In this paper the concept of duality has been introduced for Invex Programming Problem in infinite dimensional Hilbert Spaces. A generalization of the concept of Wolfe-duality has been proposed for such class of problems. Some important theorems regarding the characterization of the dual problems have also been discussed

Chatterjee, Sandip

Mukherjee, R. N.

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 30, No 3 (2015), 255–262DUALITY IN INVEX PROGRAMMING PROBLEM IN HILBERT SPACESandip Chatterjee and R.N.MukherjeeAbstract. In this paper the concept of duality is introduced for the invex programmingproblem in infinite dimensional Hilbert spaces. A generalization of the concept of Wolfe-duality is proposed for such class of problems. Some important theorems regarding thecharacterization of the dual problems are also discussed.Keywords: Invexity, Wolfe-Duality, KKT Conditions.1. IntroductionIn the last few decades, generalized monotonicity, duality and optimality condi-tions in the theory of invex optimization have been discussed rigorously by sev-eral authors. In most of the cases, such problems have been characterized in Rn[1, 2, 3, 4, 5, 6, 7, 12, 14, 16, 18, 19, 22]. Chatterjee and Mukherjee [16] very recentlygeneralized the concept of invex optimization fromRn to an arbitraryHilbert Space.Such problems are known as the invex programming problem (IPP). This papergeneralizes the concept of Wolf-duality for such class of infinite dimensional op-timization problems. Some important theorems regarding the characterization ofthe dual problems, e.g., weak duality, strong duality and strict converse duality,have been proved in this generalized structure. This generalization allows us tostudy a wider class of infinite dimensional optimization problems and their duals.2. PrerequisitesDefinition 2.1. A subset C of Rn is convex [11] if for every pair of points x1, x2 in C,the line segment[x1, x2] = {x : x = αx1 + βx2, α ≥ 0, β ≥ 0, α + β = 1}belongs to C.The set C is said to be invex [18] if there is a vector function η : C × C → Rn suchthatReceived March 30, 2015; Accepted April 19, 20152010 Mathematics Subject Classification. Primary 49; Secondary J50, J52255256 Sandip Chatterjee, R.N.Mukherjeex1 + λη(x1, x2) ∈ C ∀ x1, x2 ∈ C and ∀λ ∈ [0, 1]Definition 2.2. Let C be an open convex set in Rn and let f be real valued anddiﬀerentiable on C. Then, f is convex [11] iff (x) − f (y) ≥ 〈∇ f (y), x − y〉 , ∀ x, y ∈ C.The function f is said to be invex [18] if there is a vector function η : C × C→ Rnf (x) − f (y) ≥ 〈∇ f (y), η(x, y)〉 , ∀ x, y ∈ C.Definition 2.4. Let X and Y be two normed vector spaces. A continuous lineartransformation A: X → Y is said to be the Fre´chet (Strong) derivative [23] off : X→ Y at x, if for every  > 0, ∃ δ > 0 such that,‖ f (x + h) − f (x) − Ah ‖Y≤  ‖ h ‖X, for all hwith ‖ h ‖X≤ δWhen the derivative exists it is denoted by Df (x).Proposition 2.1. [23] Let X be a vector space and Y be a normed space. Let S bea transformation, mapping an open set D ⊂ X into an open set E ⊂ Y and let Pbe a transformation, mapping E into a normed space Z. Let T = PS, S is Fre´chetdiﬀerentiable at x ∈ D and P is Fre´chet diﬀerentiable at y = S(x) ∈ E. Then, T isFre´chet Diﬀerentiable at x and DT(x) = DP(y)DS(x).Remark. [11] It is to be noted that in Rn, Df (x) = ∇ f (x).Let H1 and H2 be two Hilbert spaces with some archimedean ordering “ ≥ ”and X ⊆ H1 is an open set. Let f : X → H2 be a diﬀerentiable(Fre´chet) function,η : X × X→ H2 be a vector function and e ∈ H2 such that ‖e‖H2 = 1.Definition 2.5. [16] The function f is said to be (η, e) − invex if(2.1) f (x) − f (y) ≥ 〈Df (y) , η(x, y)〉e, ∀ x, y ∈ X,where Df (y) stands for the Fre´chet derivative of f at y.The function f is said to be strictly (η, e)-invex if there exists e ∈ H2 with ‖e‖H2 = 1,such thatf (x) − f (y) > 〈D( f (y), η(x, y)〉e, ∀x, y ∈ X, x  y.Remark. It is to be noted that, ifH1=H2=Rn, e = (1, 1, 1, . . . , 1) and η(x, y) = (x− y),then f becomes a convex function in Rn. The norm in this case can be taken as(n)− 12 -multiple of the usual Euclidean norm.LetH1 andH2 be two realArchimedeanorderedHilbert spaces. Letφ : H1 → H2and f : H1 → H2 be (η, e) − invex functions (i.e., both the functions are invex withrespect to the same η and e) such thatf (x) = f (y)⇒ (x − y) ∈ Ker f .Let us consider the following program:Duality in Invex Programming Problem in Hilbert Space 257Min φ(x)s.t. f (x) = y x ∈ H1 , y ∈ H2x ≥ θH1 .Let usdenote theprogramby IP(H1,H2,φ, f ) or simplyby IP(if there is no confusion).In our further discussion, we shall refer this problem as invex primal.Example 2.1. Detection Filter Problem (Fortmann, Athans) [20]. Following is anexample of an IP:Min{−〈u, x〉 : u ∈ L2[0,T]}s.t. 〈u, st〉 − 〈u, s〉 ≤ 0 δ ≤ |t| ≤ T−〈u, st〉 − 〈u, s〉 ≤ 0 δ ≤ |t| ≤ T‖u‖ ≤ 1Where ‖ · ‖ and 〈·, ·〉 denotes the usual norm and inner product in L2-space, and s isa signal function with the assumption that the energy of s equals to 1, i.e., ‖s‖2 = 1.KKT Conditions. x∗ ∈ H1 is a solution of IP iﬀ there exist a scalar λ∗ ≥ 0 such thati) λ∗ f (x∗) = θii)D(φ(x∗) + λ∗ f (x∗)) = θiii)λ∗ ≥ 03. Fritz-John ConditionLet H1 and H2 be two real Archimedean ordered Hilbert spaces and I be an openinvex set in H1. Let f , , h : I→ H2 be diﬀerentiable (Fre´chet) (η, e) − invex functionswith respect to same η(·, ·) and e. Let us consider the following IPP:Min f (x)s.t. (x) ≤ θH2h(x) = θH2x ≥ θH1 .(3.1)The following theorem is a generalization of the very well known Fritz-Johnconditions. Under the assumption of invexity, the conditions are not only necessarybut also suﬃcient. The proof of the necessity of the conditions is motivated byMcShane [11]. In our discussion, whenever we consider topology we mean weaktopology.Theorem 3.1. x∗ ∈ I is a solution of IPP (3.1) iﬀ there exist non-zero scalars λ, μ andν such that(i) λ(x∗) = θ(ii)λDf (x∗) + μD(x∗) + νDh(x∗) = θ.Proof. Let K be a strictly increasing diﬀerentiable real valued function defined onH2 such that, K(x) > 0 whenever x > θ and K(x) = 0 elsewhere. It is to be noted258 Sandip Chatterjee, R.N.Mukherjeethat, DK(x) > 0 for x > θ. Since  is continuous and I is open , there exist an 0 > 0such that B(θ, 0) ⊂ I and (x) ≤ θ for x ∈ B(θ, 0). Now, let us define a function(3.2) F(x, p) = ‖ f (x)‖ + ‖x‖2 + p{K((x))+ ‖h(x)‖2}, x ∈ I and p ∈ Z+.We assert that, for each  satisfying 0 <  < 0, there exist a positive integerp() such that, for x with ‖x‖ = , F(x, p()) > θ. If not, then there would exist an ′with 0 < ′ < 0 such that, for each positive integer p, there exist a vector xp with‖xp‖ = ′ and F(xp, p) ≤ θ. Hence form (3.2),(3.3) ‖ f (x)‖ ≤ −{‖x‖2 + p{K(xp) + ‖h(xp)‖2}}Now since ‖xp‖ = ′ and since S(0, ′) = {y : ‖y‖ = ′} is weakly compact, then thereexist sub-sequences, which we relabel as xp and p ,and a point x0 with ‖x0‖ = ′,such that, xp → x0. Since f ,  and h are continuous, f (xp) → f (x0); (xp) →(x0); h(xp) → h(x0). Therefore, dividing (3.3) by −p and letting p → ∞ we get,K((x0)) + ‖h(x0)‖2 = 0. Hence by definition of K(·, ·), (x0) ≤ θ and h(x0) = θ.Thus x0 is a feasible vector. Now, by a suitable aﬃne transformation x∗ can beassumed as θ and f (x∗) = f (θ) = 0. Therefore, f (x0) ≥ f (θ) = 0. Now, from (3.3),‖ f (xp)‖ ≤ −(′)2 < 0, which is a contradiction. Hence the assertion is true.Again, for each  ∈ (0, 0), the function F(·, p()) is continuous on the closed ballB(0, ). Since B(0, ) is weakly compact, F(·, p()) attains its minimum on B(0, ) atan interior point x of B(0, ). Hence(3.4) DF(x, p()) = 0Now let us assume thatL() = 1 + (p()DK((x)))2 + (p()D(‖h(x)‖))2(3.5)λ() =D(‖ f (x)‖)√L()(3.6)μ() =⎧⎪⎪⎨⎪⎪⎩p()DK((x))√L()if (θ) = θ0 else(3.7)ν() =p()D(‖h(x‖)√L().(3.8)It is to be noted that λ() ≥ 0,and μ(), ν() ≥ θ.Now, from (3.2), (3.4) and (3.5), we get(3.9) λ()Df (x) +D(‖x‖2)√L()+ μ()D(x) + ν()Dh(x) = 0.Let → 0 through a sequence of values k. Then since ‖x‖ < , we have(3.10) xk → θ, λ(k)→ λ, μ(k)→ μ, ν(k)→ ν.Duality in Invex Programming Problem in Hilbert Space 259Therefore from (6) and (7) we get λDf (θ) + μD(θ) + νDh(θ) = θ and from thedefinition of μ, μ(θ) = θ. This proves the necessity of the conditions.Let us now consider on the suﬃciency of the conditions. Since f is (η, e) − invex,f (x) − f (x∗) ≥ 〈Df (x∗), η(x, x∗)〉e= −〈μD(x∗) + νDh(x∗), η(x, x∗〉e= −{μ〈D(x∗), η(x, x∗〉e + ν〈Dh(x∗), η(x, x∗〉e}≥ −{μ((x)− (x∗)) + ν(h(x) − h(x∗))}= −μ(x)≥ θwhich proves the suﬃciency of the conditions.It is quite obvious that using any constraint qualification assuring the positivityof λ, we can obtain the generalization of the very popular Karush-Kuhn-Tuckerconditions stated earlier.4. Duality in IPPDefinition 4.1. The generalized Wolfe dual for the invex primal is defined asMaxu,λ( f (u) + λ(u))subject to D( f (u) + λ(u)) = θλ ≥ 0u ∈ H1 and λ is a scalar.This problem will be referred to as invex dual denoted by ID in further discussion.Theorem 4.1. Strong Duality. Under the condition of a suitable constraint qualifi-cation [18] for IP, if x◦ is minimal for IP, then (x◦,λ◦) is minimal for ID, where λ◦ isgiven by the KKT Conditions and f and  are (η, e) − invex.Proof: Let (u,λ) be any vector feasible for ID. Then,( f (x◦) + λ◦(x◦)) − ( f (u) + λ(u))= f (x◦) − ( f (u) + λ(u))≥ 〈Df (u) , η(x◦, u)〉e − λ(u)=−〈λD(u) , η(x◦, u)〉e − λ(u)=λ{−〈D(u) , η(x◦, u)〉e} − λ(u)≥ λ{(u) − (x◦)} − λ(u)=−λ(x◦)≥ θTherefore, (x◦, λ◦) is maximal in the dual problem and since λ◦(x0)=0, theextreme of the two problems are same.Theorem 4.2. Weak Duality. Let x be feasible for IP and (u, λ) be feasible for ID,then we have f (x) ≥ f (u) + λ(u).Proof. From the invexity assumption, we have( f (x) − f (u) − 〈Df (u), η(x, u)〉e)+ λ((x)− (u) − 〈D(u), η(x, u)〉e) ≥ θ260 Sandip Chatterjee, R.N.Mukherjeewhich further implies( f (x) − ( f (u) + λ(u))) ≥ 〈Df (u), η(x, u)〉e+ λ〈D(u), η(x, u)〉e− λ(u)= 〈Df (u) + λD(u), η(x, u)〉e− λ(u)= −λ(u)≥ θ.Theorem 4.3. Strict Converse Duality. Let f and  are (η, e) − invex. Let x ∗ beoptimal for IP and (x¯, λ¯) be optimal for ID. If a suitable constraint qualification[18]is satisfied for IP and f is strictly invex at x¯, then x∗ = x¯.Proof. Let x∗  x¯. By the strong duality theorem, there exists λ∗ such that (x∗, λ∗) isoptimal for ID.Hence,f (x∗) = f (x∗) + λ∗(x∗) = f (x¯) + λ¯(x¯)(4.1)Now by strict invexity of f we get,f (x∗) − f (x¯) > 〈Df (x¯, η(x∗, x¯)〉e(4.2)And by invexity of g with λ¯ ≥ 0 we get,λ¯(x∗) − λ¯(x¯) ≥ 〈λ¯D(x¯), η(x∗, x¯)〉e.(4.3)Adding 4.2 and 4.3 we get,( f (x∗) − f (x¯)) + (λ¯(x∗) − λ¯(x¯)) ≥ θ.But, since λ¯(x∗) ≤ θ, we have f (x∗) − f (x¯) − λ¯(x¯) > θ, i.e. f (x∗) − f (x¯) > λ¯(x¯),which contradicts 3.1. Therefore x∗ = x¯.REFERENCES1. A. Ben-Israel, B. Mond : What is Invexity?. Journal of Australian MathematicalSociety, Ser. B, 28(1) (1986), 1-9.2. A. Ben-Tal: On Generalized Means and Generalized Convex Functions. Journal ofOptimization Theory and Applications, 21 (1977), 1-13.3. B.D. Craven: Invex functions and constrained local minima. Bulletin of the AustralianMathematical Society, 24 (1981), 1-20.4. B.D. Craven, B.M. Glover: Invex functions and duality. 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Prentice Hall, New Jersey.1976.14. R. Pini: Invexity and generalized convexity, Optimization. 22(4) (1991), 513-525.15. R.T. Rockafellar: Convex Analysis. Princeton University Press, Princeton, 1970.16. SandipChatterjee, R.N.Mukherjee: Invexity and a Class of Constrained OptimizationProblems in Hilbert Spaces. Facta Universitatis (Nisˇ): Ser. Math. Inform., 29(4) (2014),337-342.17. S. Boyd, L. Vendenberghe: Convex Optimization. CambridgeUniversity Press, Cam-bridge, 2004.18. S.K. Mishra, G. Giorgi: Invexity and Optimization. Springer-Verlag, Heidelberg,2008.19. S. Mititelu: Invex Sets and Preinvex Functions, Journal of Advanced MathematicalStudies, 2(2) (2009), 41-52.20. TE. Fortmann, M. Athans: Filter design subject to output sidelobe constraints: theo-ritical considerations. Journal of Optimization Theory and Applications, 14 (1974),179-198.21. V. Barbu, T. Precupanu: Convexity and Optimization in Banach Spaces. Springer, NewYork, 2012.22. V. Jeyakumar, B. Mond: On generalized convex mathematical programming. Journal ofAustralian Mathematical Society Series B, 34 (1992), 43-53.23. Y. Benyamini J. Lindenstrauss: Geometric Non-Linear Functional Analysis Volume 1.American Mathematical Society, Colloquium Publications 48, 2000.Sandip ChatterjeeDepartment of MathematicsHeritage Institute of TechnologyKolkata-700107, West BengalIndiafunctionals@gmail.comR.N.MukherjeeDepartment of Mathematics262 Sandip Chatterjee, R.N.MukherjeeUniversity of BurdwanBurdwan, West BengalIndiarnm bu math@yahoo.co.in

DUALITY IN INVEX PROGRAMMING PROBLEM IN HILBERT SPACE

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DUALITY IN INVEX PROGRAMMING PROBLEM IN HILBERT SPACE

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