GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

2005,25B(2):359-366

GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE 1

)

Huang Hui ( *~ Department of Mathematics, Zhongshan University, Guangzhou 510275, China Department of Mathematics, Yunnan University, Kunming 650091, China u

Abstr-act

A sufficient condition on the existence of a global weak sharp minima for

general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris' result[lj to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.

Key words

Global weak sharp minima, completeness, convex function

2000 MR Subject Classification

1

90C31, 90C25, 49J52

Introduction

Let X be a metric space and f : X -+ R U{ +oo} be a proper function. We assume t~at f is bounded below and denote by >. the infimum of f on X. Consider the following mathematical programming problem:

min{f(x) I x EX}.

(1.1 )

Let S>. denote the solution set of (1.1), that is, S>. = {x E X I f(x) = A}. we say that f has global weak sharp minima if S>. i- 0 and there exists T > 0 such that Tdist(x, S>.) :::; f(x) - A,

'V x E X.

(1.2)

Weak sharp minima occur in many optimization problems, and have important applications in the convergence analysis of some algorithms. In recent years, the study of weak sharp minima for functions has received a growing interest. We refer the reader to the paper [1, 2, 3, 4] for more details. Among these, Burke and Ferris [1] studied global weak sharp minima in H", Studniarski and Ward [3] studied local weak sharp minima in H": In this paper, we mainly consider global weak sharp minima in infinite dimensional space. Throughout the paper, N denote the set of natural numbers. Let X be a real normed space and let X* be its topological dual. The associated closed unit balls are denoted by B x - . 1 Received September 3, 2002; revised April 22, 2003. The research was supported by the National Natural Science Foundation of China(10361008) and the Natural Science Foundation of Yunnan Province(2003A002M).

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B(x, T) denote the ball with center at x and of radius T. If A c X, then the polar of A is defined by the set

AO = {x* E X* I (x*,x)::; 1, \Ix E A}. The indicator function for A is given by if x E A,

0,

otherwise.

+00, If C

c X*,

the support function is given by

'l/Jc(x)

=

sup{(x*,x) I x* E C}.

Let f : X --+ R U{+oo} be a lower semicontinuous convex function. The contingent directional derivative and the usual directional derivative of f at a point x E dom(f) in the direction hEX are respectively given by

r; x; h) =

. f f(x

li

im In

+ th') t

h'_h

t->o+

and

- f(x)

f'(x; h) = lim f(x + th) - f(x).

t

t->O+

The subdifferential of

f at x is given by

of (x)

=

{x* E X*

I (x*,y -

x) ::; f(y) - f(x),

\ly EX}.

The generalized directional derivative of f is the support function of of(x),

Let A

Xk

--+

c X, the tangent cone at a point x E A is defined by T(x I A) = {y E X I \lxk E A with x,\ltk E (0,+00) with tk --+ O+,::JYk --+ Y, such that Xk +tkYk E A}. The polar of the

tangent cone is called the normal cone and is denoted by

N(x I A) = T(x I A)o. Finally, the contingent cone to A at a point x E A is given by

K(x I A) = {y E X

I ::Jtk

--+

O+,Yk

--+

y,withx+tkYk E A}.

2 Characterizations for Convex Function to Have Global Weak Sharp Minima The following theorem provides a global weak sharp minima for a general function(without the continuity as assumption). Theorem 2.1 Let X be a metric space and f : X --+ R U {+oo} be a function such that A := inf{f(x) I x E X} > -00. Let T > 0 and 0 ::; p < 1 be constants. Suppose that

S>.

=

{x E Xlf(x)

=

A} -I-

0 and that for each x E X \ S>., there exists x' EX \ S>. such that

dist(x', ~>.) ::; pdist(x, S>.)

and

Tdist(x, x') ::; f(x) - f(x'),

No.2

Huang: GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

361

then for each x EX, Tdist(x, SA) ::::: f(x) - A.

(2.1)

Proof It suffices to show that for each x E X \ SA , (2.1) holds. For each x E X \ SA' let XQ = x, by the given condition, there is Xl E X \ SA such that

Suppose that for Xk-l E X \ SA' there is Xk E X \ SA such that

Then for Xk E X \ SA' by the given condition, there is Xk+l EX \ SA such that

Thus, by induction, we obtain a sequence {z.,} in X \ SA such that XQ = x,

for all n E

N. Hence for all n

E

N, n

Tdist(x, SA) ::::: T(dist(x, Xn)

+ dist(x n , SA)) ::::: LTdist(Xi-l, Xi) + Tpndist(x, SA) i=l

n

:s:: L(f(Xi-d - f(Xi)) + Tpndist(x, SA) :s:: f(x) - A + Tpndist(x, SA) i=l

-

as f(x n ) 2: A. It follows that .

1

TdlSt(X, SA) ::::: -l--(f(x) - A), _ pn

V n EN.

By letting n ~ +00, then (2.1) holds. Remark In [5], Ng and Zheng assumed that X is a normed space, and our proof is quite different from theirs. Lemma 2.2 Let X be a Banach space and S be a nonempty closed convex subset of X, let e E (0,1). Then for each x E X \ S, there exist s E Sand s" E X* with II s* 11= 1 such that s* E &dist(s, S) and (s", x - s) 2: edist(x, S).

n

Proof Let x E X \ Sand d := dist(x, S). Then d > 0 and S int(B(x, d)) = ¢. By the separation theoreml''l, there exists x* E X* with II x* 11= 1 such that

sup{(x*,m) 1m E S}::::: inf{(x*,m) 1m E B(x,d)} = (x*,x) - d. Choose a positive e sufficiently small such that e

< 1 and d - 2yfi(d + e + Vi) 2: ed.

(2.2)

Next, choose yES n B(x, d + c), so (x*, y - x) :::::11 y - x II::::: d + c. For any in E S, we have

(z", m - y) = (z", m - x)

+ (z", x -

y) ::::: 'l/Js(m) - 'l/Js(y)

+ c,

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that is, x* E oc'l/Js(y), By the Brondsted-Rockafellar Theoremlll, there exist s E dom('l/Js) = S and w* E o'l/Js(s) such that lis - yll :::; yE Let s* = II:: II (by (2.3) w* Ilx* II = 1, we obtain that Ilx* - s*11 :::; [z" - w*11

i-

0, since

E

and

(2.3)

Ilx* - w*11 :::; yE.

< 1 and Ilx* II = 1 ), then II s* II = 1. By (2.3) and

w* Ilw*1111 :::; yE + Illw*ll- 11:::; yE + Ilw* - x*11 :::; 2yE.

+ Ilw* -

(2.4)

Noting thatj]» - yll :::; d + E, it follows from (2.2), (2.3) and (2.4) that (s", x - s) :::: (x*, x - s) - Ilx* - s" II (11x - yll + Ily - sll) :::: d - 2yE(d

+ E + yE)

:::: ed.

It remains to show that s* E odist(s, S), that is,

(s", u - s) :::; dist( u, S) - dist(s, S),

'
Noting that s E S, we need only to show that (s*,u-s):::;dist(u,S),

'
(2.5)

Let M = {z E X I (s", z) :::; (s", s)}. Let u be an arbitrary point of X. If u E M, one has

(s", u - s) :::; 0 :::; dist(u, S).

(2.6)

If u E X \ M, it is easy to verify that

(s", u - s)

=

dist(u, M).

Since SCM, dist(u,M):::; dist(u,S). Therefore, (s*,u-s) :::;dist(u,S).

(2.7)

(2.6) and (2.7) imply that (2.5) holds. The following theorem provides equivalent conditions for a lower semicontinuous convex function to have weak sharp minima, generalizing Burke and Ferris' resultjl , Thm.2.2] in the case of convex function from finite dimensional space to infinite dimensional space. Theorem 2.3 Let X be a Banach space and f : X ---+ R U {+oo} be a proper lower semicontinuous convex function such that A := inf{f(x) I x E X} > -00. Let T > 0 and SA = {x E X I f(x) = A}. If SA i- 0, then the following statements are equivalent: i. Tdist(x, SA) :::; f(x) - A, '
3. For all hEX and x E SA'

No.2

Huang: GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

4. The inclusion

TBxholds for any x E SA; 5. The inclusion

TB x-

nN(x

1

363

SA) C 8f(x)

n[ U N(x I SA)]

U 8f(x)

C

xES A

xES A

holds; 6. For any y E X and () E (0,1),

f'(p;y-p) 2: ()Tdist(y,SA)' where p E F(}(y) := {s E SA dist(tx + (1 - t)s, SA) 2: ()tdist(x, SA)' 'Vt E (0, I)}. Proof [1 =} 2] Let x E SA' The hypothesis guarantees that for any t E (0, +00) and 1

h' E X,

+ th') -

f(x

f(x) 2: Tdist(x + th', SA)'

which implies that

f(x

+ th') -

f(x)

.::......:...----'---"----=--~

t

dist(x + th', SA) - dist(x, SA) > T----'----.:........:-"------'---'----'--'-'-

t

2: T dist(x + th, S~) - dist(x, SA) _ Tlih' _ hll. By taking lower limits on both sides as h'

-+

hand t

0+ and applying [8, Cor.2] we obtain

-+

that Observe that if f is lower semicontinuous and convex, then SA is nonempty closed and convex, and it follows that T(x SA) = K(x 1 SA)[9]. Hence 1

t: (x; h) 2: Tdist(h, K(x [2

=}

1

SA)).

3] The result is obtained by the equivalence T(x

I

SA) = K(x

1

SA) and definition

fO(x; h) 2: f-(x; h). [3

=}

4] We recall from [10, Thm. 3.1] that if K C X is a nonempty closed convex cone,

then dist(x, K)

= 'lj/'KonBx_ (x).

The result now follows from the fact that fO(x;·) = 'l/Ja!ex) (.). [4 =} 1] It only need to show that for each x E X \ SA' Tdist(x, SA)

:s; f(x) - A.

Let () E (0,1), by Lemma 2.2, there exist s E SA and s" E 8dist(s, SA) with Ils*11 = 1 such that

(s*, x - s) 2: ()dist(x, SA). It follows from Ils*11 = 1 and s* E 8dist(s, SA) that 78* E TBX- nN(s SA)' and the condition 1

implies that TS*

E

8f(s). Hence, {hdist(x, SA)

:s; (78*, X

-

s) :s; f(x) - f(s) = f(x) - A.

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The result now follows by letting B -> 1- . [1 =} 4] Let x E SA and x* E TE x * n N(x 1 SA)' Theorem 1 in [8] implies that x* E T8dist(x, SA)' By assumption, we obtain

(z", y - x) :::; Tdist(y, SA) - Tdist(x, SA) :::; f(y) - A = f(y) - f(x),

'v'y

E X,

this means that x* E 8f(x). [4 {::} 5] Similar to the proof of Lemma 2.1 in [1]. 6] We first show that Pe(Y 1 SA) =I- 0. Let y E X. If y E SA' then y E Pe(Y). If y E X \ SA' then by Lemma 2.2, there exist s E SA and s* E 8dist(s, SA) with 118* II = 1 such that [1

=}

(8*, Y - s) Then for each t

E (0,1),

~

Bdist(y, SA)'

we have

tBdist(y, SA) :::; (s", ty

+ (1 - t)s -

s) :::; dist(ty + (1 - t)s, SA)'

that is, s E Pe(y 1SA)' Now let y E X and p E Pe(y 1SA)' By assumption, one has

f(ty

+ (1 - t)p)

that is,

~

f(p)

+ Tdist(ty + (1 - t)p, SA)

~

f(p)

+ TBtdist(y, SA)'

f(p+t(y-p)) -f(p) > B di t( S) t - T IS y, A'

The result now follows by letting t

->

0+.

[6 =} 1] Let y E X. Then for each B E (0,1) and p E Pe(Y 1SA)' one has

!'(PiY-P) ~ BTdist(y, SA)' Since

(2.8)

f is convex, it follows that !'(PiY-P)

=

lim f(p+t(y-p))-f(p) :::;f(y)-f(p)=f(y)-A.

t--->O+

t

(2.9)

(2.8) and (2.9) imply that

Letting B -> 1-, one has Tdist(y, SA) :::; f(y) - A.

Remark [1 {==} 6] and [1 {==} 4] in our proof depend on techniques of Banach space, which is very different from Burke and Ferris' one.

3 Global Weak Sharp Minima and the Completeness of a Metric Space Let X be a metric space, let S(X) := {f 1 f be a proper lower semicontinuous function from

X to R U {+oo} satisfying inf{f(x) I x E X} = 0, and there exist a decreasing sequence{A~} converging to 0 and a constant "r > 0 such that lim Tfdist(x, S{ ) :::; f(x) 'v'x E X, where S( = {x E

XI

st = {x E X I f(x) = O}.

n---+oo

f(x):::; An}. For each f E S(X), let

n

No.2

Huang: GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

Theorem 3.1

Let X be a metric space, suppose that for each

Tfdist(x, st)

::;

f(x)

f

365

E S(X),

'Vx E X,

(3.1)

then X is complete. Proof We established the proof in four steps. Step 1 Suppose that {x n } is a Cauchy sequence in X, define

f(x) = lim dist(x n, x)

'V x E X.

n->CX)

Since I dist(x n, x) - dist(x m , x) I::; dist(xn,x m ) , it follows that {dist(xn,x)} is a Cauchy sequence in R. Thus for each x E X, f(x) < 00. This implies that f is well defined. Step 2 We will show that f is continuous in X and inf f(x) = O. Since xEX

1dist(x n, x) letting ti ----+ +00, one has that hence continuous. Since

dist(x n, y)

I::; dist(x, y),

1f(x) - f(y) I::; dist(x, y).

0::; inf f(x) ::; f(x n) = xEX

This implies that f is Lipschitz and

lim dist(x m , x n),

m->CX)

noting that {x n} is a Cauchy sequence in X, it follows that inf f(x) •

Step 3 We will show that

f

xEX

= o.

E S(X). It remains to show that for each x E X,

lim dist(x,S{)::; 2f(x).

m---+oo

(3.2)

171

If f(x) = 0, then (3.2) holds trivially. It suffice to show that (3.2) holds when f(x) > O. Since [z.,} is a Cauchy sequence in X, there exists N 1 EN such that 1 dist(xp , xq) < m

dist(xq,x) < 2f(x).

and

whenp,q > N 1 . It follows that f(x q) = lim dist(xp,xq)::; 1.-, that is, xq E S{. Thus p---+oo

m

171

dist(x, S{) ::; dist(x, x q) < 2f(x), >n

it follows that lim dist(x, S{) ::; 2f(x).

m---+oo

171

Step 4 We will show that {x n } converges to some u in X. By assumption and f E S(X) we know (3.1) holds, then st -I- 0. Let y, z E S{, then there exists N2 EN such that rn

dist(xN 2, y) < f(y)

+

1 -m

::;

2 -m

It follows that dist(y, z) ::; dist(xN2' y)

dist(xN2' z) < f(z)

and

+ dist(xN2, z) <

1

2

+ -m ::; -. m

~, this implies that 4

diamS~ := sup{dist(y, z) I y, z E S~ } ::; -. m m m

By Xn

st -I- 0 and {~} converging to 0, there exists u E X such that {u} st c S~, this and =

E

S{(n 2 N 1 ) imply that lim dist(xn,u)::; 1.-. Letting m m

n---+oo

m

lim dist(x n, u) = 0,

n->CX)

it means that {x n } converges to u in X, thus X is complete.

----+ 00,

one has that

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Acknowledgements The author would like to thank the referees for valuable comments. The author is also thankful to my advisor, Professor Zheng Xi-yin for many helps OP this paper. References 1 Burke J V, Ferris C. Weak sharp minima in mathematical programming. SIAM J Control Optim, 1993, 31: 1340-1359 2 Khayyal F A, Kyparosis. Finite convergence of algorithms for nonlinear programs and variational inequalities. J Optim Theory Appl, 1991, 70: 319-332 3 Studniarski M, Ward D E. Weak sharp minima: Characterizations and sufficient conditions. SIAM J Control Optim, 1999, 38: 219-236 4 Ward D E. Characterizations of strict local minima and necessary conditions for weak sharp minima. J Optim Theory Appl, 1994, 80: 551-571 5 Ng K F, Zheng X Y. Error bounds for lower continuous functions in normed spaces. SIAM J Optim, 2001, 12: 1-17 6 Richard B H. Geometric functional analysis and its applications. Berlin: Spring-Verlag, 1975. 15 7 Phelops R R. Convex functions, monotone operators and differentiability. Berlin: Spring-verg, 1989. 51 8 Burke J V, Ferris M C, Qian M. On the Clarke subdifferential of the distance function to a closed set. J Math Anal Appl, 1992, 166: 199-213 9 Clarke F H. Optimization and nonsmooth analysis. New York: John willey and sons, 1990. 55 10 Burke J V, Han S P. A Gauss-Newton approach to solving generalized inequalities. Math Oper Res, 1986, 11: 1197-1211