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Higher-Order Generalized Invexity and Duality in Mathematical Programming
Journal of Mathematical Analysis and Applications 247, 173᎐182 Ž2000. doi:10.1006rjmaa.2000.6842, available online at http:rrwww.idealibrary.com on
Higher-Order Generalized Invexity and Duality in Mathematical Programming Shashi K. Mishra Department of Mathematics, Raja Balwant Singh College, Bichpuri, Agra 283105, India
and Norma G. Rueda Department of Mathematics, Merrimack College, North Ando¨ er, Massachusetts 01845 Submitted by William F. Ames Received September 23, 1999
In this paper, we introduce the concepts of higher-order type-I, pseudo-type-I, and quasi-type-I functions and establish various higher-order duality results involv䊚 2000 Academic Press ing these functions.
1. INTRODUCTION Consider the nonlinear programming problem
Ž P.
Minimize
f Ž x.
subject to
g Ž x . G 0,
Ž 1.1.
where f : R n ª R and g: R n ª R m are twice differentiable functions. The Mangasarian second-order dual w3x is
Ž MD.
Maximize
f Ž u . y y T g Ž u . y 12 pT ⵜ 2 f Ž u . y y T g Ž u . p
subject to ⵜ f Ž u . y y T g Ž u . q ⵜ 2 f Ž u . y y T g Ž u . s 0, y G 0. 173 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
174
MISHRA AND RUEDA
Mangasarian w3x formulated the following higher-order dual by introducing two differentiable functions h: R n = R n ª R and k: R n = R n ª R m :
Ž HD1.
Maximize
f Ž u . q h Ž u, p . y y T g Ž u . y y T k Ž u, p .
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . . , y G 0.
Ž 1.2. Ž 1.3.
ⵜp hŽ u, p . denotes the n = 1 gradient of h with respect to p and ⵜp Ž y T k Ž u, p .. denotes the n = 1 gradients of y T k with respect to p. If h Ž u, p . s pT ⵜf Ž u . q 12 pT ⵜ 2 f Ž u . p and k Ž u, p . s pT ⵜg Ž u . q 12 pT ⵜ 2 g Ž u . p then ŽHD1. becomes ŽMD.. Mond and Zhang w5x obtained duality results for various higher-order dual programming problems under higher-order invexity assumptions. They considered the following dual to ŽP.:
Ž HD.
Maximize
f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p .
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . . ,
Ž 1.4.
yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . F 0,
i s 1, 2, . . . , m, Ž 1.5.
y G 0.
Ž 1.6.
In this paper, we will give more general invexity-type conditions, such as higher-order type-I, higher-order pseudo-type-I, and higher-order quasitype-I conditions, and establish various duality results under these conditions. Mond and Zhang w5x proved duality results between ŽP. and ŽHD. assuming that there exists a function : R n = R n ª R n such that f Ž x . y f Ž u . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . q h Ž u, p . y pT Ž ⵜp h Ž u, p . .
Ž 1.7. and g i Ž x . y g i Ž u . F i Ž x, u . ⵜp k i Ž u, p . Ž x, u . q k i Ž u, p . ypT Ž ⵜp k i Ž u, p . . ,
i s 1, 2, . . . , m,
Ž 1.8.
where ␣ : R n = R n ª Rq _ 04 and i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positive functions.
DUALITY IN MATHEMATICAL PROGRAMMING
175
Combining the concept of type-I functions w2x and conditions Ž1.7. and Ž1.8. when hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, we say that Ž f, yg i ., i s 1, 2, . . . , m, is V-type I at the point u with respect to functions , ␣ , and i , if f Ž x . y f Ž u . G ␣ Ž x, u . ⵜf Ž u . Ž x, u . and yg i Ž u . F i Ž x, u . ⵜg i Ž u . Ž x, u . ,
i s 1, 2, . . . , m.
Mond and Zhang w6x extended the notion of V-invexity to second-order and established duality theorems under generalized second-order V-invexity conditions. If Ž f, yg i ., i s 1, 2, . . . , m, satisfies conditions Ž1.7. and Ž1.8. with hŽ u, p . s pT ⵜf Ž u. q 12 pT ⵜ 2 f Ž u. p and k i Ž u, p . s pT ⵜg i Ž u. q 12 pT ⵜ 2 g i Ž u. p then Ž f, yg i . is said to be second-order V-type I. 2. MANGASARIAN HIGHER-ORDER DUALITY THEOREM 2.1 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽHD1.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that T
f Ž x . y f Ž u . G Ž x, u . ⵜp h Ž u, p . q h Ž u, p . y pT Ž ⵜp h Ž u, p . . Ž 2.1. and T
yg i Ž u . F Ž x, u . ⵜp k i Ž u, p . q k i Ž u, p . y pT Ž ⵜp k i Ž u, p . . , i s 1, 2, . . . , m, then infimum ŽP. G supremum ŽHD1.. Proof. f Ž x . y f Ž u . y h Ž u, p . q y T g Ž u . q y T k Ž u, p . T
G Ž x, u . ⵜp h Ž u, p . y pT Ž ⵜp h Ž u, p . . q y T g Ž u . q y T k Ž u, p . ,
by Ž 2.1. ,
T
s Ž x, u . ⵜp Ž y T k Ž u, p . . y pT Ž ⵜp y T k Ž u, p . . q y T g Ž u . q y T k Ž u, p . , G 0,
by Ž 1.2. ,
by Ž 1.3. and Ž 2.2. .
The following strong duality theorem is similar to w5, Theorem 2x.
Ž 2.2.
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MISHRA AND RUEDA
THEOREM 2.2 Žstrong duality.. Let x 0 be a local or global optimal solution of ŽP. at which a constraint qualification is satisfied, and let h Ž x 0 , 0 . s 0,
k Ž x 0 , 0 . s 0,
ⵜp h Ž x 0 , 0 . s ⵜf Ž x 0 . ,
ⵜp k Ž x 0 , 0 . s ⵜg Ž x 0 . .
Ž 2.3.
Then there exists y g R m such that Ž x 0 , y, p s 0. is feasible for ŽHD1., and the corresponding ¨ alues of ŽP. and ŽHD1. are equal. If Ž2.1. and Ž2.2. are satisfied for all feasible Ž x, u, y, p ., then x 0 and Ž x 0 , y, p s 0. are global optimal solutions for ŽP. and ŽHD1., respecti¨ ely. Remark 2.1. 1, 2, . . . , m, then and Mond w2x to
If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s Ž2.1. and Ž2.2. become the conditions given by Hanson define a type-I function. If h Ž u, p . s pT f Ž u . q 12 pT ⵜ 2 f Ž u . p,
and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž2.1. and Ž2.2. become the second-order type-I conditions given by Hanson w1x when p s q s r.
3. MOND᎐WEIR HIGHER-ORDER DUALITY THEOREM 3.1 Žweak duality.. Let x be feasible for ŽP. and let Ž u, y, p . be feasible for ŽHD.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that f Ž x . y f Ž u . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . q h Ž u, p . y pT Ž ⵜp h Ž u, p . .
Ž 3.1. and yg i Ž u . F i Ž x, u . ⵜp k i Ž u, p . Ž x, u . q k i Ž u, p . y pT Ž ⵜp k i Ž u, p . . , i s 1, 2, . . . , m,
Ž 3.2.
where ␣ : R n = R n ª Rq _ 04 and i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positi¨ e functions, then infimum ŽP. G supremum ŽHD..
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DUALITY IN MATHEMATICAL PROGRAMMING
Proof. Since Ž u, y, p . is feasible for ŽHD., we have yyi g i Ž u . y yi k i Ž u, p . q pT ⵜp Ž yi k i Ž u, p . . G 0,
i s 1, 2, . . . , m.
Since x is feasible for ŽP., then by Ž3.2. and yi G 0, we obtain
i Ž x, u . ⵜp Ž yi k i Ž u, p . . Ž x, u . G 0,
i s 1, 2, . . . , m.
Since i Ž x, u. ) 0, we have ⵜp Ž yi k i Ž u, p . . Ž x, u . G 0,
i s 1, 2, . . . , m,
hence ⵜp Ž y T k Ž u, p . . Ž x, u . G 0.
Ž 3.3.
By Ž3.1., it follows that f Ž x . y f Ž u . y h Ž u, p . q pT ⵜp h Ž u, p . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . s ␣ Ž x, u . ⵜp Ž y T k Ž u, p . . Ž x, u . , G 0,
by Ž 1.4. ,
by Ž 3.3. and ␣ Ž x, u . ) 0.
THEOREM 3.2 Žstrong duality.. Let x 0 be a local or global optimal solution of ŽP. at which a constraint qualification is satisfied, and let conditions Ž2.3. be satisfied. Then there exists y g R m such that Ž x 0 , y, p s 0. is feasible for ŽHD. and the corresponding ¨ alues of ŽP. and ŽHD. are equal. If also Ž3.1. and Ž3.2. are satisfied for all feasible Ž x, u, y, p ., then x 0 and Ž x 0 , y, p s 0. are global optimal solutions for ŽP. and ŽHD., respecti¨ ely. Proof. It follows on the linear of w5, proof of Theorem 5x. Remark 3.1. If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, then Ž f, yg i ., i s 1, 2, . . . , m, satisfying conditions Ž3.1. and Ž3.2., is V-type I, and the higher-order dual ŽHD. reduces to the Mond᎐Weir dual:
Ž D.
Maximize
f Ž u.
subject to ⵜf Ž u . y ⵜy T g Ž u . s 0 yi g i Ž u . F 0, y G 0. If h Ž u, p . s pT Ž u . q 12 pT ⵜ 2 f Ž u . p
i s 1, 2, . . . , m,
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MISHRA AND RUEDA
and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž f, yg i ., i s 1, 2, . . . , m, satisfying conditions Ž3.1. and Ž3.2., is second-order V-type I, and the higher-order dual ŽHD. reduces to the second-order Mond᎐Weir dual:
Ž 2D.
Maximize
f Ž u . y 12 pT ⵜ 2 f Ž u . p
subject to ⵜf Ž u . q ⵜ 2 f Ž u . p s ⵜy T g Ž u . q ⵜ 2 y T g Ž u . p, yi g i Ž u . y 12 pT ⵜ 2 yi g i Ž u . p F 0,
i s 1, 2, . . . , m,
y G 0. The conditions Ž2.1. and Ž2.2. are special cases of the conditions Ž3.1. and Ž3.2., where ␣ Ž x, u. s 1 and i Ž x, u. s 1, i s 1, 2, . . . , m. We can also show that ŽHD. is a dual to ŽP. under weaker conditions. THEOREM 3.3 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽHD.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that T
Ž x, u . ⵜp h Ž u, p . G 0 «
f Ž x . y f Ž u . y h Ž u, p . q pT ⵜp h Ž u, p . G 0
Ž 3.4.
and m
y Ý i Ž x, u . yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . 4 G 0 is1
«
T
Ž x, u . ⵜp Ž y T k Ž u, p . . G 0,
Ž 3.5.
where i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positi¨ e functions, then infimum ŽP. G supremum ŽHD.. Proof. Since Ž u, y, p . is feasible for ŽHD., then by Ž1.5., yyi g i Ž u . y yi k i Ž u, p . q pT ⵜp Ž yi k i Ž u, p . . G 0,
i s 1, 2, . . . , m.
Since x is feasible for ŽP. and i Ž x, u. ) 0, it follows that m
y Ý i Ž x, u . yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . 4 G 0. is1
By Ž3.5., we obtain T
Ž x, u . ⵜp Ž y T k Ž u, p . . G 0.
DUALITY IN MATHEMATICAL PROGRAMMING
179
Using Ž1.4., it follows that T
Ž x, u . ⵜp h Ž u, p . G 0. Therefore by Ž3.4., we have f Ž x . G f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p . . Remark 3.2. If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, then Ž3.4. becomes the condition for f to be pseudo-type I Žsee w7x., and if s 1, Ž3.5. becomes the condition for yg to be quasi-type I w7x. If h Ž u, p . s pT ⵜf Ž u . q 12 pT ⵜ 2 f Ž u . p and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž3.4. becomes the condition for f to be second-order pseudo-type I w4x and if s 1, Ž3.5. becomes the condition for yy T g to be second-order quasi-type I w4x. Strong duality between ŽP. and ŽHD. holds if Ž3.1. and Ž3.2. are replaced by Ž3.4. and Ž3.5., respectively. THEOREM 3.4 Žstrict converse duality.. Let x 0 be an optimal solution of ŽP. at which a constraint qualification is satisfied. Let condition Ž2.3. be satisfied at x 0 , and let conditions Ž3.4. and Ž3.5. be satisfied for all feasible Ž x, u, y, p .. If Ž xU , yU , pU . is an optimal solution of ŽHD., and if, for all x / xU
Ž x, xU . ⵜp h Ž xU pU . G 0 T
«
f Ž x . y f Ž xU . y h Ž xU , pU . q pU T ⵜp h Ž xU , pU . ) 0, Ž 3.6.
then x 0 s xU ; i.e., xU sol¨ es ŽP. and f Ž x 0 . s f Ž xU . q h Ž xU , pU . y pU T ⵜp h Ž xU , pU . . Proof. We suppose that x 0 / xU and exhibit a contradiction. Since x 0 is a solution of ŽP. at which a constraint qualification is satisfied, it follows by strong duality that there exists y 0 g R m such that Ž x 0 , y 0 , p s 0. solves ŽHD. and the corresponding values of ŽP. and ŽHD. are equal. Therefore, f Ž x 0 . s f Ž xU . q h Ž xU , pU . y pU T ⵜp h Ž xU , pU . .
Ž 3.7.
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MISHRA AND RUEDA
Since Ž xU , yU , pU . is feasible for ŽHD., we have that yyi g i Ž xU . y yiU k i Ž xU , pU . q pU T ⵜp Ž yiU k i Ž xU , pU . . G 0 for i s 1, 2, . . . , m. Since x is feasible for ŽP. and i Ž x 0 , xU . ) 0, it follows that 0
m
y Ý i Ž x 0 , xU . yiU g i Ž xU . q yiU k i Ž xU , pU . y pU T ⵜp Ž yiU k i Ž xU , pU . .
½
is1
5
G 0. By Ž3.5., we obtain T
Ž x 0 , xU . ⵜp Ž yU T k Ž xU , pU . . G 0, and then, by Ž1.4., T
Ž x 0 , xU . ⵜp h Ž xU , pU . G 0. From Ž3.6., it follows that f Ž x 0 . y f Ž xU . y h Ž xU , pU . q pU T ⵜp h Ž xU , pU . ) 0, which is a contradiction to Ž3.7..
4. GENERAL HIGHER-ORDER MOND᎐WEIR DUALITY In this section we consider the following general Mond᎐Weir type higher-order dual to ŽP. as in w5x,
Ž M-WHD.
f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p .
Max
y
Ý
yi g i Ž u . y
igI 0
qpT ⵜp
yi k i Ž u, p .
Ý igI 0
Ý
yi k i Ž u, p .
igI 0
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . .
Ý igI␣
yi g i Ž u . q
Ý igI␣
yi k i Ž u, p . y pT ⵜp
Ý
yi k i Ž u, p . F 0,
igI␣
␣ s 1, 2, . . . , r , y G 0,
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DUALITY IN MATHEMATICAL PROGRAMMING
where I␣ : M s 1, 2, . . . , m4 , ␣ s 0, 1, 2, . . . , r with Dr␣s0 I␣ s M and I␣ l I s ⭋, if ␣ /  . In w5x, it is shown that ŽM-WHD. is a dual to ŽP. under the conditions
Ž x, u .
T
ⵜp h Ž u, p . y ⵜp
žÝ
yi k i Ž u, p .
igI 0
«
f Ž x. y
Ý
/
G0
yi g i Ž x . y f Ž u . y
ž
igI 0
y h Ž u, p . y
ž
Ý
Ý
yi k i Ž u, p .
igI 0
qpT ⵜp h Ž u, p . y ⵜp
yi g i Ž u .
igI 0
žÝ
/
/
yi k i Ž u, p .
igI 0
/
G0
Ž 4.1.
and
Ý
yi g i Ž x . y
igI␣
Ý
yi g i Ž u . y
Ý
igI␣
«
yi k i Ž u, p . q pT ⵜp
igI␣ T
Ž x, u . ⵜp
žÝ
žÝ
yi k i Ž u, p . G 0
igI␣
yi k i Ž u, p . G 0,
/
igI␣
␣ s 1, 2, . . . , r .
/
Ž 4.2.
We can generalize Ž4.1. and Ž4.2. under which ŽM-WHD. is a dual to ŽP., to generalized type-I conditions, i.e., pseudo-type-I and quasi-type-I conditions. Since the proof follows along the lines of the one in w5x, we state the theorem without proof. THEOREM 4.1 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽM-WHD.. If for all feasible Ž x, u, y, p .
Ž x, u .
T
ⵜp h Ž u, p . y ⵜp
žÝ
yi k i Ž u, p .
igI 0
«
f Ž x. y
Ý
G0
yi g i Ž x . y f Ž u . y
ž
igI 0
y h Ž u, p . y
ž
/
Ý
Ý
yi k i Ž u, p .
igI 0
qpT ⵜp h Ž u, p . y ⵜp
yi g i Ž u .
igI 0
žÝ
igI 0
/
/
yi k i Ž u, p .
/
G0
Ž 4.3.
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MISHRA AND RUEDA
and y
Ý
yi g i Ž u . y
igI␣
«
yi k i Ž u, p . q pT ⵜp
Ý igI␣
T
Ž x, u . ⵜp
žÝ ./
yi k i Ž u, p . G 0
igI␣
žÝ
igI␣
yi k i Ž u, p
G 0,
/
␣ s 1, 2, . . . , r , Ž 4.4.
then infimum ŽP. G supremum ŽM-WHD.. Remark 4.1. If I0 s ⭋ and Ii s i4 , i s 1, 2, . . . , m Ž r s m., then ŽMWHD. becomes ŽHD. and the conditions Ž4.3. and Ž4.4. reduce to the conditions Ž3.4. and Ž3.5., respectively.
REFERENCES 1. M. A. Hanson, Second order invexity and duality in mathematical programming, OPSEARCH 30 Ž1993., 313᎐320. 2. M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming 37 Ž1987., 51᎐58. 3. O. L. Mangasarian, Second- and higher-order duality in nonlinear programming, J. Math. Anal. Appl. 51 Ž1975., 607᎐620. 4. S. K. Mishra, Second order generalized invexity and duality in mathematical programming, Optimization 42 Ž1997., 51᎐69. 5. B. Mond and J. Zhang, Higher order invexity and duality in mathematical programming, in ‘‘Generalized Convexity, Generalized Monotonicity: Recent Results’’ ŽJ. P. Crouzeix et al., Eds.., pp. 357᎐372, Kluwer Academic, DordrechtrNorwell, MA, 1998. 6. B. Mond and J. Zhang, Duality for multiobjective programming involving second order V-invex functions, in ‘‘Proceedings of Optimization Miniconference’’ ŽB. M. Glover and V. Jeyakumar, Eds.., pp. 89᎐100, 1995. 7. N. G. Rueda and M. A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 Ž1988., 375᎐385.
Higher-Order Generalized Invexity and Duality in Mathematical Programming Shashi K. Mishra Department of Mathematics, Raja Balwant Singh College, Bichpuri, Agra 283105, India
and Norma G. Rueda Department of Mathematics, Merrimack College, North Ando¨ er, Massachusetts 01845 Submitted by William F. Ames Received September 23, 1999
In this paper, we introduce the concepts of higher-order type-I, pseudo-type-I, and quasi-type-I functions and establish various higher-order duality results involv䊚 2000 Academic Press ing these functions.
1. INTRODUCTION Consider the nonlinear programming problem
Ž P.
Minimize
f Ž x.
subject to
g Ž x . G 0,
Ž 1.1.
where f : R n ª R and g: R n ª R m are twice differentiable functions. The Mangasarian second-order dual w3x is
Ž MD.
Maximize
f Ž u . y y T g Ž u . y 12 pT ⵜ 2 f Ž u . y y T g Ž u . p
subject to ⵜ f Ž u . y y T g Ž u . q ⵜ 2 f Ž u . y y T g Ž u . s 0, y G 0. 173 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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MISHRA AND RUEDA
Mangasarian w3x formulated the following higher-order dual by introducing two differentiable functions h: R n = R n ª R and k: R n = R n ª R m :
Ž HD1.
Maximize
f Ž u . q h Ž u, p . y y T g Ž u . y y T k Ž u, p .
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . . , y G 0.
Ž 1.2. Ž 1.3.
ⵜp hŽ u, p . denotes the n = 1 gradient of h with respect to p and ⵜp Ž y T k Ž u, p .. denotes the n = 1 gradients of y T k with respect to p. If h Ž u, p . s pT ⵜf Ž u . q 12 pT ⵜ 2 f Ž u . p and k Ž u, p . s pT ⵜg Ž u . q 12 pT ⵜ 2 g Ž u . p then ŽHD1. becomes ŽMD.. Mond and Zhang w5x obtained duality results for various higher-order dual programming problems under higher-order invexity assumptions. They considered the following dual to ŽP.:
Ž HD.
Maximize
f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p .
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . . ,
Ž 1.4.
yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . F 0,
i s 1, 2, . . . , m, Ž 1.5.
y G 0.
Ž 1.6.
In this paper, we will give more general invexity-type conditions, such as higher-order type-I, higher-order pseudo-type-I, and higher-order quasitype-I conditions, and establish various duality results under these conditions. Mond and Zhang w5x proved duality results between ŽP. and ŽHD. assuming that there exists a function : R n = R n ª R n such that f Ž x . y f Ž u . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . q h Ž u, p . y pT Ž ⵜp h Ž u, p . .
Ž 1.7. and g i Ž x . y g i Ž u . F i Ž x, u . ⵜp k i Ž u, p . Ž x, u . q k i Ž u, p . ypT Ž ⵜp k i Ž u, p . . ,
i s 1, 2, . . . , m,
Ž 1.8.
where ␣ : R n = R n ª Rq _ 04 and i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positive functions.
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175
Combining the concept of type-I functions w2x and conditions Ž1.7. and Ž1.8. when hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, we say that Ž f, yg i ., i s 1, 2, . . . , m, is V-type I at the point u with respect to functions , ␣ , and i , if f Ž x . y f Ž u . G ␣ Ž x, u . ⵜf Ž u . Ž x, u . and yg i Ž u . F i Ž x, u . ⵜg i Ž u . Ž x, u . ,
i s 1, 2, . . . , m.
Mond and Zhang w6x extended the notion of V-invexity to second-order and established duality theorems under generalized second-order V-invexity conditions. If Ž f, yg i ., i s 1, 2, . . . , m, satisfies conditions Ž1.7. and Ž1.8. with hŽ u, p . s pT ⵜf Ž u. q 12 pT ⵜ 2 f Ž u. p and k i Ž u, p . s pT ⵜg i Ž u. q 12 pT ⵜ 2 g i Ž u. p then Ž f, yg i . is said to be second-order V-type I. 2. MANGASARIAN HIGHER-ORDER DUALITY THEOREM 2.1 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽHD1.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that T
f Ž x . y f Ž u . G Ž x, u . ⵜp h Ž u, p . q h Ž u, p . y pT Ž ⵜp h Ž u, p . . Ž 2.1. and T
yg i Ž u . F Ž x, u . ⵜp k i Ž u, p . q k i Ž u, p . y pT Ž ⵜp k i Ž u, p . . , i s 1, 2, . . . , m, then infimum ŽP. G supremum ŽHD1.. Proof. f Ž x . y f Ž u . y h Ž u, p . q y T g Ž u . q y T k Ž u, p . T
G Ž x, u . ⵜp h Ž u, p . y pT Ž ⵜp h Ž u, p . . q y T g Ž u . q y T k Ž u, p . ,
by Ž 2.1. ,
T
s Ž x, u . ⵜp Ž y T k Ž u, p . . y pT Ž ⵜp y T k Ž u, p . . q y T g Ž u . q y T k Ž u, p . , G 0,
by Ž 1.2. ,
by Ž 1.3. and Ž 2.2. .
The following strong duality theorem is similar to w5, Theorem 2x.
Ž 2.2.
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THEOREM 2.2 Žstrong duality.. Let x 0 be a local or global optimal solution of ŽP. at which a constraint qualification is satisfied, and let h Ž x 0 , 0 . s 0,
k Ž x 0 , 0 . s 0,
ⵜp h Ž x 0 , 0 . s ⵜf Ž x 0 . ,
ⵜp k Ž x 0 , 0 . s ⵜg Ž x 0 . .
Ž 2.3.
Then there exists y g R m such that Ž x 0 , y, p s 0. is feasible for ŽHD1., and the corresponding ¨ alues of ŽP. and ŽHD1. are equal. If Ž2.1. and Ž2.2. are satisfied for all feasible Ž x, u, y, p ., then x 0 and Ž x 0 , y, p s 0. are global optimal solutions for ŽP. and ŽHD1., respecti¨ ely. Remark 2.1. 1, 2, . . . , m, then and Mond w2x to
If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s Ž2.1. and Ž2.2. become the conditions given by Hanson define a type-I function. If h Ž u, p . s pT f Ž u . q 12 pT ⵜ 2 f Ž u . p,
and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž2.1. and Ž2.2. become the second-order type-I conditions given by Hanson w1x when p s q s r.
3. MOND᎐WEIR HIGHER-ORDER DUALITY THEOREM 3.1 Žweak duality.. Let x be feasible for ŽP. and let Ž u, y, p . be feasible for ŽHD.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that f Ž x . y f Ž u . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . q h Ž u, p . y pT Ž ⵜp h Ž u, p . .
Ž 3.1. and yg i Ž u . F i Ž x, u . ⵜp k i Ž u, p . Ž x, u . q k i Ž u, p . y pT Ž ⵜp k i Ž u, p . . , i s 1, 2, . . . , m,
Ž 3.2.
where ␣ : R n = R n ª Rq _ 04 and i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positi¨ e functions, then infimum ŽP. G supremum ŽHD..
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Proof. Since Ž u, y, p . is feasible for ŽHD., we have yyi g i Ž u . y yi k i Ž u, p . q pT ⵜp Ž yi k i Ž u, p . . G 0,
i s 1, 2, . . . , m.
Since x is feasible for ŽP., then by Ž3.2. and yi G 0, we obtain
i Ž x, u . ⵜp Ž yi k i Ž u, p . . Ž x, u . G 0,
i s 1, 2, . . . , m.
Since i Ž x, u. ) 0, we have ⵜp Ž yi k i Ž u, p . . Ž x, u . G 0,
i s 1, 2, . . . , m,
hence ⵜp Ž y T k Ž u, p . . Ž x, u . G 0.
Ž 3.3.
By Ž3.1., it follows that f Ž x . y f Ž u . y h Ž u, p . q pT ⵜp h Ž u, p . G ␣ Ž x, u . ⵜp h Ž u, p . Ž x, u . s ␣ Ž x, u . ⵜp Ž y T k Ž u, p . . Ž x, u . , G 0,
by Ž 1.4. ,
by Ž 3.3. and ␣ Ž x, u . ) 0.
THEOREM 3.2 Žstrong duality.. Let x 0 be a local or global optimal solution of ŽP. at which a constraint qualification is satisfied, and let conditions Ž2.3. be satisfied. Then there exists y g R m such that Ž x 0 , y, p s 0. is feasible for ŽHD. and the corresponding ¨ alues of ŽP. and ŽHD. are equal. If also Ž3.1. and Ž3.2. are satisfied for all feasible Ž x, u, y, p ., then x 0 and Ž x 0 , y, p s 0. are global optimal solutions for ŽP. and ŽHD., respecti¨ ely. Proof. It follows on the linear of w5, proof of Theorem 5x. Remark 3.1. If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, then Ž f, yg i ., i s 1, 2, . . . , m, satisfying conditions Ž3.1. and Ž3.2., is V-type I, and the higher-order dual ŽHD. reduces to the Mond᎐Weir dual:
Ž D.
Maximize
f Ž u.
subject to ⵜf Ž u . y ⵜy T g Ž u . s 0 yi g i Ž u . F 0, y G 0. If h Ž u, p . s pT Ž u . q 12 pT ⵜ 2 f Ž u . p
i s 1, 2, . . . , m,
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and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž f, yg i ., i s 1, 2, . . . , m, satisfying conditions Ž3.1. and Ž3.2., is second-order V-type I, and the higher-order dual ŽHD. reduces to the second-order Mond᎐Weir dual:
Ž 2D.
Maximize
f Ž u . y 12 pT ⵜ 2 f Ž u . p
subject to ⵜf Ž u . q ⵜ 2 f Ž u . p s ⵜy T g Ž u . q ⵜ 2 y T g Ž u . p, yi g i Ž u . y 12 pT ⵜ 2 yi g i Ž u . p F 0,
i s 1, 2, . . . , m,
y G 0. The conditions Ž2.1. and Ž2.2. are special cases of the conditions Ž3.1. and Ž3.2., where ␣ Ž x, u. s 1 and i Ž x, u. s 1, i s 1, 2, . . . , m. We can also show that ŽHD. is a dual to ŽP. under weaker conditions. THEOREM 3.3 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽHD.. If, for all feasible Ž x, u, y, p ., there exists a function : R n = R n ª R n such that T
Ž x, u . ⵜp h Ž u, p . G 0 «
f Ž x . y f Ž u . y h Ž u, p . q pT ⵜp h Ž u, p . G 0
Ž 3.4.
and m
y Ý i Ž x, u . yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . 4 G 0 is1
«
T
Ž x, u . ⵜp Ž y T k Ž u, p . . G 0,
Ž 3.5.
where i : R n = R n ª Rq _ 04 , i s 1, 2, . . . , m, are positi¨ e functions, then infimum ŽP. G supremum ŽHD.. Proof. Since Ž u, y, p . is feasible for ŽHD., then by Ž1.5., yyi g i Ž u . y yi k i Ž u, p . q pT ⵜp Ž yi k i Ž u, p . . G 0,
i s 1, 2, . . . , m.
Since x is feasible for ŽP. and i Ž x, u. ) 0, it follows that m
y Ý i Ž x, u . yi g i Ž u . q yi k i Ž u, p . y pT ⵜp Ž yi k i Ž u, p . . 4 G 0. is1
By Ž3.5., we obtain T
Ž x, u . ⵜp Ž y T k Ž u, p . . G 0.
DUALITY IN MATHEMATICAL PROGRAMMING
179
Using Ž1.4., it follows that T
Ž x, u . ⵜp h Ž u, p . G 0. Therefore by Ž3.4., we have f Ž x . G f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p . . Remark 3.2. If hŽ u, p . s pT ⵜf Ž u. and k i Ž u, p . s pT ⵜg i Ž u., i s 1, 2, . . . , m, then Ž3.4. becomes the condition for f to be pseudo-type I Žsee w7x., and if s 1, Ž3.5. becomes the condition for yg to be quasi-type I w7x. If h Ž u, p . s pT ⵜf Ž u . q 12 pT ⵜ 2 f Ž u . p and k i Ž u, p . s pT ⵜg i Ž u . q 12 pT ⵜ 2 g i Ž u . p,
i s 1, 2, . . . , m,
then Ž3.4. becomes the condition for f to be second-order pseudo-type I w4x and if s 1, Ž3.5. becomes the condition for yy T g to be second-order quasi-type I w4x. Strong duality between ŽP. and ŽHD. holds if Ž3.1. and Ž3.2. are replaced by Ž3.4. and Ž3.5., respectively. THEOREM 3.4 Žstrict converse duality.. Let x 0 be an optimal solution of ŽP. at which a constraint qualification is satisfied. Let condition Ž2.3. be satisfied at x 0 , and let conditions Ž3.4. and Ž3.5. be satisfied for all feasible Ž x, u, y, p .. If Ž xU , yU , pU . is an optimal solution of ŽHD., and if, for all x / xU
Ž x, xU . ⵜp h Ž xU pU . G 0 T
«
f Ž x . y f Ž xU . y h Ž xU , pU . q pU T ⵜp h Ž xU , pU . ) 0, Ž 3.6.
then x 0 s xU ; i.e., xU sol¨ es ŽP. and f Ž x 0 . s f Ž xU . q h Ž xU , pU . y pU T ⵜp h Ž xU , pU . . Proof. We suppose that x 0 / xU and exhibit a contradiction. Since x 0 is a solution of ŽP. at which a constraint qualification is satisfied, it follows by strong duality that there exists y 0 g R m such that Ž x 0 , y 0 , p s 0. solves ŽHD. and the corresponding values of ŽP. and ŽHD. are equal. Therefore, f Ž x 0 . s f Ž xU . q h Ž xU , pU . y pU T ⵜp h Ž xU , pU . .
Ž 3.7.
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MISHRA AND RUEDA
Since Ž xU , yU , pU . is feasible for ŽHD., we have that yyi g i Ž xU . y yiU k i Ž xU , pU . q pU T ⵜp Ž yiU k i Ž xU , pU . . G 0 for i s 1, 2, . . . , m. Since x is feasible for ŽP. and i Ž x 0 , xU . ) 0, it follows that 0
m
y Ý i Ž x 0 , xU . yiU g i Ž xU . q yiU k i Ž xU , pU . y pU T ⵜp Ž yiU k i Ž xU , pU . .
½
is1
5
G 0. By Ž3.5., we obtain T
Ž x 0 , xU . ⵜp Ž yU T k Ž xU , pU . . G 0, and then, by Ž1.4., T
Ž x 0 , xU . ⵜp h Ž xU , pU . G 0. From Ž3.6., it follows that f Ž x 0 . y f Ž xU . y h Ž xU , pU . q pU T ⵜp h Ž xU , pU . ) 0, which is a contradiction to Ž3.7..
4. GENERAL HIGHER-ORDER MOND᎐WEIR DUALITY In this section we consider the following general Mond᎐Weir type higher-order dual to ŽP. as in w5x,
Ž M-WHD.
f Ž u . q h Ž u, p . y pT ⵜp h Ž u, p .
Max
y
Ý
yi g i Ž u . y
igI 0
qpT ⵜp
yi k i Ž u, p .
Ý igI 0
Ý
yi k i Ž u, p .
igI 0
subject to ⵜp h Ž u, p . s ⵜp Ž y T k Ž u, p . .
Ý igI␣
yi g i Ž u . q
Ý igI␣
yi k i Ž u, p . y pT ⵜp
Ý
yi k i Ž u, p . F 0,
igI␣
␣ s 1, 2, . . . , r , y G 0,
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where I␣ : M s 1, 2, . . . , m4 , ␣ s 0, 1, 2, . . . , r with Dr␣s0 I␣ s M and I␣ l I s ⭋, if ␣ /  . In w5x, it is shown that ŽM-WHD. is a dual to ŽP. under the conditions
Ž x, u .
T
ⵜp h Ž u, p . y ⵜp
žÝ
yi k i Ž u, p .
igI 0
«
f Ž x. y
Ý
/
G0
yi g i Ž x . y f Ž u . y
ž
igI 0
y h Ž u, p . y
ž
Ý
Ý
yi k i Ž u, p .
igI 0
qpT ⵜp h Ž u, p . y ⵜp
yi g i Ž u .
igI 0
žÝ
/
/
yi k i Ž u, p .
igI 0
/
G0
Ž 4.1.
and
Ý
yi g i Ž x . y
igI␣
Ý
yi g i Ž u . y
Ý
igI␣
«
yi k i Ž u, p . q pT ⵜp
igI␣ T
Ž x, u . ⵜp
žÝ
žÝ
yi k i Ž u, p . G 0
igI␣
yi k i Ž u, p . G 0,
/
igI␣
␣ s 1, 2, . . . , r .
/
Ž 4.2.
We can generalize Ž4.1. and Ž4.2. under which ŽM-WHD. is a dual to ŽP., to generalized type-I conditions, i.e., pseudo-type-I and quasi-type-I conditions. Since the proof follows along the lines of the one in w5x, we state the theorem without proof. THEOREM 4.1 Žweak duality.. Let x be feasible for ŽP., and let Ž u, y, p . be feasible for ŽM-WHD.. If for all feasible Ž x, u, y, p .
Ž x, u .
T
ⵜp h Ž u, p . y ⵜp
žÝ
yi k i Ž u, p .
igI 0
«
f Ž x. y
Ý
G0
yi g i Ž x . y f Ž u . y
ž
igI 0
y h Ž u, p . y
ž
/
Ý
Ý
yi k i Ž u, p .
igI 0
qpT ⵜp h Ž u, p . y ⵜp
yi g i Ž u .
igI 0
žÝ
igI 0
/
/
yi k i Ž u, p .
/
G0
Ž 4.3.
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MISHRA AND RUEDA
and y
Ý
yi g i Ž u . y
igI␣
«
yi k i Ž u, p . q pT ⵜp
Ý igI␣
T
Ž x, u . ⵜp
žÝ ./
yi k i Ž u, p . G 0
igI␣
žÝ
igI␣
yi k i Ž u, p
G 0,
/
␣ s 1, 2, . . . , r , Ž 4.4.
then infimum ŽP. G supremum ŽM-WHD.. Remark 4.1. If I0 s ⭋ and Ii s i4 , i s 1, 2, . . . , m Ž r s m., then ŽMWHD. becomes ŽHD. and the conditions Ž4.3. and Ž4.4. reduce to the conditions Ž3.4. and Ž3.5., respectively.
REFERENCES 1. M. A. Hanson, Second order invexity and duality in mathematical programming, OPSEARCH 30 Ž1993., 313᎐320. 2. M. A. Hanson and B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Programming 37 Ž1987., 51᎐58. 3. O. L. Mangasarian, Second- and higher-order duality in nonlinear programming, J. Math. Anal. Appl. 51 Ž1975., 607᎐620. 4. S. K. Mishra, Second order generalized invexity and duality in mathematical programming, Optimization 42 Ž1997., 51᎐69. 5. B. Mond and J. Zhang, Higher order invexity and duality in mathematical programming, in ‘‘Generalized Convexity, Generalized Monotonicity: Recent Results’’ ŽJ. P. Crouzeix et al., Eds.., pp. 357᎐372, Kluwer Academic, DordrechtrNorwell, MA, 1998. 6. B. Mond and J. Zhang, Duality for multiobjective programming involving second order V-invex functions, in ‘‘Proceedings of Optimization Miniconference’’ ŽB. M. Glover and V. Jeyakumar, Eds.., pp. 89᎐100, 1995. 7. N. G. Rueda and M. A. Hanson, Optimality criteria in mathematical programming involving generalized invexity, J. Math. Anal. Appl. 130 Ž1988., 375᎐385.