ON DUALITY FOR NONSMOOTH LIPSCHITZ OPTIMIZATION PROBLEMS

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1 Yugoslav Journal of Oeratons Researh Vol 9 (2009), Nuber, 4-47 DOI: /YUJOR09004P ON DUALITY FOR NONSMOOTH LIPSCHITZ OPTIMIZATION PROBLEMS Vasle PREDA Unversty of Buharest, Buharest reda@f.unbu.ro Mruna BELDIMAN Insttute of Matheatal Statsts and Aled Matheats, Roanan Aadey, Buharest Anton BĂTĂTORESCU Unversty of Buharest, Buharest Reeved: Deeber 2007 / Aeted: June 2009 Abstrat: We resent soe dualty theores for a non-sooth Lshtz vetor otzaton roble. Under generalzed nvexty assutons on the funtons the dualty theores do not requre onstrant qualfatons. Keywords: Nonsooth Lshtz vetor otzaton, Frtz John tye neessary otzaton ondtons, dualty theores.. INTRODUCTION We shall ntrodue soe defntons used n ths artle and forulate a vetor otzaton roble together wth ts Mond-Wer dual. n The real n-densonal vetor sae wll be denoted by R and we wll use the n followng onventons for any two vetors xy, R : x< y x < y,,..., n, x y x y,,..., n, and x y, x< / y s the negaton of x< y.

2 42 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz Throughout ths aer we wll denote to a real Banah sae by X, the toologal dual of X by X, and the value of a funton ξ n X at d by ξ, d. We wll onsder ths funton for the defntons that follow: ϕ : X R Defnton. (Clarke []) The funton ϕ s loally Lshtz f for any x X there ( ) exst a neghborhood N x of x and a onstant 0 suh that for any ϕ( y) ϕ( z) K y z. x K > y, z N( x) Defnton.2 (Clarke []) The generalzed dretonal dervatve of a loal Lshtz funton ϕ at x n the dreton d s denoted by o ϕ( y+ td) ϕ( y) ϕ ( xd ; ) = lsu. t y x t 0 Defnton.3 The Clarke generalzed subgradent of a loally Lshtz funton ϕ at x s denoted by o ϕ( x) = ξ X ϕ ( x; d) ξ, d, d X. Defnton.4 (see also Gorg and Guerraggo [2]) Let us onsder: η : X X X, ρ R, d : X X R +. We say that: ϕ s ( η, ρ) -seudonvex f for x, y X, ϕ o ( x ; η( yx, )) ρ dyx (, ) ϕ( y ) ϕ( x ), or, equvalently, for x, y X, ξ ϕ( x), ϕ( y) < ϕ( x) ξ, η( y, x) < ρd( y, x ). ϕ s ( η, ρ) -quasnvex f for x, y X, o ϕ( y) ϕ( x) ϕ ( x; η( y, x)) ρd( y, x) or, equvalently, for x, y X, ξ ϕ( x), ϕ( y) ϕ( x) ξ, η( y, x) ρd( y, x ) ϕ s strtly ( η, ρ) -seudonvex f for x, y X, wth x y, ϕ o ( x ; η( yx, )) ρ dyx (, ) ϕ( y ) > ϕ( x ), x

3 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz 43 or, equvalently, for x, y X, wth x y, and ξ ϕ( x), ϕ( y) ϕ( x) ξ, η( y, x) < ρd( y, x ). For the rest of our resentaton we wll onsder the followng loally Lshtz funtons: f : X R,,...,, g : X R,,...,. We an defne the vetor otzaton roble (VP): n f ( x) : = ( f ( x), f ( x),..., f ( x)), (VP) 2 subet to: g ( x) 0,,...,, and ts Mond-Wer vetor dual roble (VD): ax f ( v ), VD) μ f v λ g = = v (0.) subet to: 0 ( ) + ( ), g v λ ( ) 0,,...,, (0.2) ( μ,..., μ, λ,..., λ ) 0. (0.3) Defnton.5 A (VP)-feasble ont x X s sad to be a weakly effent soluton for (VP) f there doesn t exst any other (VP)-feasble ont y X suh that f ( y) < f( x). In a slar anner, a weakly effent soluton for (VD) s defned. 2. DUALITY THEOREMS In ths seton we wll establsh the weak and the strong dualty relatons between the robles (VP) and (VD). Usually, see referenes [3-5, 7], the dual roble s forulated by usng the Kuhn-Tuker tye neessary otalty ondtons: μ f v λ g = = 0 ( ) + λ g ( v) = 0,,...,, ( μ,..., μ ) 0, ( λ,..., λ ) 0. ( v), Sne the equalty ondtons λ g () v = 0, and ( μ,..., μ ) 0 are not resent n the stateent of the roble (VD), we do not requre any onstrant

4 44 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz qualfaton for our dualty results by usng Frtz-John tye neessary otalty ondtons and (strt) seudonvexty assutons on the funtons. Theore 2. (Weak Dualty) Suose that the funtons f are ( η, ρ )-seudonvex, {,..., }, and g are strtly ( η, ρ )-seudonvex, {,..., }. Then, for any feasble soluton x of (VP) and any feasble soluton (, v μ, λ ) of (VD), suh that μρ + λρ 0, f( x) </ f( v), where μ = ( μ,..., μ ) R and = = λ λ λ = (,..., ) R. Proof: Let us suose that, on the ontrary, there exsts a (VP)-feasble soluton x and a (VD)-feasble soluton (,, ) v μ λ suh that f ( x) < f ( v), for all,...,. (0.4) We wll rove that the strt nequaltes (0.4) ontradt the nluson (0.). Sne the funtons f are ( η, ρ )-seudonvex, for any f (), v,...,, ξ ξ, η( x, v) < ρ d( x, v ). (0.5) We shall onsder these two ases: Case : λ = 0. Fro (0.3) and (0.5) we get μξ, η( xv, ) < μρd( xv, ) 0 = = for any f (). v ξ Ths ontradts the nluson (0.). λ Let Case 2: 0. g ( v) 0, for all M. Sne g ( x) 0, t follows g ( x) g ( v), for all M. M = {,..., λ > 0 }. Fro (0.2) Relaton (0.4) les x v, and fro the strt ( η, ρ )-seudonvexty of ξ, η( x, v) < ρ d( x, v ) g for all M and any ξ g (). v Sne λ = 0 for all M, λξ, η( x, v) < λρ d( x, v ) (0.6) = = for any g (), v ξ,...,.

5 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz 45 On the other hand, the nequalty (0.5) les that μξ, η( x, v) μρd( x, v ) (0.7) = = for any ξ f (). v Cobnng nequaltes (0.6) and (0.7), we obtan μξ + λξ, η( xv, ) < μρ + λ ρ d( xv, ) 0 = = = = for any ξ f () v and ξ g (). v Ths ontradts the nluson (0.). Theore 2.2 (Strong Dualty) Let x be a weakly effent soluton for (VP). Then, there exst μ R and λ R suh that ( x, μ, λ ) s a feasble soluton for (VD) and the obetve values of robles (VP) and (VD) are equal. Moreover, f all funtons f are ( ηρ, )-seudonvex, g are strtly ( η, ρ )-seudonvex and ( x, μ, λ ) s a weakly effent soluton for (VD). Proof: Let x be a weakly effent soluton for (VP) and let us defne [ ] h( x) = ax f ( x) f ( x). μρ + λρ 0, then = = Followng the Mna s aroah [6], we an easly hek that x s an otal soluton of the followng salar otzaton roble: n { hx ( ) subet to gx ( ) 0 }. that Fro Theore 6.. n [] we get that there exst μ R and 0 μ hx ( ) + λ g( x), = λ g ( x) = 0,,...,, ( μ, λ,..., λ ) 0. By Prooston n [] we obtan { } hx ( ) o f( x),..., = = τξ τ =, τ 0, ξ f( x). = = λ R suh Thus, there exst μ R and λ R suh that

6 46 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz μ f x λ g = = 0 ( ) + λ g ( x) 0,,...,, ( μ,..., μ, λ,..., λ ) 0, ( x),.e., ( x, μ, λ ) s a feasble soluton for (VD) and learly the values of the obetve funton of (VP) and (VD) are equal. If the funtons f are ( ηρ, )-seudonvex and g are strtly ( η, ρ )- seudonvex, then t follows fro Theore 2. that f( x) </ f( v) for any (VD)-feasble soluton (, v μ, λ ), n artular ths s true for ( x, μ, λ ), that eans that ( x, μ, λ ) s a weakly effent soluton of (VD). Exale 2. Let us onsder the followng funtons: where 2 2 ( ) =, 2( ) =, ( ) =, f x x f x x g x x x R. These funtons are obvously loally Lshtz and {} f ( x) =, f ( x) = 2 x, g( x) = 2 x. 2 We onsder the vetor otzaton roble where n {( f ( x), f 2 ( x)) x P} (VP) { R } [ ] P = x g( x) 0 =,, and ts Mond-Wer dual roble where { 2 2 ax ( f ( v), f ( v)) ( v, μ, μ, λ) D} (VD) μ + μ2(2 v) + λ(2 v) = D = (, v μ, μ2, λ) R λ( v ) 0 ( μ, μ2, λ) 0 If we take η ( x, y) = x y and ρ = 0, then f, f are ( η, ρ) -seudonvex and g s strtly ( η, ρ) -seudonvex. Let us denote 2 3 { R μ μ2 λ R μ μ2 λ } V = v (,, ), s.t. ( v,,, ) D = (,0]. It s easy to verfy that for any x P and any (, v μ, μ2, λ) D ( f ( x), f ( x)) </ ( f ( v), f ( v)) 2 2

7 V. Preda, M. Beldan, A.Bătătoresu / On Dualty for Nonsooth Lshtz 47 or, equvalently, for any [,] 2 2 ( x, x ) ( v, v ) x and any v V, </ (0.8) whh eans weak dualty between (VP) and (VD)., 0 s the soluton of all weakly effent soluton of (VP). Sne Moreover, [ ] for any [, 0] v v v 3 v v v v V there exsts ( μ, μ, λ ) R, suh that (, v μ, μ2, λ ) D, t 2 v v v follows fro (0.8) that ( v, μ, μ2, λ ) s a weakly effent soluton of (VD). Thus, strong dualty holds between (VP) and (VD). REFERENCES [] Clarke, F.H., Otzaton and Nonsooth Analyss, Wley-Intersene, New York, NY, 983. [2] Gorg, G., and Guerraggo, A., Varous tyes of nonsooth nvexty, Journal of Inforaton and Otzaton Senes, 7 (996) [3] Gorg, G., and Guerraggo, A., The noton of nvexty n vetor otzaton: sooth and nonsooth ase, J.P. Crouzex, J.E. Martnez-Legaz, and M. Volle (eds.), Generalzed Monotonty: Reent Results, Kluwer Aade Publshers, Dordreht, Holland, , 998. [4] Jeyakuar, V., and Mond, B., On generalzed onvex atheatal rograng, J. Austral. Math. So., 34B (992) [5] Lee, G.M., Nonsooth Invexty n Multobetve Prograng, J. Inforaton & Otzaton Senes, 5 (994) [6] Mna, M., Weak Pareto-otal neessary ondtons n a nondfferentable ultobetve rogra on a banah sae, J. Ot. Theory Al., 4 (983) [7] Mshra, S.K., and Mukheree, R.N., On generalzed onvex ultobetve nonsooth rograng, J. Austral. Math. So., 38B (996) [8] Mond, B., and Wer, T., Generalzed onavty and dualty, n: S. Shable and W.T. Zeba, (eds.), Generalzed Conavty n Otzaton and Eonos, Aade Press, New York, NY, 98,