This article was downloaded by: [Universitaetsbibliothek Giessen]On: 01 November 2014, At: 01:24Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optimization: A Journal ofMathematical Programming andOperations ResearchPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gopt20

(Φ1, Φ2) Optimality And Duality UnderDifferentiablityRita. Pini a & Chanchal. Singh ba Istituto di Metodi Quantitativi per le Scienze Economichee Aziendali , Universitàdeyli Studi di Milano , Via Sigieri 6,Milano, 20135, Italyb Department of Mathematics , St. Lawrence University ,Canton, New York, 13617, U.S.A.Published online: 30 Mar 2007.

To cite this article: Rita. Pini & Chanchal. Singh (1997) (Φ1, Φ2) Optimality And Duality UnderDifferentiablity, Optimization: A Journal of Mathematical Programming and Operations Research,41:2, 101-116, DOI: 10.1080/02331939708844329

To link to this article: http://dx.doi.org/10.1080/02331939708844329

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms& Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

(@,, Q,) OPTIMALITY AND DUALITY UNDER DIFFERENTIABILITY

RITA PINT" and CHANCHAL SINGHb

"Istituto di Metodi Qlrantitatici per le Scienze Economiche e Azicw/uli, Cni'ersitd degli Studi di Miluno, Via S i g i e ~ i 6 , 20135 Milnrlo, Italy;

hDepartiizerlt of Mathematics, St . Lawrence Unicersiry. Caijtcn. Ncu. York 1361 7 , C.S.A

(@,, @,)-convexity is applied to develop optimality conditions of Fritz John type and Kuhn-Tucker type under differentiabihty for a minimization problem w t h peal valued objective and inequality constraints. A dual of the blond-Weir type is consicered and a number of tveak and strong duality results are established. Weak and strong duality theof-ems are also glven in the framework of Wolfe duality.

Keytcotd~: Nonlinear programming: Fritz John conditions: Kuhn-Tucker mnditions: weak and strong duality

.t~lutl~eri~ntic.s Suhjecr Clilssificcxtion 1 99 1 : Primary: 90C30; Secondary: 26B25. 49A55. 49B99. 52A40

1. INTRODUCTION

In recent years, many generalizations of convexity have appeared in the literature aiming at applications to duality theory and optima- lity conditions. For earlier contributions. a reader ma) consult Avriel et (11. [I], Mangasarian [6] and Schaible and Ziem3a [ l l ] . For more recent contributions, the reader may consult a survey article by Pini and Singh [8] covering the last ten years of development in this area of research. Recently. Pini and Singh 191 introducecl (@,,@,)- convex functions and studied some of their properties, both in the differentiable and nondifferentiable setting. They also showed that some of the well known classes of generalized convex functions (e.g. D

ownl

oade

d by

[U

nive

rsita

etsb

iblio

thek

Gie

ssen

] at

01:

24 0

1 N

ovem

ber

2014

B-vex functions [ 2 ] , geodesic convex functions [lo]. invex functions L3] etc.) form subclasses of the class of (a,, @,)-convex functions.

The purpose of this article is to apply the (@,. @,)-convexity to develop duality theory and optimality conditions in the differentiable setting. In Section 2 we reproduce some of the basic concepts develop- ed by Pini and Siagh [9] to understand this work. In Section 3, a minimization problem with scalar valued objective and inequality constraints is introduced. In this section, Kuhn-Tucker type and Fritz John type problenls are defined under (a,, @,)-convexity and assum- ing differentiability. It is pointed out that our statement of the Kuhn- Tucker problem (and similarly of the Fritz John problem) is more general than the classical ones found for Instance in Mangasarian 161. Wc establish Kuhn-Tucker type and Fritz John type sufficiency and Krlhn-Tucker type necessary conditions. In Section 4. we introduce a Mond-Weir [7] type dual for. the minimization problem introduced in Section 3 and establish a number of weak and strong duality results. Treatment of Wolfe dual for the same minimization problem intro- duced in Section 3 is presented in Section 5. Again a number of weak and strong duality theorems are proved. The main instrument in establishing these duality results is the concept of (d l , 4,)-strict con- vexity defined in Section 2.

In the following, I ) will denote a subset of RE, The following conven- tions for equalities and inequalities will be used. If .u, j, E R", then

us 1. IS the negatlon of u < y. Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

we will also assulne that Qi, is continuous with respect to ;..

D E F ~ I T ~ O N 2.2 [9] A function f t F is said to be

(concave) on D if

for all x, 3. E D, 2 E [0, 11

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

R. PIXI AND C. SINGH

;. ~ ( 0 . I] we have

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

( @ , . Q 2 ) OPTIMALITL A N D DUALITY

3. OPTIMALITY CONDITIONS U N D E R DIFFERENTIABILITY

Consider the problem

where f : D -+ R. y = (q,. y2. . . . . y,): D -t R'" are differentiable function:, on D. Let D,, be the set of feasible solutions of (P).

For our statements of Kuhn-Tucker problem [5] and Fritz Joha problem [4], and the duality results, we fix once for all the vector valued function $: D x D x F" -+ Rm,

where . . . ,$,, are related to some @,. ... ,GI, as in Relation 2.4.

6 -Kuhn-Tucker Problem: Find .f ED,,. zi E R'" (if they exist) such that

q ( . ~ ) 6 0 (ii)

If such a (.?, zi) exists. we say it is a solution of the $-Kuhn-Tucker problem.

Remark 3.2 The statement of the a-Kuhn-Tucker Problem is more general than the classical one, where, instead of (iii), we have fig ( 2 ) = 0. This follows easily from the following

Proof Consider the inequalities

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

R. PIN1 AND C. SINGH

Therefore

Gi(x,.Y5jU,gi) - $.(?,4,0>gi) $i(s, % gi\ = liin I .p 6 0.

7,-0- /L

COROLLARY 3.1 Let .\"ED, iieRrn satisfi, the conditions q(.?) 0 , ii 2 0. uy( .? ) = 0; tlzen, .for. ever? ,fclnsihle solution s of ( P ) .

Proof From iig(i) = 0, q ( ~ ) $ 0 and .i E DO, we have that ~f 6, = 0, then G,$,(x, .I, y,) = 0, if G, > 0, we have that q, ( Y ) = 0, and therefore, from Lemma 3.1.

thereby proving the result.

( ' 1 ) the&Kuhrl-Tutker prohlem has cr aolutlon (7, U), ( b ) j is (4, , $ ~ ~ ) - s t r ~ r l j ~ C O I Z C ~ Y (it Y for 011 Y E DO, (c) q, is ($,, 6,)-conre* ut T ( i = 1 . . . . , I H 1 for d l x E DO.

Then 7 is un optmlul solution oj ( P I .

Proof By the$-Kuhn-Tucker Condlt~on (1) we have, for any feasible solutlon u,

from assumptions (b) and (c), we obtain, for any s # 2

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

(a,. a2) OPTIMALITY A N D DUALITY

Or ( b y Kuhn-Tucker Condition (iii)).

If max (f(s).,f ( f ) ) = f (s), we are done. If max ( , f ( s ) , f ( 2 ) ) = f ( s ) >,f'(.u), then, b y (a.l)(iii),

which. dividing both sides by i and taking the limit as 7. + O . gives

contradicting (3.1). Hence max ( f (.\-). f (-7): =,f(s) and the proof is comp- lete.

The following results can be established using similar arguments as in the proof of Theorem 3.1.

Tlwri 2 is ~rri optiinal solrrtior~ of Prohlelll ( P ) .

( [ I ) 2 is mi op t i i~~n l sol~rtion ofpr.obleili ( P ) ; ( h ) g xrtisfies one of rlw cori,str.triiit q ~ d j f j ~ ~ ~ i t i o i ~ s .rfirterl iii

Aiaiiprsar.ian [6].

Tlien t h i w csists a tr E R'" suc~l~ t l ~ i t ( 2 , t r ) is rr sol~rtiori of' thc +-Kuhrl- T w k e r pr~)hleil~.

Proof R y Theorem 7.3 4 in Manpasarian 161, there exist:; a N E It1'' Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

108 R. PIN1 A N D C. S l N G H

such that

From Corollary 3.1 if follows that 1';' E i $ i ( . ~ , f, yi) 6 0, thereby proving that (S, 21) is a solution of the$-Kuhn-Tucker problem.

Next we define a modified Fritz John problem and establish a Fritz John sufficiency result. 6 -Fritz John Problem: Find .? E Do, r; E R, r E R'" (if they exist) such that

r,Vf (s) + r Vg(s) = 0 (i)

g(Y) 6 0, (ii)

(I-,, F ) 3 0 (iv)

If such a (2. I-,, I;) exists. we say that it is a solution of the $-Fritz John Problem.

THEOREM 3.4 (Fr i t z Jolm Suffrciencj.). Suppose tlzut

( a ) the $-Fritz Jolin problel?~ has a solution ( 2 , Fo, F); ( h ) .for (111 x E DO either f is ( 4 , , 4 , )-strictly c o n w s u t .?, & > 0 and gi

is ( 4 , , $i)-c~oncrs ( i = 1,2, . . . ,171) at 2, or f is ( 4 , , 4,)-concex at .?, gi is ($,, $ i ) - c o ~ ~ ~ v . ~ ( i # j ) and y j is ($ , , $j)-strictlj. conces unrl r; > 0.

Then \- is an optimal soltltion of Problenl ( P ) .

Proof By hypothesis (a) it follows that

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

(a),. a)2) OPTIMALITY AND DUALITY

which implies, by hypothesis (b).

In particular,

Now. by (3.2) and Fritz John Condition (iv), it follows th: t I-, > 0. Therefore, again by (3.2), it follows that

Arguing the same way as in the proof of Theorem 3.1, we ;ondude that f is an optimal solution of Problem (P).

4. MOND-WEIR DUALITY

In this section we consider a Mond-Weir [7] type dual and establish some duality results. Mond and Weir called this dual 0 2 , we call it D l .

( D l ) maximize f ' ( j )

subject to V j ( J , ) + uVg( j ) 3 0, u,g, (J.) 9 0 ( i = 1,2,. . . ,m), 11 9 0. 1 1 6 Rm.

Let X = { ( y , u) E D x Rm : V f ( y ) + u V g ( ~ ) = 0, u,g,(y) 3 C .

be the set of feasible solutions of the dual problem ( D l ) . Let us first state the following

LEMMA 4.1 Let 2 , (J, 6) he feasible po i~z t s f o r ( P ) and ( [ ) I ) . Theiz cy U , $ , ( X , 4',g,) 6 0. D

ownl

oade

d by

[U

nive

rsita

etsb

iblio

thek

Gie

ssen

] at

01:

24 0

1 N

ovem

ber

2014

110 R. PIN1 AND C. SINGH

Proof If iii = 0. then MI $i(.f, J. Y i ) = 0. Assume that iii > 0: in this case. ( ~ ~ ( 3 ) 6 0 and gi(J) 2 0. An application of Lemma 3.1 shows that $i (T. J. gi) $ 0, therefore

THEOREM 4.1 (Weak Dual i ty) . Suppose tlzut

( a ) sen,; ( h ) (7, U) E Yl; ( c ) f is ($,, $, )-strir~t/y cowex at J for .Y = 2; ( d ) each g i ( i = 1 . ... , in) is (@,, 4,)-coi~ces a t j for s = 2.

Proof By the dual constraint we have that

By hypothesis (c), the above equation implies that

Taking into account Lemma 4.1, we get

Now by (2.l)(iii)

If max jf(Y). f ( t ) J =f (Y), we are done. If max ( f (f) , f if))- = f (J) > f (f). then. by (2.1)(1it).

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

(@,. 0,) OPTIMALITY AND DUALITY 11 1

which. dividing both sides by 1. and taking the limit as i + 0'. gives

But (4.2) contradicts (4.1). Hence, max ( f (s), f (y)) = f (2 ) and the proof is now complete. The following result can be established usi ~g similar arguments as in the proof of Theorem 4.1.

THEOREM 4.2 ( Weak Dudit!,). Suppose that

( a ) S E Do: ( h ) (,F, i i ) € Y1; ( e ) f is ( 4 1 , $2) -con~e .x at 7 for x = Y; ( d i ertch gi ( i # j ) ( 4 , . 4,)-conrex at j f i v x = 2: ( e ) iij > 0 and y j is ($,, q5,)-strictlj. concerv at !.for x = 2 .

COROLLARY 4.1 Let 2 E Do irnd (j, ii) E Y,; i f f ( F ) = f (i ) and us- sumptions o f the W ~ a l i Duality Theorein hold, then 2 is optinru1,for ( P ) rrnd (J, 11) is o p t i ~ i l f i)r ( D l ). IIZ particul~ir, if.? is feasihlejilr ( P ) m d there exists ii such tlznt (3 , i i ) is fha.sihle,for ( D l ) , then 2 is optirrul.for ( P ) m d ( 2 , i i ) is an optimal solution fbr ( D l ).

Proof Follows easily from the weak duality theorem involved in the statement.

THEOREM 4.3 (Strony Duality). Si~ppose that

( a l I 1s a11 optzvzal solutron of ( P I . ( h ) q satlsjies an , one of the tonstramt yuul$cat~on~, stated 111

Mangrrmrm [ h ]

The11 t l~ere exists ii such that ( 2 , i i ) is an optiinal .solutior~,fi~r ( D l ) .

Proof From the Kuhn-Tucker Necessary Conditions. h e r e exists 6 3 0 such that V f (.?) + ii Vg(F) = 0, iig(.?) = 0. Since ii 3 0, g(2) 6 0, it follows that Gigi(?) = 0, therefore (2 . ii) is feasible for I : D ~ ) . From Theorem 4.1. for any feasible ( F , 17) we have that f ( 2 ) 3 f (2). thereby concluding that (.?. ii) is an optimal solution for ( D l ) .

THEOREM 4.4 (Strict Dualit!,). Suppose that

( a ) 2 is (in optimal s01uti011 o f Prohlern ( P ) ; Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

( b ) y suti.Sfies aiz!. one qf the constraint qualifications stilted in Mculgasarirln [6];

(c) h!~potlzeses qf' Tlzeorem 4.1 are satisfied; ( d ) (j, C) is nn optinla1 soltrtioii o f tlze d ~ u / problem ( D l ).

Proof Suppose 2 # j. Since 2 is an optimal solution of Problem (P). there exists ieRn', G 3 0. such that (2. u) is an optimal solution of the dual Problem (Dl). From the fact that (j, 2) is an optimal solution of (Dl). we have that f (\-I = f (J) and

By hypothesis (c). we get

since C,g,(j) 2 0 for every i = 1,2, ... ,n1, and g, (2) 6 0, we have by Lemma 4.1

On the other hand,

Therefore

$,if. I., f ,go.

Since (4.3) and (4.4) contradict each other, we must have 2 = J.

5. WOLFE DUALITY

In this section we consider the following Wolfe dual of Problem (P) and establish some duality results under somewhat different~weaker D

ownl

oade

d by

[U

nive

rsita

etsb

iblio

thek

Gie

ssen

] at

01:

24 0

1 N

ovem

ber

2014

(Q,. 0,) OPTIMALlTY AND DUALITY 113

hypotheses than the ones used in the dual ( D l ) set up of the previous section:

( 0 2 ) maximize f (j.) + uq (y) Subject to V j 0) + uVg(j . ) = 0, J E D . u 3 0, tr E Rm.

Let

Y2 is the set of feasible solutions of Problem ( 0 2 ) .

THEOREM 5.1 (R/t.uk Dziulity). Suppose thut

Proof By hypothesis (b) and (c) we have

I f f (.u) < f ( j ) + Ug(j). from hypothesis (a) and the fact that f i 22 0,

Hence,

max ( f (Y) + fig(?), f ( J ) + Ug(J) J = f ( j ) + f ig(J) .

Therefore, by Lemma 3.1

contradlctlng (5.1); hence f ( T ) # f ( J ) + Uq(7). The conclusi~n now follows. D

ownl

oade

d by

[U

nive

rsita

etsb

iblio

thek

Gie

ssen

] at

01:

24 0

1 N

ovem

ber

2014

114 R. PIN1 AND C. SINGH

COROLLARY 5.1 Suppose that assumptions ( a ) - ( c ) of Theorem 5.1 are satisfied, and f ( f ) = f ( j ) + i g ( j ) . T l m 2 is optinml for ( P ) and ( J , U) is optinzal f i ~ r ( 0 2 ) .

Proof The conclusion follows easily from Theorem 5.1.

THEOREM 5.2 Suppose that

( a ) 2 is oprinial jbr ( P ) ; ( h ) (1 satisfies one qf' the co~istrrrinr quuI$cutions; ( c i J'+ Fg is ( 4 , , 4,)-s tr ict ly conces ut ?,for ecerj. (J?, C) E Y,, and,for

erery 2 E DO.

Proof By hypothesis (a), (b) and Theorem 3.3, there exists UER"' , i s 0 such that (Y, 12) is a feasible solution of (D2), and

Let ( j , 2') be a feasible solution for ( 0 2 ) . Then, by Theorem 5.1.

Therefore. by (5.31,

i.e. (.f, 6 ) is an optimal solution of ( 0 2 ) .

THEOREM 5.3 (Strict Duali t j ) . Suppose that

( a ) 2 is optinzul Jbr ( P ) ; ( b ) LJ sutisfies one of the construint yuulificutions; ( c ) ,f + Cg is $,)-strictly c o m e s at : for ecerj. (f, C) E Y,, and fbr

ull X E Do.

Proof Suppose that 2 # 2. By Theorem 5.2, there exists i 3 0, GER" such that ( f , u) is an optimal solution of ( 0 2 ) ; hence, by hypothesis ( d l ,

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14

I@, .@2) Of'TIMALlTY AND DUALITY 115

Also, by the Kuhn-Tucker Necessary Conditions (Theorem 3.3). uq(2) = 0. Therefore (5.4) reduces to

Since S 2 0 and g(.T) $0. .Scj(F) $ 0. Hence, by (5.5),

By hypothesis (d) ,

Now apply hypothesis (c) to have

Moreover, from Lemma 3.1 and (5.6),

Now, (5.8) contradicts (5.7). Hence 2 = 2

References

[ I ] Avriel, M.. Diewert. W. E.. Schaible. S. and Zang, I. (1987). Genrrtrlr~id C'orwc~cirj. Plenum Press. New York, NY.

[2] Bector. C. R . and Singh. C. (1991). B-Vex Functions. J.O.TA.. 71. 23; 253. [3] Hanson, M. A. (1981). O n Sufficiency of Kuhn-Tucker Conditions. J . ~tliith. Anal.

.4ppl., 80, 545-550. [4j John, F. (19481. Extremum Problems with Inequalities as Subsidiary Conditions.

in Studies and Essays. Courunt .4nniaersn~y Volurne. edited by K . 0. Friedrichs. 0 . E. Neugbauer and J . J. Stoker. Interscience Publishers. New York. 187- 204.

[ 5 ] Kuhn, H. W. and Tucker, A. W. (1951). Nonlinear Programming. in Pr~ceeelir~gs of the Second Berkelej Sjnlpusiunz on Mathematical Statistics uncl Prohihility. edited by J . Neyman. Uni>ersity of California Press, Berkeley. CA. 431-492.

[6] Mangasarian, 0. (1969). .Vonlinee~r Programming, McGraw Hill B m k Company. New York. NY.

[7I] Mond. B. and Weir. T. (1981). Generalized Concavity and Duality. 1 1 , Genei~trilzrci Concacity in Oprirnizrrtiorz and Economics. edited by S. Schaible and 'A'. T. Ziernba, Academic Press. 263-279.

[8] Pini. R. and Singh. C. (1996). A Survey of Recent (19851995) A d v ~ r c e s in Gene- ral i~ed Convexity with Applications to Duality and Optimality Con~litions. Oyti- nlization (To appear), D

ownl

oade

d by

[U

nive

rsita

etsb

iblio

thek

Gie

ssen

] at

01:

24 0

1 N

ovem

ber

2014

1 16 R. PIN1 AND C. SlNGH

[9] Pini, R. and Singh, C. 11996). (@,.@,)-Convexity, Optimizil~ion (To appear). [lo] Rapcsak. T. (1987). Geodesic Convexity in Nonlinear Optimization. J o ~ w ~ n i of

Optlnliztrtion Theor), nnd Applicaiions. 69, 1 6 9 1 8 3 , [ I 11 Schaible, S. and Ziemba. W. T. ieds) 11981). Gmeraiiied Conrrrrir~ 111 Optitni;atio~~

am1 E ~ ~ r w m i c , s . Acadmiic Press, New York. NY. [I21 Wolfe. P. (1961). A Duality Theory for Nonlinear Programming. Q~rartrrlj. of"

Applied ,t~lathi.~iltrrics, 19. 239-244.

Dow

nloa

ded

by [

Uni

vers

itaet

sbib

lioth

ek G

iess

en]

at 0

1:24

01

Nov

embe

r 20

14