A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem

A Weaker Constraint Qualification of Globally Convergent

Homotopy Method for a Multiobjective

Programming Problem

Guangming Yao1, Wen Song2

1_{Department of Mathematics, Clarkson University, Potsdam, USA} 2_{Department of Mathematics, Harbin Normal University, Harbin, China}

Email: [email protected], [email protected]

Received September 29, 2012; revised January 9, 2013; accepted January 16, 2013

ABSTRACT

In this paper, we prove that the combined homotopy interior point method for a multiobjective programming problem introduced in Ref. [1] remains valid under a weaker constrained qualification—the Mangasarian-Fromovitz constrained qualification, instead of linear independence constraint qualification. The algorithm generated by this method associated to the Karush-Kuhn-Tucker points of the multiobjective programming problem is proved to be globally convergent.

Keywords: Multiobjective Programming Problem; Homotopy Method; KKT Condition; Efficient Solution; MFCQ

1. Introduction

Let be the -dimensional Euclidean space, and let and denote the nonnegative and positive , respectively. For any two vectors _{1 2}

n

 n

n 

 n



 _n

, n



, ,



y y y 

,n



y

and _{1 2} in , we use the following conventions: _{i i} . Similarly, we can define , and



, , ,z zn

iff

y



_n

z i

z z



 

y z

,

y z

, y 1, 2,

  

z yz.

Consider the following multiobjective programming problem (MOP)

 

 

 

min

. . 0, 0,

f x

s t g x  h x 

where











T 1 2

T

1 2 1 2

, , , , ,

, , , , , , ,

n

p

m

x f f f f

g g g g h h h h

 

 



 





T

.

s

We assume that all f ii, 1, , , p gj,j



1, 2, , .



I and

are twice continuously differentiable functions, where

, k

h kJ



1, 2, ,



,

I   m J  s

Let

 

 



 





 



 



1

1

: 0,

: 0 ,

: 1 ,

: : 0 .

n

n

p p

i i

j

x g x h x

x g x

I x j I g x

 



 

    

   

 

    

 

  











0 ,

It is well known that if x is an efficient solution of

(MOP), under some constraint qualifications, such as the Kuhn and Tucker constraint qualification (see Ref. [2]) or the Abadie constraint qualification (see Ref. [3]), then the following Karush-Kuhn-Tucker (KKT) condition at

x for (MOP) holds (see Refs. [4,5]):

 

 

 

 

 

T T T

0 0

0

f x g x u h x v

Ug x

h x 

     

 _



 _



(1)

where p\ 0 ,

 

m, ,

u v

  s and



1 2



diag , , , m .

U u u  u

We say that x is a KKT point of (MOP) if it satisfies

the KKT condition.

Since the remarkable papers of Kellogg et al. (Ref. [6])

and Chow et al.(Ref. [7]) have been published, more and

more attention has been paid to the homotopy method. As a globally convergent method, the homotopy method (or path-following method) now becomes an important tool for numerically solving nonlinear problems include- ing nonlinear mathematical programming and comple- mentarily problems (see Refs. [3,4]).

In 1988, Megiddo (see Ref. [8]) and Kojima et al. (see

methods, one of the main ideas is numerically tracing the center path generated by the optimal solution set of the so-called logarithmic barrier function. Usually, the strict convexity of the logarithmic barrier function or non- emptiness and boundedness of the feasible set (see Ref. [10]) are needed. In 1997, Lin, Yu and Feng (see Ref. [11]) presented a new interior point method—combined homotopy interior point method (CHIP method)—for convex nonlinear programming without such assump- tions. Subsequently, Lin, Li and Yu (see Ref. [12]) generalized CHIP method to general nonlinear program- ming where, instead of convexity condition, they used a

more general “normal cone condition”.

In 2003, Lin, Zhu and Sheng (see Ref. [13]) general- ized CHIP method to convex multiobjective program- ming(CMOP) with only inequality constraints. Instead of (CMOP), they considered an associated non-convex non- linear scalar optimization problem and constructed the homotopy mapping.

In Refs. [1,14], we considered a combined homotopy interior point method for the multiobjective programm- ing (MOP) under the condition linearly independent constraint qualification (LICQ). To find a KKT point of (MOP), we construct a homotopy as follows











 

 



 





 

 

 







 



T T T 0

0

0 0

2 5 2 5 0

1

1

0 , ,

1 1 p i

i

f x g x u h x v x x

h x

H _{Ug x U g x}

e

  

   _

    



 _{       } 

 

 

 

   _ 

 

 _{ } 

  _  _   

   







(2)

where 0



0_, 0_{, ,}0 0



m

 

_{0 ,}



_{1,1, ,1}



T p_,

x u v R e R

  



       



, , ,



p m s,

 

0,1 ,

x u v R R

   



    



_{2 5 2 5}



 

₀ 2 5



 

₀ 2 5

 

₀ 2 5



1 , p , and 1 , p .

2 5        

Let

  















0

1 _{0 ,} p m s _{0,1 : ,} 0_,

H H

     

 



    0 .

Let n

A be a nonempty closed set and xA.

We recall that the Fréchet normal cone of A at x is

defined as

 

lim sup , 0 .

A

n A

x x

x x x

N x x

x x

 

 

   



_  

   

  

We used the following basic assumptions which are commonly used in that literature:

(A1)  is nonempty (Slater condition) and bounded;

(A2) (LICQ)   x , the matrix

 

 

 



T T



, j :

h x g x j I x

  

is a matrix of full column rank; (A3) Normal condition:

 



ˆ



 

, .

x x N_ x x

     

It is well known that if condition (A2) holds, then

 

 

 

 

 

ˆ _{: 0, ,} s _{. (3)}

i i j j i

i I x j J

N_ x u g x v h x u i I x v

 

 

 

       

 







 

,

We have proved the following convergence result in Ref. [1].

Theorem 1.1 (Convergence of the method) Suppose

,

i j

f g and k are twice continuously differentiable

functions such that the conditions (A1), (A2), and (A3) hold. Then for almost all

h

0 m s_,

R R

 



     

the zero-point set H_01

 

0 of the homotopy map (2)

contains a smooth curve 0



0,1 ,



p m s

R



  

   

which starts from



0,1 . As

 

0

 

0 0

p m s

T  R_   of _ is nonempty, and the

-x component of every point in T is a KKT point of

(MOP).

Recently, many researchers extended and improved the results in Ref. [1] to convex multiobjective pro- gramming problem, see Ref. [14-17]. The purpose of this paper is to show that Theorem 1.1 remains true under the condition MFCQ instead of LICQ. The paper is organi- zed as following. In Section 2, we prove the existence and convergence of a smooth homotopy path from almost any interior initial point _0 _{to a solution of the}

KKT system of (MOP) under the condition MFCQ.

2. Main Results

We need the following elementary condition.

(A2′) (MFCQ) For every x , the following

conditions hold:

 hj

 

x , ,jJ are linear independent;

 there exists a n

pR such that

 

 

0,g_i x p iI x and h_j

 

x p0,jJ.

Clearly, condition (A2) implies (A2′). It is also known

that if (A2′) holds, then (3) remains valid.

By using an analogue argument as in Ref. [1], we can prove the following two theorems.

Theorem 2.1 Suppose that and conditions

(A1), (A2′) hold. Then for almost all initial points

  

 

0 m _{0 , 0}

 



     is a regular value of H_0

and 0

 

1 ₀

H_ consists of some smooth curves. Among

them, a smooth curve, say _0 starts from

 

0_{,1 .}



Theorem 2.2 Suppose that and conditions

(A1), (A2′) hold. For a given if 0 is a regular value of

  

0

    m s,



  

0

H

 , then the projection of the

smooth curve _0 on the  component is bounded.

We next prove that _0 is a bounded curve.

Theorem 2.3 (Boundedness) Suppose that the con- ditions (A1), (A2′), and (A3) hold. Then for a given

0 m _,

 



    s

if 0 is a regular value of H_0,

then _0 is a bounded curve.

Proof: By Theorem 2.2, it is sufficient to show that the component of smooth curve is bounded. Suppose

 

u v,

-that there exists a sequence





,





0, k

k _

   

such that

, ,

k k

k

x x    

and



k, k







,

u v   k 

where x , 0,1 , 0.

 

 

m s R 

 _{Since closed unit}

circle of is compact, without loss of generality we can assume that







 



,

lim , .

,

k k

k k k

u v

u v u v

 

 

Clearly, u0, ,



u v 



1. By (2), we have







 

 



 





T T

T ₀

1

0,

k k k k

k

k k k

k

f x g x

h x v x x

 



   

   

u



(4)

 

0

 

0 _0.

k k k

U g x U g x (5)

Let

 





 





1

2

: lim ,

: lim .

k i k

k j k

I x i I u

I x j J v









   

   

By (5), we know I1

   

x I x .

Rewrite (4) as





 

 

 

 

 

 





T

T ₀

1

0.

k k k k

k i i

i I x

k k k k k

i i k

i I x

f x u g x

u g x h x v x x

 











 

   

 

 

     

 







(6)

Divide (6) by



k, k



u v and let k , since



k, k



u v  , (6) becomes

 

T









 

T

1 0

I I

g x _ u h x v ,

     (7) where

 





 

 





 









: ,

: .

I i

I i

g x g x i I x

u u i I x

  

  

  

 

1) If uI 0,

 _{ then 0,and}

 

_{0. By} u h x Tv

(A2′), v0. This is a contradiction with



u v, 



1.

2) If uI 0,

_{ we consider the following two cases:}

1. If _



0,1



, we know



1



u_I0 because of

0.

I

u  By (A2′), there exists a nonzero vector

such that

,

n pR

 

T

 

T

T _{0 and} T _0.

I

p g x  p h x 

,

(8) This, together with (7), implies that

which is a contradiction.





1  uI 0  

 

2. If 1, by (7) and (A2′), we know v0. So, 1

u  since



u v, 



1.

Because of (5), I₁

   

x I x  . Thus





1, ,

k

i .

x  u   k   i I

Without loss of generality, we can assume that





0, 1 ,and lim .

k

k I

I k _k k

I

u

u k N

u 



    u

Hence u0, u 1.

a) If I2

 

x

 _{ , then}



k



_{is bounded. We may}

v

assume lim k .

kv v Divide (6) by



1



k k uI



 and let ,

 







 







T

T ₀

1

lim 0.

1

I

k k k

k k

k

k I

g x u

h x v x x

u 

 





 

 

   



 

 

 (9)

This implies that



1



lim

1 k

k

k uI



 _

exists. Indeed,

 





 

T ₀

T ₀

lim

.

k k k

k

k h x v x x

h x v x x

 

 

  

 

 

   





If

 

T ₀

0,

h x v x x

  

that is

 

T ₀

.

h x v x x

  

By condition (A3), 0_.

xx This is impossible since

. So

x 

 

T ₀ 0.

h x v x x

   

By (9), we then have that





1 lim

1 k

k

k uI



 _ exists.

Assume



1



lim .

1 k

k

k uI

 

 _ 

Then 0. 0

If  , (9) becomes

 

T

 

T ₀

.

I u

g x h x v x



  

    x

This contradicts to condition (A3). If 0, (9) becomes

 

T 0. I

g x u

 

By (A2′), there exists a nonzero vector such

that

n pR

 

_T

 

T

0, and 0.

I I

g x p p g x u

   

Thus u0, which contradicts u 1.

b) If 2 , without loss of generality, we can

assume that

 

I x  

















 

1 ,

ˆ ˆ

lim , ,

1 ,

k k k I

k k k

k I

u v

u v u v

 





 

where

 

ˆ ˆ, = 1. Since







1 k, k



as

k u vI k

 ,

     (10)

we divide (4) by





1



k, k



k u vI



 , and let k , we have that

 

T

 

T

ˆ ˆ 0.

I

g x u h x v

    (11) If uˆ 0, then

 

T

ˆ 0.

h x v

 

By condition (A2′), ˆ 0.v This is a contradiction

since

 

u vˆ ˆ, 1.

If ˆ 0,u by (A2′), there is a nonzero vector

such that

n

pR

 

T

 

T

T _0,and T _0.

I

p g x  p h x 

This, together with (11), implies This is a contradiction.

ˆ 0.

u

Therefore, _0 is a bounded curve.

By an analogue argument as in Ref. [1], it is easy to show the following result.

Theorem 2.4 (Convergence of the method) Suppose that the conditions (A1),(A2′), and (A3) hold. Then for almost all 0    ms, the zero-point set

 

0

1 ₀

H_ of the homotopy map (2) contains a smooth curve





0 0,1 ,

p m



 

    which starts from

 

0,1 . As

0,

  the limit set

 

0 p m

 

0

T    of _0

is nonempty, and every point in T is a solution of (1).

Therefore, Theorem 2.4 shows that for almost all

 

0 p m s _{0 ,}

 



    the homotopy Equation (2) generates a smooth curve _0 starts from

 

0_,1



which is called the homotopy path, the limit set

 

0

 

0 0

p m s

T  R   of _ is nonempty, and

the x-component of every point in is a KKT point of

(MOP), the

T

-component

 of the homotopy path is the solution of (1) as  goes to 0.

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