Exact Penalty Functions in Constrained Optimization

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SIAM J. CONTROL AND OPTIMIZATION

(C) 1989 Society for Industrial and Applied Mathematics

Vol. 27, No. 6, pp. 1333-1360, November 1989

006

EXACT PENALTY FUNCTIONS IN CONSTRAINED OPTIMIZATION* G. DI PILL0

AND

L. GRIPP0:[:

Abstract. In this paper formal definitions of exactness for penalty functions are introduced and sufficient conditions for a penalty function to be exact according to these definitions are stated, thus providing a unified framework for the study of both nondifferentiable and continuously differentiable penalty functions. In this framework the best-known classes of exact penalty functions are analyzed, and new results are established concerning the correspondence between the solutions of the constrained problem and the unconstrained minimizers of the penalty functions.

Key

words, exact

penalty functions, nonlinear programming, constrained optimization

AMS(MOS) subject classifications. 49D30, 49D37, 90C30

1. Introduction. A considerable amount of investigation, both from the theoretical and the computational point of view, has been devoted to methods that attempt to solve nonlinear programming problems by means of a single minimization of an unconstrained function. Methods of this kind are usually termed exact penalty methods, as opposed to the sequential penalty methods, which include the quadratic penalty method and the method of multipliers (see, e.g., [4], [23], and [26]). We can subdivide exact penalty methods into two classes: methods based on exact penalty functions and methods based on exact augmented Lagrangian functions. In our terminology, the term "exact penalty function" is used when the variables of the unconstrained problem are in the same space as the variables of the original constrained problem, whereas the term "exact augmented Lagrangian function" is used when the unconstrained problem has to be minimized on the product space of the problem variables and of the multipliers. Exact penalty functions can be subdivided, in turn, into two main classes: nondifferentiable exact penalty functions and continuously differentiable exact penalty functions. Nondifferentiable exact penalty functions were introduced for the first time in [39] and have been widely investigated in recent years (see, e.g., [1], [2], [5]-[10], [22], [25], [29], and [35]). Continuously differentiable exact penalty functions were introduced in [24] for equality constrained problems and in [28] for problems with inequality constraints; further contributions have been given in [14], [15], and [34]. Exact augmented Lagrangian functions were introduced in 11 and 12] and have been further investigated in [3], [4], [19]-[21], [31], and [38]. In this paper we restrict our attention to exact penalty functions, with the aim of providing a unified framework which applies both to the nondifferentiable and to the continuously differentiable case. We start from the introduction of formal definitions of various kinds of exactness that attempt to capture the most relevant aspects of the notion of exactness in the context of constrained optimization. This is motivated by the fact that in thecurrent Received by the editors June 6, 1988; accepted for publication (in revised form) November 11, 1988. This research was partially supported by the National Research Program on "Modelli e Algoritmi per l’Ottimizzazione," Ministero della Pubblica Istruzione, Rome, Italy. Dipartimento di Informatica e Sistemistica, Universit di Roma "La Sapienza," Via Eudossiana 18, 00184 Rome, Italy. $ Istituto di Analisi dei Sistemi ed Informatica del Consiglio Nazionale delle Ricerche, Viale Manzoni

30, 00185 Rome, Italy. 1333


1334

G. DI PILLO AND L. GRIPPO

literature the term exact penalty function seems to be used without a definite agreement on its meaning. In particular, as noted in [29], most of the literature on this subject is mainly concerned with conditions that ensure that the penalty function has a local (global) minimum at a local (global) minimum point of the constrained problem. On the other hand, since the penalty approach is an attempt to solve a constrained problem by the minimization of an unconstrained function, this characterization is fully satisfactory only when both the constrained problem and the penalty function are convex. In the nonconvex case, the study of converse properties appears to be of greater interest, as they ensure that local (global) minimizers of the penalty function are local (global) solutions of the constrained problem. Moreover, again in the nonconvex case, a distinction has to be made between properties of exactness pertaining to global solutions and properties pertaining to local solutions. It will be shown that, for the same penalty function, different kinds of exactness can be established under different regularity requirements on the problem constraints. Finally, the correspondence between the constrained and the unconstrained minimization problem can only be established with reference to a compact set containing the problem solutions, and this must be carefully taken into account in the analysis of the properties of exactness. The formal definitions mentioned so far constitute the basis for the development of sufficient conditions for a penalty function to be exact according to some specified notion of exactness. In particular, we establish sufficient conditions which apply both to the nonditterentiable and to the continuously ditterentiable case, thus providing a unified framework for the analysis and the construction of exact penalty functions. In this framework, we consider the best-known classes of exact penalty functions, and we provide a complete analysis of their properties, recovering known results and establishing new ones. The paper is organized as follows. Section 2 contains the problem statement, basic notation, and preliminary results. In 3 we formalize the definitions of various kinds of exactness of penalty functions, which are classified as weak exactness, exactness, strong exactness, and global (weak, strong) exactness. Section 4 deals with nonditterentiable penalty functions: we analyze the properties of lq exact penalty functions as well as those of the globally exact nondifferentiable penalty function considered in [16]. In 5 we study continuously differentiable exact penalty functions, and we introduce a globally exact differentiable penalty function for mixed equality and inequality constrained problems by extending the results given in [15]. Computational aspects are beyond the scope of this paper. We refer, e.g., to [3], [4], [9], 18], [21], [27], [28], [33], [34], [36], and [37] for some algorithmic applications of exact penalty functions.

2. Problem statement, basic notation and preliminary results. The problem considered here is the general nonlinear programming problem: minimize f(x)

(P)

,

-

subject to g(x) [m, h [R" and the feasible set

P, p =< n

h(x) O, are continuously differentiable functions

y := {x e": g(x) 0,

We denote by

-

g(x) - 0}.

We adopt the following terminology. The linear independence constraint qualification (LICQ) holds at xR if the p are linearly independent. gradients Vgi(x), Io(x), Vh(x),j= 1,. The Mangasarian-Fromovitz constraint qualification (MFCQ) holds at x N if Vh(x),j= 1,...,p are linearly independent and there exists a zR such that

.,

V gi(x)’ z < O,

Io(x)

Vhi(x)’z=O

j= l,

p.

It can be shown, by using the theorems of the alternative [32] that the MFCQ can be restated as follows. if there exist no ui, The MFCQ holds at x p such Io(x), and vj, j 1,

,

"

that

p

i

2lo(x

u,Vg,(x)+ y vvh(x)=O, U

(ui,

6

Io(x),

O,

Io(x), v,

j

1,

p) O.

In some cases we shall make use of a stronger constraint qualification, which is stated in the following equivalent formulations. The extended Mangasarian-Fromovitz constraint qualification (EMFCQ) holds at x p are linearly independent and there exists a z Nn such that ifVh(x),j 1,

,

V gi(X)’ Z < O, V h(x)’ z O,

I+(x) j

l,

p.

1336



The EMFCQ holds at x Nn if there exist no

ui,

/+(x), and v.i,j 1,

, p such

that P

2 uVg(x)+ j=l 2 vVh(x)=O, il+(x) I+(x), ui >= O,

_

p) O. j 1," (ui, 6 I+(x), v, It can be noted that the LICQ implies the MFCQ and that the EMFCQ implies the MFCQ. It is known that if ff is a local solution of problem (P) and if the MFCQ holds at then e that is, there ex.ist K-T multipliers (h,/2) associated with :. We recall that a nonempty set is called an isolated set of if there exists

,

-,

*

*

of Y( and such that if is contained in the interior a closed set Y( such that x e Y(-*, then x % Isolated sets of local minimum points possess the property stated in the following lemma, which is proved in [23]. be an isolated compact set of local minimum points of problem LEMMA 1. Let to the local minimum value f*; then there exists a compact set corresponding (P), c and for any point x Y( fq such that if x *, then f(x) > f*. We also state the following lemma, which for q >_-2 is an obvious consequence of

* , *

,

.

,

the equivalence of the norms I1" IIq and I1" Ilq-1 on LEMMA 2. Let q 1 0 such that for all z ’, we have"

Iz, lq-’ lllzllq

-.

i=1

Proof The assertion follows from a more general result on positive homogeneous [3 continuous functions ([30, Thm. 5.4.4]). In the sequel we shall be concerned with compact perturbations of the feasible set. In particular, we shall consider the case in which o% is compact and-there exists a m, /3 > 0, such that the set a vector/3 (ao, a’)’ with ao := 6 0t {x "" g(x)= 0 for all d ". If : is a critical point of F and F is differentiable at we have 7F(Y)=0; in this case we say that ff is a stationary point of F. Finally, we denote by (:; p) the open ball around ff with radius p > 0.

.,

u-:=

,

.

.,

,

s,

1337


EXACT PENALTY FUNCTIONS

3. Definitions of exactness for penalty functions. Roughly speaking, an exact penalty function for problem (P) is a function F(x; e), where e > 0 is a penalty parameter, with the property that there is an appropriate parameter choice such that a single unconstrained minimization of F(x; e) yields a solution to problem (P). In particular, we require that there is an easy way for finding correct parameter values by imposing that exactness is retained for all e ranging on some set of nonzero measure. More specifically, we take e (0, e*] where e*> 0 is a suitable threshold value. In practice, the existence of a threshold value for the parameter e, and hence the possibility of constructing the exact penalty function F(x; e), can only be established with reference to some compact set @. Therefore, instead of problem (P) we shall consider the following problem.

(P)

minimize f(x),

"

x

.

71 @,

It can be observed that if such that f’l where @ is a compact subset of c @, then problem () and problem (P) are equivalent. We denote by and L, respectively, the set of global solutions and the set of local solutions of problem (P), that is: oT := {x f3 @" f(x) 0 such that, for all e (0, e*], (e) and, moreover, any local unconstrained minimizer of problem (Q) is a local solution of problem (P), that is:

(e)

_

,

for all e

(0, e*].

It must be remarked that the notion of exactness given in Definition 2 does not require that all local solutions of problem (P) in correspond to local minimizers of the exact penalty functions. A one-to-one correspondence of local minimizers does not seem to be required, in practice, to give a meaning to the notion of exactness, since the condition o7 q3(e) ensures that global solutions of problem (P) are preserved. However, for the classes of exact penalty functions considered in the sequel, it will be shown that this correspondence can be established, also, at least with reference to isolated compact sets of local minimizers of problem (P) contained in Thus, we can also consider the following definition. DEFINITION 3. We say that the function F(x; e) is a strongly exact penalty function for problem (P) with respect to the set @ if there exists an e*> 0 such that, for all e(0, e*], q= (e), (e)_ and, moreover, any local solution of problem (P) is a local unconstrained minimizer of F(x; e), that is" belonging to

.

,,

. .

N

(e) for all e (0, e*].

The properties considered in the preceding definitions do not characterize the behavior of F(x; e) on the boundary of Although this may be irrelevant from the conceptual point of view in connection with the notion of exactness, it may assume a considerable interest from the computational point of view, when unconstrained descent methods are employed for the minimization of F(x; e). In fact, it may happen that there exist points of such that a descent path for F(x; e) that originates at some of This implies that the sequence of points these points crosses the boundary of produced by an unconstrained algorithm may be attracted toward a stationary point of F(x; e) out of or may not admit a limit point. Therefore, it could be difficult to construct minimizing sequences for F(x; e) which are globally convergent on toward the solutions of the constrained problem. In order to avoid this difficulty, it is necessary to impose further conditions on F(x; e), and we are led to introduce the notion of global exactness of a penalty function. DEFINITION 4. The function F(x; e) is said to be a globally (weakly, strongly) exact penalty function for problem (P) with respect to the set @ if it is (weakly, strongly) 0@ there exists a neighborhood exact and, moreover, for any e > 0 and for any we have" (, p) such that if {Xk}C and limk_ xg

.

,

lim inf F(Xk; e) > F(x; e), k-eo

for all x (; p) f3 The condition given above excludes the existence of minimizing sequences for F(x; e) originating in that have limit points on the boundary. In fact, we can state the following proposition.

1339



.

PROPOSITION 1. Let F(x; e) be a (weakly, strongly) globally exact penalty function with respect to the set 9 and let {Xk} c be a sequence such that F(Xk+I’, e)< F(Xk" e). Then, any limit point of {Xk} belongs to Proof By the compactness of 9 there exists a subsequence, which we relabel {Xk}, 9. Reasoning by contradiction, assume that 09. Then, recalling such that Xk-> Definition 4, we have, for sufficiently large values of j, lim infk-,oo F(Xk’, e) F(xj’, e), which contradicts the assumption F(Xk; e)j. 4. Sufficient comlitions for exactness. In this section we state sufficient conditions which imply that a penalty function F(x; e) possesses some of the properties of exactness considered in the preceding section. Everywhere below we suppose that the following assumption holds. that Assumption (A1). Any global solution of problem () belongs to the set is: We note that Assumption (A1) concerns the selection of the set 9 and implies c that it can be satisfied, in particular, by a proper choice of 9, whenever the global solutions of problem (P) belong to a bounded subset of Let Y( be the subset of ffl where the function F(x; e) takes the same values off(x), that is"

,

=.

.

;

(1)

Y{:={xof-)’F(x;e)=f(x)foralle>O}.

The next theorem establishes a sufficient condition for F(x; e) to be a weakly exact penalty function in the sense of Definition 1. THEOREM 1. Let F(x; e) be such that the following conditions are satisfied. (al) For any e > O, the function F(x; e) admits a global minimum point on a set such that c_c_ ; c_ 9. 9 (a2) If { ek} and {xk} c__ g’ are sequences such that limk_oo ek 0, limk_,oo xk and lim supk_,oo F(xk, ek) 0 such that, for all (a4) For any e (0, e()], if %(e) and x %(e) is a global minimum point of problem (Q) -< o(: ), we have x satisfying x Then, F(x; e) is a weakly exact penalty function for problem (P) with respect to the

.

set 9.

Proof We show first that there exists an e*>0 such that, for all e (0, e*] we have 3(e) and qa(e) c_ Recalling condition (al), we have that, for any given e > 0, there exists a point x* such that: *" e) min F( x, F(x; e). xG We prove, by contradiction, that there exists an e* > 0 such that, for all e_e (0, e*] the point x* is a global solution to problem (). Suppose that this assertion is false. Then, for any integer k there must exist an ek ko such that, for all k -> k, x e a(e), e _-< e(:) and [[x - ll -< (), where e(2) and o-(2) are the numbers for all k _-> k, so that, considered in condition (a4). Therefore (a4) implies that x by (2), we obtain"

=,

’"

k >= k

f(x,) F(x; e) such that for be concluded that there exists (0, e*] any global minimizer to On the other hand, by of (). solution is a on global problem x* F(x; e) and hence, for all in of solutions are () the problem global Assumption (A1), is a global minimizer for problem (Q). e (0, e*], we have that x* Now let Thus we have proved that for e(0, e*], (e) and (e)___ we have condition e (0, e*] and let x be any point in (e) c_ By (a3) (e) c__ ff{ so that:

Hence, x ff

is both a

.global

.

.

f(x) F(x; e).

(3) If

is another global minimizer of Problem

f(g)

(4)

_

(’), again by (a3), we have

F(X; e).

Therefore, as f(x)=f(g), (3) and (4) imply that F(X; e)= F(x; e) and this proves that X is a global solution to problem (Q). Thus, (e) for all e (0, e*] and this U completes the proof. A short discussion of conditions (a)-(a4) is in order. Condition (a) requires the existence of a global minimizer of F(x’, e) on the set @ and the case When Two cases are of interest" the case @, recalling that @ is compact, the existence of a global minimizer is ensured by the continuity of we must specify some further condition on the behavior of F(x; e) F(x; e); if on 0. We shall address this problem later in connection with sufficient conditions for global exactness. Condition (a) indicates the role played by the penalty parameter e it requires, in particular, that, as e goes to zero, if the penalty function remains bounded from above, the constraints are satisfied in the limit. With regard to (a3), we may note that this condition is satisfied whenever, for all e>0"

.

.

,

F(x; e) =f(x), for x

o f-I

.

.

1341



.

In fact, in this case we have, by (1), that Y{ -71 The different classes of exact penalty functions considered in the sequel are associated with different characterizations of Y{. In particular, in the case of nondifferentiable penalty functions, we have Y{" o% However, this requirement would be too strong to allow the construction of continuously differentiable exact penalty functions, as will be apparent from the content of 5. Thus, in the case of continuously differentiable exact penalty functions, the set Y{ turns out to be a subset of f3 containing a region where suitable necessary optimality conditions for problem (’) are satisfied. Finally, we may note from the proof of Theorem 1 that condition (a4) is of major relevance in order to establish the properties of exactness considered, since the first three conditions are usually satisfied in the case of sequential penalty functions also. We give now a sufficient condition for F(x; e) to be an exact penalty function in the sense of Definition 2, which is obtained by replacing (a4) of Theorem 1 with a stronger condition and by imposing that F(x; e) is bounded above by f(x) on the set

o

More specifically, we state the following theorem. THEOREM 2. Let F(x; e) be such that conditions (al)-(a3) of Theorem 1 are satisfied and assume further that the following conditions hold. and x (e), (as) There exists an g>0 such that, for all e (0, g], if (e) we have x (a6) F(x; e)O andx6f’l. Then, F(x; e) is an exact penalty function for problem (P) with respect to the set @, in the sense

of Definition

2.

Proof We observe first that condition (as) is stronger than condition (a4) of Theorem 1 so that the function F(x; e) is a weakly exact penalty function in the sense of Definition 1. Hence, there exists an g>0 such that for all e e (0, g] we have and this implies (e) Now let e*= min g], where g is the number (e) considered in condition (as), let e (0, e*], and assume that x e (e). Then, by (as), we have x Y{" so that we get F(x; e)=f(x). This implies that for e (0, e*] and for some p > 0 it can be written:

.

,

f(x) O; fig,(x) =0; fig,(x) < 0

and

[v(x)’a,

if(x)>O; if(x) =0; ifh(x)0 such that" ie

Io()

1349



6. Continuously differentiable exact penalty functions. In this section we study a class of continuously differentiable exact penalty functions, making use of the sufficient conditions established in 4. The key idea for the construction of continuously differentiable exact penalty functions is that of replacing the multiplier vectors (, ) which appear in the augmented Lagrangian function of Hestenes, Powell, and Rockafellar [4] with continuously differentiable multiplier functions (&(x), (x)), depending on the problem variables.

Let

:= {x 6g": Vgi(x),

Io(x), Vh(x),j 1,"

p are linearly independent}; we can consider the multiplier functions (A (x), pc(x)) introduced then, for any x in [28], which are obtained by minimizing over P the quadratic function in (A, pc) defined by: ",

"

,(;,

., x):-IlVxL(x, ;t, )11+ ,ll a(x); ,

where 3/ 0 and

G(x) := diag (g,(x)). The function (h, pc; x) can be viewed as a measure of the violation of the set of K-T necessary conditions: Vx/4X, ;t, ) 0, (x) =0.

Let N(x) be the (m+p)x(m+p) matrix defined by: og(x) oh(x)’ og(x) og(x)’

N(x) :=

v(x

Ox

Ox

oh(x) og(x)’ Ox

Ox

Ox

Ox

Oh(x) Oh(x)’ Ox

Ox

In the next proposition we recall some known results established in [28]. gand 3’ O. Then: (a) the matrix N(x) ispositive definite; PROPOSITION 12. Let a there exists (b) unique minimizer (h (x), pc(x)) of the quadratic function in (,, pc), (, pc; x), given by

-

k(x)

"

OX

(x)

oh(x)

Vf(x);

,

(c) if (g, X, t2) g x x NP is a triple such that VxL(g, fi) 0 and G(g). O, and pc () f; we have h () (d) the Jacobian matrices of h (x) and pc (x) are given by

(16) LOxJ where

R(x):

Og(x) OX

VL(x, h(x), /z(x))+

E

i=1

Og(x) OX

e’VxL(x, h(x), pc(x))’VZgi(x)


1350


s(x):

h(x) V]L(x, h(x),/z(x))+ OX

e;V,,L(x, h(x), (x))’V2hj(x) j=l

VxL(x, A (x), i(x)) := [VxL(x, A,

, =,(x)

VZxL(x, h(x), t*(x)) := [VZxL(x, h, =(x)

A(x) := diag (a,(x)) and ei (ej) denote the ith(jth) column of the m x m(p x p) identity matrix. Thus we can consider the penalty function introduced in [28], defined by p

1

W(x; e):=f(x)+h(x)’(g(x)+ Y(x; e)y(x; e))+- Ilg(x)+ Y(x; e)y(x; e)[[ 2 8

(17) 1

+,(x)’h(x)+- Ilh(x)ll

,

where

yi(x; e):= -min O, gi(x)+- hi(x)

(18)

Y(x; e):= diag (y,(x; e)). It can be verified that the function W(x; e) can also be written in the form 1

W(x; e)=f(x)+A(x)’g(x)+-Ilg(x)ll+**(x) ’h (x) 1

+-Ilh(x){{ E

-

" {min[O,ei(x)+2gi(x)]} 2

7- E I4E

.

i=l

From the above expression and the differentiability assumptions on the problem functions, it follows that W(x; e) is continuously differentiable on The expression of W(x; e) can be derived by means of the following reasoning. Consider the transformed problem: minimize f(x)

subject to g(x) + Yy

O,

h(x) O,

,

where Yi, m are slack variables and Y:= diag 1,. Define the augmented Lagrangian function for this problem:

L,(x, y, A, ; e):=f(x) + M(g(x)+

yy)+l iig(x)+ gyll2+ t,,h(x)+l [[h(x)[[2. E

E

Then, by substituting (h(x), (x)) for (A,/z) and minimizing with respect to y, we get the function W(x; e), that is,

W(x; e)=La(x,y(x; e),a(X),l(X); e)=minLa(x,y,a(X),l(X); e). y Since, by construction,

[VyLa(x, y, h,/z; e)] a=a(x)=0, t*

y=y(x; e)

1351



the gradient expression of W(x; e) can be obtained by treating formally y(x; e) as a constant vector. Thus, we have"

v W(x; )

V/(x) +

Og(x)’ Ox

Oh(x)’ a(x)+(x) Ox

20g(x)’ 2 Oh(x)’ -(g(x)+ Y(x; e)y(x; e))h(x) e

+

Ox

ox(x)’ Ox

Ox

e

(g(x)+ g(x; )y(x; ))+

o(x)’ Ox

h(x),

where OA(x)/Ox and Ol(x)/Ox are the Jacobian matrices defined in (16). Some properties of exactness of the function W(x; e) have been established in [15] for inequality constrained problems. Here we perform a more complete analysis for problems with both equality and inequality constraints, making use of the sufficient conditions given in 4. c We suppose that Assumption (A1) of 4 is satisfied, that is, and that everywhere in this section the following assumption holds. Assumption (A4). The LICQ is satisfied on Some immediate consequences of the definition of W(x; e) are pointed out in the following proposition. PROPOSITION 13. Let (2, ) be a K-T triple for problem (P), such that @. Then, for any e >0, we have" (a) g(ff)+ Y(2; e)y(ff; e)=0;

,

,

(b) W(2; e)=f(2); (c) v w(z; )=0. Proofi By Proposition 12 we have ;t()= ] and (2)=/2, so that, since (2,,/2) is a K-T triple for problem (P), we obtain VL(2, A (2),/ (2)) 0, (2) -> 0 and ;t() 0 when g()< 0. Then, (a) is satisfied by definition of y(2; e) and (b) follows directly from (17). Finally, (c) follows from (19), taking (a) into account and noting that, by assumption, h()=O and VxL(& h(),/(2)) =VL(, ,/2).

,

,

Then, we have the following proposition. PROPOSITION 14. Let 9. Then, there exist numbers e() > 0 and cr() > 0 such that, for all e6(0, e()], if x is a stationary point of W(x; e) satisfying we that is a K-T triple for problem (P). have (), (x, A(x),/(x)) IIx-ll ; e Let x definition of by y(x; then, e), we have Proof

Y2(x; e)h(x) =-2 Y2(x; e)(g(x)+ Y(x; e)y(x; e));

(20)

E

moreover, by definition of h (x) we can write:

Og(x) Ox

VxL(X, (x), (x)) -va(x) (x)

=-y2G(x)(G(x)+ yE(x; e))A(x)+ y2G(x) y2(x; e)A(x)

(21)

=-y:ZG(x)A(x)(g(x)+ Y(x; e)y(x; e))+ y2G(x) y2(x; e)A(x). Therefore, by (20) and (21) we get

Og(x) VxL(x,A(x), ix(x))=-y2G(x)(eA(x)+2YZ(x; e))(g(x)+ Y(x; e)y(x; e)), Ox

1352



so that, by (19), we can write

Og(x)

e------ V W(x; Ox

og(x)

e------ VxL(x,a(x),(x))

e)

Ox

+ (22)

+

+ OA(x)’ Og(x)(Og(X)’ -o-ox ox 2

(

og(x) Ox

2

/

oh(x)’

(g(x)+ Y(x; e)y(x;

o.(x)’)

+e h(x) Ox

Ox

Kl,(x; e)(g(x)+ Y(x; e)y(x; e))+ K,2(x; e)h(x), where / g(x) Og(x)’ Ox \ Ox

oa(x)’ (og(x) +e\ xx Ox K12(x;

--+e o(x)’

og(x) oh(x)’

e := 2--

Ox

e)) rG(x)A(x))

y2G(x) Y(x;

KI,(x; e):= 21

og(x) Ox

Ox

Ox

Now, by definition of x(x), we have oh(x) vxI4x, t (x), (x)) o Ox

and hence, by (19) we can write

(23)

e

oh(x) Ox

VW(x; e)= K2,(x; e)(g(x)+ Y(x; e)y(x; e))+ K22(x; e)h(x),

where

g21(X; E):= 2

K2(x; e) := 2

Oh(x) Og(x)’ OX

OX

Oh(x) OA(x)’

t- e OX

OX

Oh(x) -F Oh(x)’ oh(x) o(x)’ e Ox

Ox

Thus, from (22) and (23) we get, for all x

Ox

Ox

:

og(x)

(24)

e

Oh(x)

V W(x’e)=K(x’e)

g(x)+ Y(x; e)y(x; e) h(x)

L ox where K(x; e) is the matrix defined by

Let now

;

[Kl(X"

e)]

e) K2(x; K21(x; e) K22(x; e) C1 @; then, by definition of y(x; e) we have K(x;e):=

y2(.; O)=-G(:), so that, by definition of K(x; e), we get

K(;; 0) 2N(;).



1353

Therefore, since Assumption (A4) implies that N(x) is nonsingular, by continuity there exist numbers e()> 0 and tr()> 0 such that the matrix K(x; e) is nonsingular for all e [0, e()] and all x such that IIx- ll-- 0. Hence, the triple (x, A (x),/x(x)) is a K.T triple l-] for problem (P). The next proposition establishes the correspondence between stationary points of W(x; e) and K-T triples for problem (P) on the whole set @. PROPOSITION 15. Assume that the EMFCQ holds on 9. Then, there exists an e*> 0 such that, for all e(0, e*], if x is a stationary point of W(x; e), we have that (x, A (x), I (x)) is a K-T triple for problem (P). Proof The proof is by contradiction. Assume that the assertion is false. Then, for such that V W(x; e)=0, any integer k, there exists an ek =< 1/k and. a point xk but (x, A (x),/z(xk)) is not a K-T triple for problem (P). Since is compact, there exists a convergent subsequence (relabel it again {x}) @. Moreover, since V W(x; ek) 0 for all k and since e 0, such that lim_ x we have in the limit, by (19):

:

(28)

Og(:)’ Ox

(g()) + y(;; O)y(); 0))+ oh(;)’ h(.)=0, Ox

where, by definition of y(x; e) we have y,2. (; O)= -min [0, g()], It follows that (28) can be rewritten into the form: p

.

E g,())Vgi(;)-k E iel+(;)

hj(:)Vhj(;) 0,

j=l

where /+()={i" gi()=>0}. Therefore, by the EMFCQ we have gi(;)=0, ie/+(), On the other hand, by Proposition 14 and hj()=0,j= 1,...,p so that e f-) there exists an integer k such that for all k => k we have that (x, ,(x),/x(x)) is a K-T triple for problem (P) and we get a contradiction. The properties of exactness of W(x; e) are summarized in the following theorem. THEOREM 6. (a) The function W(x; e) is a weakly exact penalty function for problem (P) with respect to the set 9.


1354


(b) Assume that the EMFCQ is satisfied on 9. Then, the function W(x; e) is an exact penalty function for problem (P) with respect to the set 9; moreover, if Assumption (A2) holds, the function W(x; e) is a strongly exact penaltyfunctionforproblem (P) with respect to the set 9. Proof. Let g 9; we show first that conditions (al)-(a4) of Theorem I are satisfied. It is easily seen that (al) follows from the continuity of W(x; e) and the compact-

.

ness of With regard to condition (aE), let {ek} and {Xk}.C and assume that limk_ e k 0, limk_ Xk

be sequences such that e k

0,

lim sup W(Xk; ek) < 0. koo

By the continuity assumptions we get from (17) g() + Y(; 0)y(; 0) 0, which imply

and

_ __ h(;) =0,

f()_- 0, i=0, 1,..., m for

In particular, the next two propositions are the analogue of Propositions 13 and 14 and can be proved in a similar way. @, PROPOSrrloN 16. Let (, 12) be a K-T triple for problem (P), such that Then, for any e > O, we have:

,

(a) g(X)+ Y(ff; e)f(ff; e)=O; (b) Z(; e) =f(#); (c) vz(; e)=0.

x

.

17. Let en, there exist numbers e() > 0 and () > 0 is a stationary point of Z(x; e) satisfying for all e (0, e (2)], if x (2), we have that (x,, A (x), (x)) is a K-T triple for problem (P).

PROPOSlmOy

such that,

We now need the following lemma which is proved in [15]. LEMMA 3. Let {6)}, i= 1,..., r be r sequences of positive numbers. en, there exist an index i* and subsequences {)}, r corresponding to the same index 1, set K, such that:

lim=l 0 such that, for all e 6 (0, e*], if x is a stationary point of Z(x; e), we have that (x,A(x),tx(x)) is a K-T triple forproblem (P). Proof Reasoning by .contradiction, we assume that for any integer k, there exists an ek imply i6/+(), we can rewrite (35) into the following form" Since i6 J and i= P

iel+()

.

j=l

This implies, by the EMFCQ, that v 0 for e/+() and uj 0, for j 1,..., p. On the other hand, as l,. 1, we have either h()= 0 (if i*= 0) or g,.()+ .(; 0)= 0 (if i*e {1,..., m}). In both cases we get a contradiction to (32). Then we can conclude that Therefore, from (31) and (34), taking the limit of eVZ(xg; e) over the subsequence converging to we have

,

E

2

g’() + (" 0)

k(g’() + (" 0)): a :()

a()

V &()

+ [2 -hJ(;) Ilh()llh()]Vha()=O. a() =,

ao()

+


1358

(3.

DI PILLO AND L. (3RIPPO

Noting that, by definition of )Ti(Xk; ek), the inequality gi() < 0 implies gi() + fi/2(; 0) 0, we can write p

il+()

j=l

where now

Vi:=2

gi() + ,2.(;; O) (gi(:) +)7(.; 0)) 2 >--_0

ai()

+

a()

and

hj(:) uj:=2[l+llh()]] 2 ] ao(x’---- ao()" Therefore, again by the EMFCQ, we have v/--0, i L(), and /xj=0,j= 1,...,p which imply

gi() +)7/2(:; 0) 0,

h() 0,

1,

, rn

j=l,...,p,

:

ft. As e k 0, Proposition 17 implies that for sufficiently large values of k, so that the triple (Xk, A(Xk), tZ(Xk)) is a K-T triple for problem (P) and this yields a contradiction.

[3

We can now summarize the properties of exactness of Z(x; e) in the following theorem. THEOREM 7. (a) The function Z(x; e) is a globally weakly exact penalty function for problem (P) with respect to the set @. (b) Assume that the EMFCQ is satisfied on 9. Then, the function Z(x; e) is a globally exact penalty function for problem (P) with respect to the set 9; moreover, if Assumption (A2) holds, the function Z(x; e) is a globally strongly exact penalty function for problem (P) with respect to the set 9. Proof. By construction, we have limk_Z(Xk; e)= for any sequence {Xk}C such that Xk y 0@. Hence, by Definition 4 we have that Z(x; e) is globally (weakly, strongly) exact if it is (weakly, strongly) exact. assertion (a) can be proved along the same lines followed in the Letting proof of Theorem 6, making use of Propositions 16 and 17 in place of Propositions 13 and 14. we can proceed, as in the proof of Theorem With regard to (b), again letting 6, by employing Proposition 18 in place of Proposition 15 and making use of the inequality:

,

,

Z(x; e)

Exact Penalty Functions in Constrained Optimization

Exact Penalty Functions in Constrained Optimization

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