Second-order duality for invex composite optimization

Journal of the Egyptian Mathematical Society (2014) xxx, xxx–xxx

Egyptian Mathematical Society

Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems

ORIGINAL PAPER

Second-order duality for invex composite optimization Saroj Kumar Padhan a b

a,*

, Chandal Nahak

b

Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, India Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India

Received 24 August 2013; revised 21 December 2013; accepted 2 February 2014

KEYWORDS Invex composite function; Second-order duality; Mangasarian and Mond– Weir type duality; Weak duality; Strong duality; Converse duality

Abstract The second-order duality results for the invex composite optimization problem are studied. Its objective function is a composition of nonfinite valued differentiable invex and a vector valued functions. Several duality results are also discussed for both constrained and unconstrained optimization problems. Examples and counterexamples are illustrated to justify the present work. 2010 MATHEMATICS SUBJECT CLASSIFICATION:

ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction and preliminaries Optimization of a suitable objective function subject to appropriate constraints appear in Engineering, economics, management and operation research. Composite functions are useful when one quantity depends on a second quantity, and in turn that the second quantity depends on a third quantity. This is an extremely general situation with many real world applications. Consider the invex composite optimization problem ðIPÞ min fðxÞ s:t:

49M; 65K; 90C26

x 2 Rn ;

ð1Þ

where fðxÞ ¼ gðFðxÞÞ; g : Rm ! R [ fþ1g is differentiable invex function with respect to g; g : Rm  Rm ! Rm , domðgÞ ¼ * Corresponding author. Tel.: +91 9583489086. E-mail address: [email protected] (S.K. Padhan). Peer review under responsibility of Egyptian Mathematical Society.

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fy 2 Rm : gðyÞ < þ1g and F : Rn ! Rm is a vector valued function. Yang [1] considered the convex composite optimization problem, where the objective function was a composition of nonfinite valued lower semi-continuous convex function and the vector valued function. He also established various duality results. Hanson [2] introduced the concept of invexity and found that Kuhn–Tucker conditions are the sufficient conditions for optimality. Further, many researchers discussed, various properties, extensions, and applications of generalized invex functions [3–6]. The study of second order duality is useful due to the computational advantage over first order duality, because it gives bounds for the value of the objective function when approximations are used. Aghezzaf [7] introduced the second-order generalized convexity for vector valued functions and studied desired duality results for second-order mixed type dual problems. Ahmad [8] considered a pair of Mond–Weir type second order symmetric nondifferentiable multiobjective programs. Weak, strong and converse duality results were studied under g-pseudo-invexity assumptions. Again, Ahmad and Husain [9] introduced a class of second-order ðF; a; q; dÞ-convex functions and their generalization. Under the assumptions of ðF; a; q; dÞ-

1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2014.02.005 Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005

2

S.K. Padhan, C. Nahak

convexity, they established various duality theorems for a second-order Mond–Weir type multiobjective dual problem. Kassem [10] formulated a pair of second-order symmetric dual multiobjective nonlinear programs over arbitrary generalized cone pseudo-convex functions. Under the assumption of cone pseudo-convex functions, they proved different duality results. Jayswal and Prasad [11] derived Mond–Weir type nondifferentiable second-order fractional symmetric dual programs over arbitrary cones and expressed duality results under K-F-generalized convexity assumptions. Gupta and Kailey [12] formulated a new pair of multiobjective second-order symmetric dual programs over arbitrary cones and proved weak, strong and converse duality theorems under K-g-bonvexity assumptions. Recently, Gulati etal. [14] corrected the inconsistencies of the work of Mishra and Lai [13]. Later Ahmad and Husain [15] formulated a pair of second-order fractional programming problems and symmetric duality theorems were established. However, to the best of our knowledge, no one has yet discussed second-order duality for any optimization problems involving invex composite functions. In this paper, we redefine convex composite optimization model of Yang [1], by taking invex composite functions for both constrained and unconstrained cases. We also study the second-order duality of the said model and establish many duality results. Our present approach is similar to the pioneer work of Mangasarian [16]. Moreover, we discuss some examples and counterexamples to verify our results. For a function h : Rn ! R [ fþ1g, the conjugate function  h : Rn ! R [ fþ1g is defined by h ðx Þ ¼ supfhx; x i  hðxÞg; x2Rn

for all x 2 Rn ;

ð2Þ

where h; i denotes the usual inner product in Rn . Here we define the invexity of the composite functions. Definition 1.1. A differentiable function g : Rm ! R [ fþ1g is said to be invex at a point FðuÞ, where F : Rn ! Rm , with respect to g, if there exists g : Rm  Rm ! Rn such that gðFðxÞÞ  gðFðuÞÞ P hrgðFðuÞÞ;gðFðxÞ;FðuÞÞi; for all FðxÞ 2 Rm : It follows that every convex composite function is invex composite but the converse is not true, follows from the following counterexample 1.1. Example 1.1. Let F : R2 ! R2 ; g : R2 ! R given by Fðx1 ; x2 Þ ¼ ð2x21 x22 ; x61 x62 Þ; gðx1 ; x2 Þ ¼ x31  x2 ;   rgðx1 ; x2 Þ ¼ 3x21 ; 1 ;     rgðFðx1 ; x2 ÞÞ ¼ rg 2x21 x22 ; x61 x62 ¼ 12x41 x42 ; 1 ;   gðFðx1 ; x2 ÞÞ ¼ g 2x21 x22 ; x61 x62 ¼ 8x61 x62  x61 x62 ¼ 9x61 x62 : Now gðFðx1 ;x2 ÞÞ  gðFðu1 ;u2 ÞÞ  hrgðFðu1 ;u2 ÞÞ;ðFðx1 ;x2 Þ  Fðu1 ;u2 ÞÞi    ¼ 9x61 x62 þ 9u61 u62  12u41 u42 ;1 ; 2ðx21 x22  u21 u22 Þ;x61 x62  u61 u62   ¼ 8x61 x62 þ 8u61 u62 þ 24u41 u42 x21 x22  u21 u22 < 0; for ðx1 ;x2 Þ ¼ ð2;2Þ and ðu1 ;u2 Þ ¼ ð1;1Þ: i:e: gðFðx1 ; x2 ÞÞ is not convex at ðx1 ; x2 Þ ¼ ð2; 2Þ and ðu1 ;u2 Þ ¼ ð1; 1Þ:

Now consider g : R2 R2 ! R2 defined by gðFðx1 ; x2 Þ;  Fðu1 ; u2 ÞÞ ¼ 9Fðx1 ; x2 Þ ¼ 18x21 x22 ; 9x61 x62 . We have gðFðx1 ; x2 ÞÞ  gðFðu1 ;u2 ÞÞ  hrgðFðu1 ; u2 ÞÞ;gðFðx1 ; x2 Þ;Fðu1 ; u2 ÞÞi    12u41 u42 ;1 ; 18x21 x22 ; 9x61 x62

¼ 9x61 x62 þ 9u61 u62 

¼ 9u61 u62 þ 216u41 u42 x21 x22 P 0; for all ðx1 ;x2 Þ 2 R2 :

Hence gðFðx1 ; x2 ÞÞ is not convex, but invex with respect to the g. Definition 1.2. A differentiable function g : Rm ! R [ fþ1g is said to be pseudo-invex at a point FðuÞ, where F : Rn ! Rm , with respect to g, if there exists g : Rm  Rm ! Rm such that hrgðFðuÞÞ; gðFðxÞ; FðuÞÞi P 0 ) gðFðxÞÞ  gðFðuÞÞ P 0;

for all FðxÞ 2 Rm :

It follows that every invex composite function is pseudoinvex composite but the converse is not true, follows from the following counterexample 1.2. Example 1.2. Let us define F : R2þ ! R2þ ; g : R2þ ! R by   Fðx1 ; x2 Þ ¼ 2x1 x2 ; x31 x32 ; gðx1 ; x2 Þ ¼ x31  x2 ;   rgðx1 ; x2 Þ ¼ 3x21 ; 1 ;     rgðFðx1 ; x2 ÞÞ ¼ rg 2x1 x2 ; x31 x32 ¼ 12x21 x22 ; 1 ;   gðFðx1 ; x2 ÞÞ ¼ g 2x1 x2 ; x31 x32 ¼ 8x31 x32  x31 x32 ¼ 9x31 x32 : Consider the function g : R2þ  R2þ ! R2 defined by 8 9ðFðx1 ; x2 Þ þ Fðu1 ; u2 ÞÞ; > > > < if u1 u2 P x1 x2 ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞ ¼ 1 > > > 24 ðFðx1 ; x2 Þ  Fðu1 ; u2 ÞÞ; : if u1 u2 < x1 x2 : For u1 u2 P x1 x2 , one can easily get gðFðx1 ;x2 ÞÞ  gðFðu1 ;u2 ÞÞ  hrgðFðu1 ;u2 ÞÞ;gðFðx1 ;x2 Þ;Fðu1 ;u2 ÞÞi    12u21 u22 ;1 ; 18ðx1 x2 þ u1 u2 Þ;9ðx31 x32 þ u31 u32 Þ

¼ 9x31 x32 þ 9u31 u32 

¼ 18x31 x32  216u21 u22 ðx1 x2 þ u1 u2 Þ < 0;

and for u1 u2 < x1 x2 , it is obtained that gðFðx1 ; x2 ÞÞ  gðFðu1 ; u2 ÞÞ  hrgðFðu1 ; u2 ÞÞ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞi   ¼ 9x31 x32 þ 9u31 u32  12u21 u22 ; 1 ;    1 1 3 3  2ðx1 x2  u1 u2 Þ; x1 x2  u31 u32  24 24   3 3  231  3 3 3 3 u u  x1 x2 þ x1 x2 u1 u2  x31 x32 < 0; ¼ 24 1 2 as u1 u2 < x1 x2 and x1 ; x2 ; u1 ; u2 2 Rþ :

Hence gðFðx1 ; x2 ÞÞ is not invex. Now we check for pseudoinvexity property. When u1 u2 < x1 x2 , it is found that gðFðx1 ; x2 ÞÞ  gðFðu1 ; u2 ÞÞ ¼ 9x31 x32 þ 9u31 u32 < 0; as u1 u2 < x1 x2

and

x1 ; x2 ; u1 ; u2 2 Rþ ;

Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005

Second-order duality for invex composite optimization hrgðFðu1 ; u2 ÞÞ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞi     1  1 3 3 ¼ 12u21 u22 ; 1 ;  2ðx1 x2  u1 u2 Þ; x1 x2  u31 u32 24 24  1 3 3 ¼ ðu1 u2  x1 x2 Þ þ u u  x31 x32 24 1 2 < 0; as u1 u2 < x1 x2 and x1 ; x2 ; u1 ; u2 2 Rþ : So, gðFðx1 ; x2 ÞÞ is pseudo-invex. If u1 u2 P x1 x2 , one can get hrgðFðu1 ; u2 ÞÞ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞi      ¼ 12u21 u22 ; 1 ; 18ðx1 x2 þ u1 u2 Þ; 9 x31 x32 þ u31 u32   ¼ 216u21 u22 ðx1 x2 þ u1 u2 Þ þ 9 x31 x32 þ u31 u32 P 0; as x1 ; x2 ; u1 ; u2 2 Rþ ; gðFðx1 ; x2 ÞÞ  gðFðu1 ; u2 ÞÞ ¼

9x31 x32

P 0;

þ

9u31 u32

as u1 u2 P x1 x2

and x1 ; x2 ; u1 ; u2 2 Rþ :

Hence gðFðx1 ; x2 ÞÞ is pseudo-invex. Therefore gðFðx1 ; x2 ÞÞ is not invex but pseudo-invex with respect to the g. Definition 1.3. A differentiable function g : Rm ! R [ fþ1g is said to be quasi-invex at a point FðuÞ, where F : Rn ! Rm , with respect to g, if there exists g : Rm  Rm ! Rm such that gðFðxÞÞ  gðFðuÞÞ 6 0 ) hrgðFðuÞÞ; gðFðxÞ; FðuÞÞi 6 0;

for all FðxÞ 2 Rm :

It follows that every invex composite function is quasiinvex composite but the converse is not true, follows from the following counterexample 1.3. Example 1.3. Let us define F :   Fðx1 ; x2 Þ ¼ x1 x2 ; x31 x32 ; gðx1 ; x2 Þ ¼

R2þ

!

R2þ ; g

:

R2þ

! R by

x31

þ x2 ;  2  rgðx1 ; x2 Þ ¼ 3x1 ; 1 ;     rgðFðx1 ; x2 ÞÞ ¼ rg x1 x2 ; x31 x32 ¼ 3x21 x22 ; 1 ;   gðFðx1 ; x2 ÞÞ ¼ g x1 x2 ; x31 x32 ¼ x31 x32 þ x31 x32 ¼ 2x31 x32 : Consider the function g : R2þ  R2þ ! R2 defined by ( 2ðFðx1 ;x2 Þ þ Fðu1 ;u2 ÞÞ; if x1 x2 > u1 u2 ; gðFðx1 ;x2 Þ;Fðu1 ;u2 ÞÞ ¼ 1 ðFðx1 ;x2 Þ  Fðu1 ;u2 ÞÞ; if x1 x2 6 u1 u2 : 3 For x1 x2 > u1 u2 , it is obtained that gðFðx1 ;x2 ÞÞ  gðFðu1 ;u2 ÞÞ  hrgðFðu1 ;u2 ÞÞ;gðFðx1 ;x2 Þ;Fðu1 ;u2 ÞÞi       ¼ 2 x31 x32  u31 u32  3u21 u22 ;1 ; 2ðx1 x2 þ u1 u2 Þ;2 x31 x32 þ u31 u32 ¼

4u31 u32  6u21 u22 ðx1 x2 þ u1 u2 Þ

< 0; as x1 ;x2 ;u1 ;u2 2 Rþ ;

and for x1 x2 6 u1 u2 , one can easily see that gðFðx1 ;x2 ÞÞ  gðFðu1 ;u2 ÞÞ  hrgðFðu1 ;u2 ÞÞ;gðFðx1 ;x2 Þ;Fðu1 ;u2 ÞÞi       1  1 ¼ 2 x31 x32  u31 u32  3u21 u22 ;1 ; ðx1 x2  u1 u2 Þ; x31 x32  u31 u32 3 3   2 2  2 3 3 3 3 2 2 ¼ x1 x2  u1 u2 þ x1 x2 x1 x2  u1 u2 3 < 0; as x1 ;x2 ;u1 ;u2 2 Rþ ðequality holds when x1 x2 ¼ u1 u2 Þ:

3 Hence gðFðx1 ; x2 ÞÞ is not invex. Now we check for quasi-invexity property. For x1 x2 > u1 u2 , it is found that hrgðFðu1 ; u2 ÞÞ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞi  2 2     ¼ 3u1 u2 ; 1 ; 2ðx1 x2 þ u1 u2 Þ; 2 x31 x32 þ u31 u32 > 0; as x1 ; x2 ; u1 ; u2 2 Rþ ; gðFðx1 ; x2 ÞÞ  gðFðu1 ; u2 ÞÞ   ¼ 2 x31 x32  u31 u32 > 0;

as x1 x2 > u1 u2

and

x1 ; x2 ; u1 ; u2 2 Rþ :

So, gðFðx1 ; x2 ÞÞ is quasi-invex. When x1 x2 6 u1 u2 , one can get gðFðx1 ; x2 ÞÞ  gðFðu1 ; u2 ÞÞ   ¼ 2 x31 x32  u31 u32 6 0; as x1 x2 6 u1 u2 and x1 ; x2 ; u1 ; u2 2 Rþ ; hrgðFðu1 ; u2 ÞÞ; gðFðx1 ; x2 Þ; Fðu1 ; u2 ÞÞi    2 2  1  1 ¼ 3u1 u2 ; 1 ; ðx1 x2  u1 u2 Þ; x31 x32  u31 u32 3 3   1 ¼ u21 u22 ðx1 x2  u1 u2 Þ þ x31 x32  u31 u32 3 6 0; as x1 x2 6 u1 u2 and x1 ; x2 ; u1 ; u2 2 Rþ : Hence gðFðx1 ; x2 ÞÞ is quasi-invex. Therefore gðFðx1 ; x2 ÞÞ is not invex but quasi-invex with respect to the g. 2. Second-order duality for the invex composite optimization problem In this section we discuss the second order duality for the invex composite optimization problems for both constrained and unconstrained cases. 2.1. Unconstrained case Let us assume that F : Rn ! Rm is twice differentiable, the Jocobian of F at x; rFðxÞ is an m  n matrix and r2 Fi ðxÞ is an n  n matrix for each i ¼ 1; 2; 3; . . . ; m. The second-order dual (DP) of the primal (IP) is given by max s:t:

1 Lðu; y Þ  pT r2 Lðu; y Þp 2 ðu; y ; pÞ 2 Rn  Rm  Rn ; 

2



rLðu; y Þ þ r Lðu; y Þp ¼ 0;

ð3Þ ð4Þ

where Lðu; y Þ ¼ hy ; FðuÞi  g ðy Þ; y 2 domðg Þ and g is the convex conjugate of g. rLðu; y Þ denotes the first order derivative of L at u with respect to u. Now using the conjugate property and invexity of g, we discuss several duality (weak, strong and converse) results. Theorem 2.1 (Weak duality). Let x and ðu; y ; pÞ be feasible solutions of (IP) and (DP), respectively. If g is the conjugate of gand hrgðFðuÞÞ; gðFðxÞ; FðuÞÞi þ 12 pT r2 Lðu; y Þp P 0 then fðxÞ P Lðu; y Þ  12 pT r2 Lðu; y Þp.

Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005

4

S.K. Padhan, C. Nahak

Proof. As x and ðu; y ; pÞ are feasible solutions of (IP) and (MD), respectively. We have

 1 fðxÞ  Lðu; y Þ  pT r2 Lðu; y Þp 2 1  ¼ gðFðxÞÞ  hy ; FðuÞi þ g ðy Þ þ pT r2 Lðu; y Þp 2 1 T 2 P gðFðxÞÞ  gðFðuÞÞ þ p r Lðu; y Þp 2 ½as g is the conjugate of g 1 P hrgðFðuÞÞ; gðFðxÞ; FðuÞÞi þ pT r2 Lðu; y Þp 2 ½by invexity of g P 0: 

Theorem 2.3 (Converse duality). Let ð x; y ; pÞ be an optimal solution of (DP). Assume that ð5Þ

then x is an optimal solution of (IP). h

s:t:

fðxÞ hðxÞ 6 0;

ð6Þ

where fðxÞ ¼ gðFðxÞÞ; g : Rm ! R is differentiable invex function with respect to g and F : Rn ! Rm is vector valued function. hðxÞ ¼ lðFðxÞÞ; l : Rm ! Rs is differentiable vector valued invex function with respect to same g. According to generalized Kuhn–Tucker theorem [17], it is necessary that under certain constraint qualification, x0 to be minimal point of ðIP1Þ means, there exist k 2 Rsþ ; u 2 Rn such that the Lagrangian Lðx; kÞ ¼ fðxÞ þ hk; hðxÞi ¼ fðxÞ þ kT hðxÞ satisfies at x0 i.e. rLðx0 ; kÞ ¼ 0; T

k lðFðx0 ÞÞ ¼ 0:

ð9Þ ð10Þ

where rgðFðuÞÞ and r½kT lðFðuÞÞ denotes the 1  m gradient of g and kT l, respectively at FðuÞ and r2 gðFðuÞÞ and r2 ½kT lðFðuÞÞ denotes the m  m Hessian matrix of g and kT l, respectively at FðuÞ. As it has been seen that there is no much difference between (DP) and (MD). The only difference is that (DP) has an additional term g ðy Þ, due to the definition of the Lagrangian Lðx; kÞ of (IP1). To establish the following weak, strong and converse duality theorems, we assume that g and kT h are invex with respect to same g. Theorem 2.4 (Weak duality). Let x and ðu; k; pÞ be feasible solutions of (IP1) and (MD), respectively. If there exist constants kðu; kÞ and Kðu; kÞ such that pT r2 Lðu; kÞp P kðu; kÞkpk2 ; 2

kr Lðu; kÞk 6 Kðu; kÞ and

for all p 2 Rm ;

ð11Þ

0 < kðu; kÞ 6 Kðu; kÞ;

ð12Þ

then fðxÞ P Lðu; kÞ  12 pT r2 Lðu; kÞp, provided kpk 

Kðu; kÞ kgðFðxÞ; FðuÞÞk P 0: kðu; kÞ

ð13Þ

We have taken Euclidean norm for the vector space and the Frobenius norm is considered for matrices (throughout the paper).

kr2 Lðu; kÞkkgðFðxÞ; FðuÞÞkkpk

Consider the following constrained invex composite optimization problem min

1 Lðu; kÞ  pT r2 Lðu; kÞp 2 rLðu; kÞ þ r2 Lðu; kÞp ¼ 0; ðu; k; pÞ 2 Rn  Rs  Rm ; k P 0;

Proof. Suppose x and ðu; k; pÞ are feasible solutions of (IP1) and (MD), respectively. it is easy to see that

2.2. Constrained case

ðIP1Þ

max s:t:

Proof. As g is the conjugate of g; ðx0 ; y ; p ¼ 0Þ is a feasible solution of (DP). From the weak duality ðx0 ; y ; p ¼ 0Þ is the optimal solution of (DP). 

Proof. The proof is easy and hence omitted.

In this section we discuss Mangasarian type second order duality (MD) of the primal problem (IP1). ðMDÞ

Theorem 2.2 (Strong duality). Let x0 be an optimal solution of (IP). If g is the conjugate of g, then there exists an y 2 domðg Þ # Rm such that ðx0 ; y ; p ¼ 0Þ be feasible for (DP) and the value of the objective functions of (IP) at x0 and (DP) at ðx0 ; y ; p ¼ 0Þ are equal. If in addition, the weak duality holds between (IP) and (DP), then ðx0 ; y ; p ¼ 0Þ is the optimal solution of (DP).

hrgðFð xÞÞ; gðFðuÞ; Fð xÞÞi P 0;

2.2.1. Mangasarian type duality

ð7Þ ð8Þ

Taking g and l as twice continuously differentiable function, we study both Mangasarian and Mond–Weir type second order duality of the primal problem (IP1).

P kgðFðxÞ; FðuÞÞT r2 Lðu; kÞpk:

ð14Þ

Now

 1 fðxÞ  Lðu; kÞ  pT r2 Lðu; kÞp 2

1 P fðxÞ  fðuÞ þ kT hðxÞ  kT hðuÞ þ pT r2 Lðu; kÞp 2 ½by Eqs:ð6Þandð10Þ P < rgðFðuÞÞ; gðFðxÞ; FðuÞÞ > þ < kT rlðFðuÞÞ; 1 gðFðxÞ; FðuÞÞ > þ pT r2 Lðu; kÞp 2 ½by invexity of g and kT h 1 ¼ gðFðxÞ; FðuÞÞT r2 Lðu; kÞp þ pT r2 Lðu; kÞp ½by Eq:ð9Þ 2 P Kðu; kÞkgðFðxÞ; FðuÞÞkkpk þ kðu; kÞkpk2 ½by Eqs:ð11Þ; ð12Þandð14Þ   Kðu; kÞ ¼ kðu; kÞkpk kpk  kgðFðxÞ; FðuÞÞk kðu; kÞ P 0 ½by Eq:ð13Þ:



Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005

Second-order duality for invex composite optimization

5

Corollary 1. The weak duality Theorem 2.4 holds with the closeness condition (13) replaced by

From Eq. (18), we have

kpkkgðFðxÞ; FðuÞÞk ¼ 0: Theorem 2.5 (Strong duality). Let x0 be an optimal solution of (IP1) that satisfies Kuhn–Tucker Theorem [17] under suitable constraint qualification. Then there exists a k 2 Rsþ such that ðx0 ; k; p ¼ 0Þ be feasible for (MD) and the value of the objective functions of (IP1) at x0 and (MD) at ðx0 ; k; p ¼ 0Þ are equal. If in addition, the weak duality holds between (IP1) and (MD), then ðx0 ; k; p ¼ 0Þ is the optimal solution of (MD). Proof. The proof is similar to that of Theorem 2.2.

Proof. Suppose x and ðu; k; pÞ are feasible solutions of (IP1) and (MD), respectively.



x; k; pÞ be an optimal Theorem 2.6 (Converse duality). Let ð solution of (MD).

1  kT hðuÞ þ pT r2 Lðu; kÞp 6 0 2 ) kT hðxÞ  kT hðuÞ 6 0 ½by Eqs:ð6Þ; ð19Þandð20Þ ) hkT rhðuÞ; gðFðxÞ; FðuÞÞi 6 0 ½by quasi-invexity of kT h ) hðrfðuÞ  r2 Lðu; kÞpÞ; gðFðxÞ; FðuÞÞi 6 0 ½by Eq:ð17Þ ) hrfðuÞ; gðFðxÞ; FðuÞÞi P hr2 Lðu; kÞp; gðFðxÞ; FðuÞÞi ) fðxÞ P fðuÞ: ½by pseudo-invexity of g and Eq:ð21Þ: 

Assume that hrgðFð xÞÞ; gðFðuÞ; Fð xÞÞi P 0;

ð15Þ

then x is an optimal solution of (IP1). Proof. Assume that x is not an optimal solution of (IP1). Then there exists a feasible solution u of (IP1) such that gðFðuÞÞ < gðFð xÞÞ:

Theorem 2.8 (Strong duality). Let x0 be an optimal solution of (IP1) that satisfies Kuhn–Tucker Theorem [17] under suitable constraint qualification. Then there exists a k 2 Rsþ such that ðx0 ; k; p ¼ 0Þ be feasible for (MWD) and the value of the objective functions of (IP1) at x0 and (MWD) at ðx0 ; k; p ¼ 0Þ are equal. If in addition, the weak duality holds between (IP1) and (MWD), then ðx0 ; k; p ¼ 0Þ is the optimal solution of (MWD).

ð16Þ Proof. The proof is easy and hence omitted.

Since g is invex we have xÞÞ P hrgðFð xÞÞ; gðFðuÞ; Fð xÞÞi P 0; gðFðuÞÞ  gðFð which gives a contradiction to the strict inequality (16). Hence x is an optimal solution of (IP1). 



Theorem 2.9 (Converse duality). Let ð x; k; pÞ be an optimal solution of (MWD). Again assume that pT r2 Lð x; kÞp P 0; T

xÞÞ rLð x; kÞ P 0: gðFðuÞ; Fð

2.2.2. Mond–Weir type duality We establish the following Mond–Weir type second order duality (MWD) of the primal problem (IP1). ðMWDÞ

max

fðuÞ

s:t:

rLðu; kÞ þ r2 Lðu; kÞp ¼ 0; 1 kT hðuÞ  pT r2 Lðu; kÞp P 0; 2 ðu; k; pÞ 2 Rn  Rs  Rm ; k P 0:

gðFðxÞ; FðuÞÞ rLðu; kÞ P 0: Then fðxÞ P fðuÞ.

Then x is an optimal solution of (IP1).

ð17Þ ð18Þ ð19Þ

Theorem 2.7 (Weak duality). Let x and ðu; k; pÞ be feasible solutions of (IP1) and (MWD), respectively. Again assume that

T

ð23Þ

Proof. Suppose to the contrary x is not an optimal solution of (IP1). Then there exists a feasible solution u of (IP1) such that

The beauty of Mond–Weir duality is that the objective function of both the primal problem and the dual problem is same. That is why to study the duality relations between the primal problems and their corresponding Mond–Weir type dual problems are comparatively easy. Now assuming the pseudo-invexity of g and quasi-invexity of kT h with respect to same g, we express the following duality relations between (IP1) and (MWD).

pT r2 Lðu; kÞp P 0;

ð22Þ

gðFðuÞÞ < gðFð xÞÞ:

ð24Þ

Since ð x; k; pÞ is an optimal solution of (MWD), Eq. (18) becomes 1 kT hð xÞ  pT r2 Lð x; kÞp P 0: 2

ð25Þ

Now from Eq. (25), we have 1  kT hð xÞ þ pT r2 Lð x; kÞp 6 0 2 xÞ 6 0 ) kT hðuÞ  kT hð ½by Eqs:ð6Þ; ð19Þandð22Þ xÞ; gðFðuÞ; Fð xÞÞi 6 0 ) hkT rhð ½by quasi-invexity of kT h at the point Fð xÞ xÞ  r2 Lð x; kÞpÞ; gðFðuÞ; Fð xÞÞi 6 0 ) hðrfð ½by Eq:ð17Þ

ð20Þ ð21Þ

) hrfð xÞ; gðFðuÞ; Fð xÞÞi P hr2 Lð x; kÞp; gðFðuÞ; Fð xÞÞi xÞÞ: ) gðFðuÞÞ P gðð ½by pseudo-invexity of g at the point Fð xÞ and Eq:ð23Þ:

Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005

6 Which gives a contradiction to the strict inequality (24). Hence x is an optimal solution of (IP1).  3. Concluding remark In the present paper, a second-order dual of the invex composite optimization has been introduced. Duality results are established for the both constrained and unconstrained cases under invexity and generalized invexity assumptions. If we take gðFðxÞ; FðuÞÞ ¼ FðxÞ  FðuÞ, the convex composite optimization of Yang [1] is a particular case of our work. Mond–Weir type duality is studied to weaken the invexity requirement to generalized invexity (pseudo-invexity and quasi-invexity). Many examples and counterexamples are discussed in support of the present investigations. Acknowledgments The authors wish to thank the referees and editor in chief for their valuable suggestions which improved the presentation of the paper. References [1] X.Q. Yang, Second-order global optimality conditions for convex composite optimization, Math. Program. 81 (1998) 327–347. [2] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550. [3] A. Ben-Israel, B. Mond, What is invexity?, J Aust. Math. Soc. Ser. B 28 (1986) 1–9. [4] M.A. Hanson, B. Mond, Self-duality and Invexity, FSU Statistics Report M 716, Department of Statistics, Florida State University, Florida, 1986.

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Please cite this article in press as: S.K. Padhan, C. Nahak, Second-order duality for invex composite optimization, Journal of the Egyptian Mathematical Society (2014), http://dx.doi.org/10.1016/j.joems.2014.02.005