Keywords: duality, saddle point, complementary slackness, Karush-Kuhn-Tucker conditions, perturbation function, supporting functions.

Transcription

1 DUALITY THEORY Jørge Tid Uiversity of Copehage, Deark. Keywords: duality, saddle poit, copleetary slackess, KarushKuhTucker coditios, perturbatio fuctio, supportig fuctios. Cotets 1. Itroductio 2. Covex Prograig 3. Liear Prograig 4. Iteger Prograig 5. Geeral Matheatical Prograig 6. Coclusio Glossary Bibliography Biographical Sketch Suary The purpose of this chapter is to preset the duality theory i atheatical prograig. The atheatical setup of duality depeds o the actual proble uder study, which for exaple ay be a iteger prograig proble or a covex prograig proble. The ai itetio is to itroduce a uifyig fraework clearly exhibitig the basic ivolutory property of duality. It is the deostrated how to derive a duality theory for soe of the ost iportat classes of atheatical prograig probles. Ephasis is put o the descriptio of the iterrelatioships aog the various theories of duality. Detailed atheatical derivatios, ost of which are easily available i the textbooks etioed, have ot bee icluded. 1. Itroductio Duality is a iportat cocept i ay areas of atheatics ad its eighborig disciplies. Ecyclopedia Britaica explais the cocept as follows: I atheatics, priciple whereby oe true stateet ca be obtaied fro aother by erely iterchagig two words. Here we shall cosider duality i the cotext of optiizatio ad the two words to be iterchaged are goig to be the ters: axiu ad iiu. First soe otatio: Itroduce two arbitrary oepty sets S ad T ad a fuctio K(x, y) : S T. Cosider the followig prial proble. z 0 = ax i K( x, y). (1)

2 Let x 0 S. If z 0 is fiite ad z 0 = i K(x 0, y) the x 0 is said to be a optial solutio of the proble. (For siplicity of otatio we assue that for fixed x the ier iiizatio proble of (1) is obtaied for a arguet of y T uless it is ubouded below). Thus, the prial proble is cosidered as a optiizatio proble with respect to (w.r.t.) the variable x. We shall ext itroduce the dual proble as a optiizatio proble w.r.t. y. w0 = i ax K( x, y). (2) Let y 0 T. If w 0 is fiite ad w 0 = ax K(x, y 0 ) the y 0 is said to be a optial solutio of the proble. (Agai, for siplicity of otatio we assue that the ier axiu w.r.t. x is obtaied uless the value is ubouded above). Note that the dual proble is equivalet to ax i K( x, y), which has the for of a prial proble. By the above defiitio its dual proble is i ax K( x, y), which is equivalet to (1). This exhibits the ice property of ivolutio which says that the dual of a dual proble is equal to the prial proble. We ay thus speak of a pair of utually dual probles, i accordace with the above quotatio fro Ecyclopedia Britaica. The etire costructio ay be iterpreted i the fraework of the socalled zerosu gae with two players, player 1 ad player 2. Player 1 selects a strategy x aog a possible set of strategies S. Siilarly, player 2 selects a strategy y T. Accordig to the choice of strategies a aout K(x, y), the socalled payoff, is trasferred fro player 2 to player 1. I the prial proble (1), player 1 selects a (cautious) strategy for which this player is sure to receive at least the aout z 0. Player 2 selects i the dual proble (2) a strategy such that w 0 is a guarateed axiu aout to be paid to player 1. Iterestig cases are obtaied whe optial solutios exist for both probles such that z 0 = w 0. I this case, we speak of strog duality. I geeral, we have the followig socalled weak duality. Propositio 1: z 0 w 0. Proof: For ay x 1 S we have K(x 1, y) ax K(x, y). Hece

3 i K( x, y) i ax K( x, y). 1 Sice x 1 S is arbitrarily selected we get ax i K( x, y) i ax K( x, y), i.e. z 0 w 0. Whe strog duality does ot exist, i.e. whe z 0 < w 0 we speak of a duality gap betwee the dual probles. Closely related to strog duality is the followig otio. (x 0, y 0 ) S T is a saddle poit provided that K( x, y ) K( x, y ) K( x, y) for all ( x, y) S T. The ext propositio states the relatioship. Propositio 2: (x 0, y 0 ) S T is a saddle poit if ad oly if: (i) x 0 is a optial solutio of the prial proble, (ii) y 0 is a optial solutio of the dual proble ad (iii) z 0 = w 0. Proof: By a reforulatio of the defiitio, we get that (x 0, y 0 ) is a saddle poit if ad oly if i K( x, y ) = K( x, y ) = ax K( x, y ). This iplies that z = ax i K( x, y) K( x, y ) i ax K( x, y) = w. I geeral, by the weak duality i propositio 1 z 0 w 0. Hece (x 0, y 0 ) is a saddle poit if ad oly if ax i K( x, y) = K( x, y ) = i ax K( x, y). 0 0 This is equivalet to (i), (ii), ad (iii) i the propositio. I the above fraework of a gae, a saddle poit costitutes a equilibriu poit aog all strategies.

4 I the sequel, we shall cosider soe specific types of optiizatio probles derived fro the prial proble (1) ad the dual proble (2) by further specificatios of the sets S ad T ad the fuctio K(x, y). Let x fuctio b be a variable, f(x) : a fuctio, g(x) : a vector of costats. Let K(x, y) = f(x) + y(b g(x)) with y defied as the oegative orthat of y 0 a ultivalued ad +, i.e. T is. The prial proble the has the for axi f ( x) yg( x) + yb. (3) Observe that if for a give x a iequality of g(x) b is violated, the the ier iiizatio i the prial proble (3) is ubouded. Hece, the correspodig x ca be eglected as a cadidate i the outer axiizatio. Otherwise, if a give x S satisfies g(x) b we say that x is a feasible solutio. I this case, the ier iiizatio over y + yields f(x). Hece, the prial proble ay be coverted to ax f( x) s.t gx ( ) b x S. The otatio s.t. stads for subject to. The above is a stadard forat for a geeral atheatical prograig proble cosistig of a objective fuctio to be axiized such that the optial solutio foud is feasible. I the sequel we shall study various types of atheatical prograig probles, first by a discussio of the classical case of covex prograig. For each type we shall create the specific dual proble to be derived fro the geeral dual (2). TO ACCESS ALL THE 12 PAGES OF THIS CHAPTER, Visit: (4) Bibliography Ecyclopedia Britaica Olie, [A geeral ad widely used ecyclopedia available at all ajor libraries ad o the iteret at < Mioux M. (1986). Matheatical Prograig, Theory ad algoriths. Chichester: Wiley. [This is a classic oograph with a detailed descriptio of all ajor developets i covex ad geeral atheatical prograig]

5 Nehauser G. L. ad Wolsey L. A. (1999). Iteger ad Cobiatorial Optiizatio. Paperback Editio, Wiley. [This is a reprit of a classic oograph, which appeared i It gives a detailed presetatio of all ajor aspects of iteger prograig] Owe G. (1995). Gae Theory. 3rd editio. Acadeic Press. [A classic book o gae theory for the atheatically orieted reader] Rockafellar R. T. (1996). Covex aalysis. Reprit Editio, Priceto Uiversity Press. [This is a reprit of a ai oograph about covexity. Appeared i It gives a coplete axioatic developet of all ajor issues i the area] Schrijver A. (1986). Theory of Liear ad Iteger Prograig. Wiley. [A classic book with a coplete presetatio ad discussio of all ajor optiizatio probles of liear ad cobiatorial ature icludig graph ad etwork probles] Wolsey L. A. (1998). Iteger Prograig. Wiley. [A fudaetal textbook i the field givig a theoretical ad algorithic aalysis of all ajor areas withi iteger prograig] Biographical Sketch Jørge Tid is Professor of Operatios Research i the Departet of Statistics ad Operatios Research, Istitute for Matheatical Scieces, Uiversity of Copehage, Deark. He received his aster s degree at Uiversity of Copehage ad his seior doctor s degree at the Uiversity of Aarhus, Deark. He was previously Associate Professor at Uiversity of Aarhus, ad has held visitig positios at Corell Uiversity, Uiversité Catholique de Louvai, CaregieMello Uiversity ad Uiversity of Liköpig. He has doe research i theory, odelig ad solutio ethods i the area of optiizatio. He has particularly bee ivolved i topics at the iterface betwee atheatics ad ecooics. He is egaged i teachig progras, especially devoted to subjects regardig this iterface. His works have bee published i iteratioal scietific jourals dealig with theory ad applicatios i optiizatio, such as Matheatical Prograig, Maageet Sciece, Operatios Research Letters, Joural of Global Optiizatio ad others. He has orgaized several scietific eetigs, ad is Progra Chaira for the 18 th Iteratioal Syposiu o Matheatical Prograig, Copehage, 2003.