometry of Banach spacesControl theory

2 Convex Analysis and Optimization

Approximate dualityApproximate saddle-point theoremsExistence of exact and approximate solutionsOptimality and ε-optimality conditionsStabilityWell-posednessAlgorithms

Introduction. Aim

To extend EVP to a vector optimization problem:

Min{f (x) : x ∈ X} (VOP),

f : X → Y , Y locally convex Hausdorff space

Let D ⊂ Y be a nontrivial (D 6= {0}) pointed (D ∩ (−D) = {0}) convexcone. The relation

y , z ∈ Y , y ≤D z ⇐⇒ z − y ∈ D

defines a partial order in Y , that is compatible with the linear structureof Y : y , z, v ∈ Y , α > 0,

y ≤D z ⇒ y + v ≤D z + vy ≤D z ⇒ αy ≤D αz

D+ = {ξ ∈ Y ∗ : ξ(d) ≥ 0∀d ∈ D}

Introduction. Aim

Definition

x0 ∈ X is an efficient solution of (VOP), denoted x0 ∈ E(f ,≤D), if thereis not x ∈ X such that f (x) ≤D f (x0), f (x) 6= f (x0)

In the EVP, property (c) says that x is a strict solution of a perturbedoptimization problem

Definition

x0 ∈ X is a strict efficient solution of (VOP), denoted x0 ∈ SE(f ,≤D), ifthere is not x ∈ X\{x0} such that f (x) ≤D f (x0)

Outline

1 Introduction

2 An equivalent formulation of EVP for (SOP)

3 EVP for (VOP) with a vector-valued perturbationLoridan (1984)Németh (1986)Göpfert et al. (2000) and Gutiérrez et al. (2008)Summary

4 EVP for (VOP) with a set-valued perturbationIdea, objective and toolsOn the lower boundedness assumptionThe concept of set-valued D-metricOn the lower semicontinuity assumption(C, ε)-efficient solutions of problem (VOP)EVPs for (VOP) with a set-valued perturbation function

IntroductionAn equivalent formulation of EVP for (SOP)

EVP for (VOP) with a vector-valued perturbationEVP for (VOP) with a set-valued perturbation


Suppose that g is l.s. and l.b. For each x0 ∈ Min(g, ε) and α > 0 thereexists x ∈ X such that (a) g(x) ≤ g(x0), (b) d(x , x0) ≤ α and (c)x ∈ SMin(g + (ε/α)d(·, x))


Consider that g is l.s. and l.b. For each x0 ∈ dom(g) and γ > 0 thereexists x ∈ X such that (d) g(x)− g(x0) + γd(x0, x) ≤ 0 and (c)x ∈ SMin(g + γd(·, x))

1974 (version I) 1979 (version II)Hypoteses l.s., l.b., x0 ∈ AMin(g, ε) l.s., l.b., x0 ∈ dom(g)

Thesis (a), (b), (c) (d), (c)

If g is l.b., x0 ∈ Min(g,g(x0)− infx∈X g(x)) for all x0 ∈ X

C. Gutiérrez The Ekeland variational principle in vector optimization



Loridan (1984)Németh (1986)Göpfert et al. (2000) and Gutiérrez et al. (2008)Summary

Theorem (Loridan, 1984)

Let X be a normed space, Y = Rp, D = Rp+,

q = (q1, . . . ,qp) ∈ Rp+\{0} and ε =

∑pi=1 qi . Suppose that fi are l.s.

and l.b. for all i . If x0 ∈ AMin(∑p

i=1 fi , ε) then there exists x ∈ X suchthat(a)

∑pi=1 fi (x) ≤

∑pi=1 fi (x0),

(b) ‖x − x0‖ ≤√

ε,(c) x ∈ E(f + (1/

√ε)‖ · −x‖q,≤D)

Definition (Vályi, 1985)

x0 ∈ X is an ε-efficient solution of (VOP) in the Vályi sense withrespect to ξ ∈ D+ if x0 ∈ AMin(ξ ◦ f , ε)





Theorem (Németh, 1986)

Suppose that D is normal and closed, f is submonotone and o.l.b. Foreach q ∈ D\{0}, γ > 0 and x0 ∈ X there exists x ∈ X such that(d) f (x) + γd(x , x0)q ≤D f (x0),(c) x ∈ SE(f + γd(·, x)q,≤D)

The convex cone D is normal if

(yi ), (zi ) ⊂ D, yi ≤D zi , zi → 0⇒ yi → 0

f is submonotone if

X ⊃ (xn)→ x0, f (xk ) ≤D f (xm), k > m⇒ f (x0) ≤D f (xn)∀n

f is order lower bounded (o.l.b.) if there exists z ∈ Y such thatz ≤D f (x) for all x ∈ X





Definition (Németh, 1986)

x0 ∈ X is an ε-efficient solution of (VOP) in the Németh sense withrespect to H ⊂ D\{0} if (f (x0)− εH − D) ∩ f (X ) = ∅

Theorem (Németh, 1986)

Suppose that D is normal and closed, f is submonotone and o.l.b. Letq ∈ D\{0} and consider H = D\U, where U ⊂ Y is a neighbourhoodof zero.If x0 ∈ X is an ε-efficient solution of (VOP) in the Németh sense withrespect to H then there exists x ∈ X such that(a) f (x) + εd(x0, x)q ≤D f (x0),(b) d(x0, x)q ∈ U,(c) x ∈ SE(f + εd(·, x)q,≤D)





Theorem (Göpfert-Tammer-Zalinescu, 2000)

Let q ∈ D\(−cl(D)). Assume that f is o.l.b. and the sets{y ∈ X : f (y) + d(y , x)q ≤D f (x)} are closed for all x ∈ X

Then, for each x0 ∈ X there exists x such that(d) f (x) + d(x0, x)q ≤D f (x0)(c) x ∈ SE(f + d(·, x)q,≤D)

The result is true if D is not pointed





Definition (Kutateladze, 1979)

x0 ∈ X is an ε-efficient solution of (VOP) in the Kutateladze sensewith respect to q ∈ D\{0} if there is not x ∈ X such thatf (x) ≤D f (x0)− εq, f (x) 6= f (x0)− εq

Theorem (Gutiérrez-Jiménez-Novo, 2008)

Let q ∈ D\{0}, α > 0, ε > 0 and x0 ∈ X . Assume that the sets{x ∈ X : f (x) ≤D f (x0) + rq} are closed ∀ r ∈ R

If x0 ∈ X is an ε-efficient solution of (VOP) in the Kutateladze sensewith respect to q then there exists x ∈ X such that(a) f (x) ≤D f (x0) y f (x) 6= f (x0) if x 6= x0,(b) d(x , x0) ≤ α,(c) x ∈ SE(f + (ε/α)d(·, x)q,≤D)





Assumptions

1 Several hypotheses have been assumed on the order cone

2 Different ε-efficiency notions are considered

3 The objective function is o.l.b., or there exists an ε-efficientsolution in the Kutateladze sense

4 Usually, the lower semicontinuity hypothesis requires that certainsublevel sets are closed

Thesis. There are two types of conclusions: (a),(b),(c) or (d),(c), andthe perturbation function is single-valued

Proof. Various techniques and tools have been used: scalarization,Zorn lemma, Dancs-Hegedus-Medvegyev theorem, etc.




Idea, objective and toolsOn the lower boundedness assumptionThe concept of set-valued D-metricOn the lower semicontinuity assumption(C, ε)-efficient solutions of problem (VOP)EVPs for (VOP) with a set-valued perturbation function

Idea. In a vector optimization problem, the objective function can beimproved via any direction of the order cone. So, the perturbationfunction can be defined by using different directions. In the literature,the perturbation function has been defined by using a fixedimprovement direction:

(c) x ∈ SE(f + (ε/α)d(·, x)q,≤D) (q ∈ D\{0})

However, if one considers several improvement directions at thesame time, then one obtains stronger EVPs, since the set of pointsthat satisfy statement (c) is smaller than the previous one:

(c) x ∈⋂

q∈H

SE(f + (ε/α)d(·, x)q,≤D) (H ⊂ D\{0})





Consider the following set-valued optimization problem:

Min{G(x) : x ∈ X} (SVOP)

where G : X ⇒ Y is a set-valued mapping such that G(x0) = {y0},i.e., the image of x0 ∈ X by G is a singleton

Definition

x0 is a strict efficient solution of (SVOP), denoted x0 ∈ SE(G,≤D), ifthere is not x ∈ X\{x0} and y ∈ G(x) such that y ≤D y0

(c) x ∈ SE(f + (ε/α)d(·, x)H,≤D) (H ⊂ D\{0})





Objective. To obtain an EVP whose perturbation function is set-valued

Tools. Suitable lower boundedness and lower semicontinuityconcepts, appropriate order cones, a notion of set-valued metric andthe concept of (C, ε)-efficiency

First, conclusions (d) and (c) are obtained (version I). Then, an EVPwith conclusions (a), (b) and (c) is showed (version II), which is basedon the (C, ε)-efficiency concept. In both cases, the proofs areanalytical





Definition

1 D is normal if for each two nets (yi ), (zi ) ⊂ Y such that0 ≤D zi ≤D yi , yi → 0 then zi → 0

2 D is w-normal if is normal with respect to the weak topology

3 D is based if there exists a convex set B ⊂ Y such that 0 /∈ cl(B)and cone(B) = D





Definition(i) M ⊂ Y is order lower bounded (o.l.b.) if there exists z ∈ Y suchthat z ≤D y for all y ∈ M

(ii) M is lower bounded in the strong (resp. weak) topology, denotedt.l.b.s. (resp. t.l.b.w.), if for each neighbourhood V ⊂ Y of zero in thestrong (resp. weak) topology, there exists ρ > 0 such that M ⊂ ρV + D

(iii) (Gutiérrez-Jiménez-Novo, 2008) M ⊂ Y is order lower boundedby scalarizations (o.l.b.s) if inf{ξ(y) : y ∈ M} > −∞ ∀ ξ ∈ D+

Proposition (Gutiérrez-Jiménez-Novo, 2008)

(i) M o.l.b. =⇒ M t.l.b.s. =⇒ M t.l.b.w. =⇒ M o.l.b.s.

(ii) If M ⊂ −D and D is w-normal then M is o.l.b.s. ⇐⇒ M is t.l.b.w.





(c) x ∈ SE(f + d(·, x)H,≤D) (H ⊂ D\{0})

Definition (Gutiérrez-Jiménez-Novo, 2008)

A set-valued mapping F : X × X → 2D is a set-valued D-metric(s.v.D-metric) if:(i) F (x , y) 6= ∅, F (x , x) = {0} ∀ x , y ∈ X , 0 /∈ F (x , y) ∀ x 6= y(ii) F (x , y) = F (y , x) ∀ x , y ∈ X(iii) F (x , y) + F (y , z) ⊂ F (x , z) + D ∀ x , y , z ∈ X

Let ∅ 6= H ⊂ D\{0} be a D-convex set (H + D is convex). ThenF (x , y) = d(x , y)H is a s.v.D-metric





Let F be a s.v.D-metric, x ∈ X and γ > 0. We denote

AγFx := {z ∈ X : (f (z) + γF (z, x)− f (x)) ∩ (−D) 6= ∅}


We say that γF and problem (VOP) are lower semicontinuous (l.s.) atx ∈ X if for each (zn) ⊂ AγF

x such that (zn)→ z and f (zn) ≤D f (zm)

∀n > m, it follows that z ∈ AγFx

AγFx is the sublevel set of the set-valued mapping f + γF (·, x) at f (x)

in the vectorial sense





Let ∅ 6= C ⊂ Y\{0} be a coradiant set (∪β>1βC ⊂ C) and ε > 0


x0 ∈ X is a (C, ε)-efficient solution of (VOP), denotedx0 ∈ AE(f ,C, ε), if (f (X )− f (x0)) ∩ (−εC) = ∅


Let q ∈ D\{0}, H ⊂ D\{0} and ξ ∈ D+\{0}. The following sets arecoradiant:(i) C = {y ∈ Y : ξ(y) > 1} (Vályi, 1985),(ii) C = q + D\{0} (Kutateladze, 1979),(iii) C = H + D\{0} (Németh, 1986),(iv) C = D\(q − D) (White, 1986),(v) C = {y ∈ Y : ξ(y) > 1} ∩ D (Helbig-Pateva, 1994),(vi) If Y is normed, C = {y ∈ Y : ‖y‖ > 1} ∩ D (Tanaka, 1995)






Suppose that D is w-normal and based. Consider γ > 0 and let F bea s.v.D-metric such that

0 /∈ clw ⋃

d(x,y)≥δ

F (x , y)

, ∀ δ > 0 (1)

Let x0 ∈ X . If γF and problem (VOP) are l.s. at x0 and(f (X )− f (x0)) ∩ (−D) is o.l.b.s. then there exists x ∈ X such that(a) (f (x) + γF (x0, x)− f (x0)) ∩ (−D) 6= ∅(c) x ∈ SE(f + γF (·, x),≤D)

(f (X )− f (x0)) ∩ (−D) is o.l.b.s. iff is t.l.b.w., since D is w-normalEach s.v.D-metric F satisfies 0 /∈ F (x , y) for all x 6= y . Statement (1)is a little stronger than this property






Consider γ > 0 and a s.v.D-metric F . Assume that C is coradiant,C + D = C and

0 /∈ clw ⋃

d(x,y)≥δ

F (x , y)

, ∀ δ > 0

⋃x,y∈X

F (x , y) ⊂ cone(C)

If x0 ∈ AE(f ,C, ε), γF and problem (VOP) are l.s. at x0 and(f (X )− f (x0)) ∩ (−D\C) is o.l.b.s., then there exists x ∈ X such that(a) (f (x) + γF (x0, x)− f (x0)) ∩ (−D) 6= ∅(b) F (x0, x) ∩ (cone(C)\((ε/γ)C)) 6= ∅(c) x ∈ SE(f + γF (·, x),≤D)






Let D be a nontrivial pointed convex cone, let C ⊂ D\{0} be acoradiant set such that 0 /∈ clw (C) and let F be a s.v.D-metric. Wesay that D, C and F are compatible if C + D = C and⋃

x,y∈X

F (x , y) ⊂ cone(C)


If D, C and F are compatible, then both EVP are equivalent


References. Motivations and vector optimizationproblems

I. Ekeland. On the variational principle. J. Math. Anal. Appl. 47, 324–353,1974I. Ekeland. Nonconvex minimization problems. Bull. Amer. Math. Soc.(N.S.) 1(3), 443–474, 1979

A. Göpfert, H. Riahi, C. Tammer, C. Zalinescu. Variational Methods in Par-tially Ordered Spaces. Springer-Verlag, New York, 2003D. T. Luc. Theory of Vector Optimization. Lecture Notes in Econom. Math.Systems 319, Springer-Verlag, Berlin, 1989G. Y. Chen, X. X. Huang, X. Yang. Vector Optimization. Set-Valuedand Variational Analysis. Lecture Notes in Econom. Math. Systems 541,Springer-Verlag, Berlin, 2005S. Dancs, M. Hegedus, P. Medvegyev. A general ordering and fixed-pointprinciple in complete metric spaces. Acta. Sci. Mathem (Szegd) 46, 381–388, 1983

References. ε-efficiency concepts

C. Gutiérrez, B. Jiménez, V. Novo. On approximate efficiency in multiobjec-tive programming. Math. Meth. Oper. Res. 64, 165–185, 2006C. Gutiérrez, B. Jiménez, V. Novo. A set-valued Ekeland’s variational prin-ciple in vector optimization. SIAM J. Control Optim. 47(2), 883–903, 2008S. Helbig, D. Pateva. On several concepts for ε-efficiency. OR Spektrum16, 179–186, 1994S. S. Kutateladze. Convex ε-programming. Soviet Math. Dokl. 20, 391–393,1979A. B. Németh. A nonconvex vector minimization problem. Nonlinear Anal.10(7), 669–678, 1986T. Tanaka. Approximately efficient solutions in vector optimization. J. Multi-Criteria Decis. Anal. 5, 271–278, 1996I. Vályi. Approximate solutions of vector optimization problems, SystemsAnalysis and Simulation (A. Sydow, M. Thoma, and R. Vichnevetsky eds.),Akademie-Verlag, Berlin, 246–250, 1985D. J. White. Epsilon efficiency. J. Optim. Theory Appl. 49(2), 319–337, 1986

References. EVP for (VOP) with a vector-valuedperturbation

G. Y. Chen and X. X. Huang. A unified approach to the existing three typesof variational principles for vector valued functions. Math. Methods Oper.Res. 48(3), 349–357, 1998A. Göpfert, C. Tammer, and C. Zalinescu. On the vectorial Ekeland’s vari-ational principle and minimal points in product spaces. Nonlinear Anal.39(7), 909–922, 2000C. Gutiérrez, B. Jiménez, V. Novo. A set-valued Ekeland’s variational prin-ciple in vector optimization. SIAM J. Control Optim. 47(2), 883–903, 2008G. Isac. The Ekeland’s principle and the Pareto ε-efficiency. Lecture Notesin Econom. Math. Systems 432, Springer-Verlag, Berlin, 148–163, 1996P. Loridan. ε-Solutions in vector minimization problems. J. Optim. TheoryAppl. 43(2), 265–276, 1984A. B. Németh. A nonconvex vector minimization problem. Nonlinear Anal.10(7), 669–678, 1986C. Tammer. A generalization of Ekeland’s variational principle. Optimization25(2–3), 129–141, 1992

References. EVP for (VOP) with a set-valuedperturbation

E. M. Bednarczuk, M. Przybyla. The vector-valued variational principle inBanach spaces ordered by cones with nonempty interiors. SIAM J. Optim.18(3), 907–913, 2007C. Gutiérrez, B. Jiménez, V. Novo. A set-valued Ekeland’s variational prin-ciple in vector optimization. SIAM J. Control Optim. 47(2), 883–903, 2008

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