polar of the cone of feasible directions. They are derived through an extension of Valadier's formula for the subdifferential of a sup-function. Finally, a stability ...
Journal of Mathematical Sciences, Vol. 116, No. 4, 2003
A SUP-FUNCTION APPROACH TO LINEAR SEMI-INFINITE OPTIMIZATION M. A. Goberna, M. A. L´ opez, and M. I. Todorov
UDC 517.977
In this paper, we consider linear semi-infinite programming problems and the possibility of having no active constraint at a boundary point of the corresponding feasible set. This phenomenon causes serious troubles when one intends to construct the cone of feasible directions at this point. The main feature of our approach consists of exploiting the (sub)differential properties of the sup-function that allows replacing infinitely many constraints with a unique convex constraint. In the case where the set of active constraints is empty, this sup-function can present a quite abnormal behavior. To face this situation, two families of relaxed active constraint sets are introduced in the paper, and the relationship between them is studied in detail. Under the so-called strong Slater condition, we obtain two different formulas for the polar of the cone of feasible directions. They are derived through an extension of Valadier’s formula for the subdifferential of a sup-function. Finally, a stability result is given for the most relevant mapping in our context. 1. Introduction Consider the linear semi-infinite programming problem (LSIP) Inf c x s.t.
at x ≤ bt , t ∈ T,
(P )
where c, x, and at are vectors in the Euclidean space Rn , bt ∈ R, and c and at denote the corresponding transposed vectors. The system of constraints of problem (P ) is denoted by σ, i.e., σ = {at x ≤ bt , t ∈ T }. The solution set of σ (also called the feasible set of (P )) is denoted by F . We say that σ is consistent if F is nonempty. For the LSIP, some “strict” solutions have special relevance. Assume that there exist x0 ∈ Rn and ε > 0 such that at x0 ≤ bt − ε for all t ∈ T . In this case, we say that x0 is a strong Slater point (an SS element) of σ and σ satisfies the SS condition. Most numerical approaches to problem (P ) are based on the (partial) knowledge of the so-called cone of feasible directions at each iterate-point provided by the algorithm. In our convex context, this cone, at the point x ∈ F , is D(F ; x) := {d ∈ Rn | x + λd ∈ F for a certain λ > 0}. Obviously, D(F ; x) = Rn if x is an interior point of F . Thus, this cone needs to be investigated only at the boundary points of the solution set. We conjecture that the cone of feasible directions at a point x of the boundary of F must be related with the binding constraints at this point. In fact, if we consider the set of indices associated with the active (or binding) constraints at x T (x) := {t ∈ T | at x = bt }, we can prove that D(F, x) ⊂ A(x)o , where A(x) is the so-called active cone at x defined as the convex cone spanned by the set {at | t ∈ T (x)}, and A(x)o is the polar cone of A(x). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 95, Optimization and Related Topics–4, 2001. c 2003 Plenum Publishing Corporation 1072–3374/03/1164–3359 $ 25.00
3359
Moreover, in relation with problem (P ), the Karush–Kuhn–Tucker condition (KKT condition) holds at the point x if −c ∈ A(x). The KKT condition at x ∈ F implies the optimality of this point. To obtain the KKT condition as a necessary optimality condition, some constraint qualifications are needed. In [5], the following constraint qualification is considered: The constraint system of (P ) is said to be locally FM at x ∈ F if D(F, x)o = A(x) (see also [10]). In [3], this property is studied in the context of convex semi-infinite programming, and its relationship with extended versions of the Slater constraint qualification is analyzed in detail. Also, in [9], different constraint qualifications for the convex SIP are comparatively studied. In LSIP, the set of active constraints at a point of the boundary of the feasible set can be empty. In this case, the inclusion D(F, x) ⊂ A(x)o = Rn is trivial (it does not provide any local information), and the fulfillment of the local FM property as well as of the constraint qualifications considered in [9] is precluded. This paper deals with the possibility of having no active constraint at the boundary of the feasible set, and the main feature of our approach consists of exploiting the (sub)differential properties of the sup-function f (x) := sup{at x − bt | t ∈ T }. For this, two families of quasi-active constraint sets are introduced in Sec. 2. The sets in one of these families are always nonempty at any point of the boundary of the feasible set, and they play a crucial role in this theory. The relationship between these families is studied here and, in particular, it is shown in Sec. 3 that the members of both families are mutually embedded under certain assumptions relative to the boundedness for some of them and the fulfillment of the SS condition. Also, in Sec. 3, and again under the same assumptions, two different formulas for D(F, x)o are derived through an extension of Valadier’s formula for the subdifferential of the sup-function f (x). Finally, a stability result is given for the most relevant mapping in our context. Let us introduce the notation used in the paper. Given a nonempty set X in the Euclidean space p R , we denote the convex hull of X by conv(X), the conical convex hull of X by cone(X), and the polar cone of X by X o (i.e., X o = {y ∈ Rp | y x ≤ 0 for all x ∈ X}). R+ (X) represents the set {λx | for every pair λ ≥ 0 and x ∈ X}. It is assumed that cone ∅ = {0n }, where 0n denotes the null-vector in Rn . By dim(X), we denote the dimension of the affine hull of X. From the topological standpoint, int(X), cl(X), and bd(X) represent the interior, the closure, and the boundary of X, respectively. The Euclidean norm is represented by · , whereas B is the corresponding open unit ball. Finally, lim means lim . r
r→∞
(T ) R+ ,
By we denote the cone of all functions λ : T → R+ such that λt = 0 for all t ∈ T , except, maybe, for a finite subset of subscripts, denoted by supp λ. 2. Two Sets of Quasi-Active Constraints Since for a LSIP, the set T (x) can be empty, we consider the set of ρ-active constraints at x ∈ bd(F ) T (x; ρ) := {t ∈ T | at x ≥ bt − ρ}, with ρ > 0. Obviously, if x ∈ F , T (x) =
T (x; ρ).
ρ>0
Associated with the ρ-active constraint set at x, we introduce the ρ-active gradient set at x ∈ bd(F ) V (x; ρ) := {at | t ∈ T (x; ρ)} = {at | t ∈ T and at x ≥ bt − ρ}. In [7], different results are derived via some properties of the set W (x; γ) := {at | t ∈ T and at y = bt for a certain y ∈ x + γB}, where γ > 0 and x ∈ bd(F ). Actually, W (x; γ) is the set of gradients of constraints that are active at the points of a γ-ball around x. Some relevant properties of this set established in [7] are as follows: 3360
(i) If x ∈ bd(F ), then W (x; γ) = ∅ for all γ > 0. (ii) If x is an extreme point of F , then dim(W (x; γ)) = n for all γ > 0. The boundedness of W (x; γ) is also related with the properties of the sup-function f (x) := sup{at x − bt | t ∈ T }. f is a proper closed convex function and F = {x ∈ Rn | f (x) ≤ 0}. If f (x) < 0 and f is continuous at x, then a positive scalar δ can be found such that f (x) ≤ 0 if x ∈ x + δB, i.e., x ∈ int(F ). Thus, if f is continuous at x and x ∈ bd(F ), then f (x) = 0. Obviously, if f (x) < 0, then the point x is an SS element of σ. The following example shows that the SS condition itself is not sufficient to preclude the possibility of f (x) < 0 at a point x ∈ bd(F ). Example 1. In R, we consider the system σ = {tx ≤ 1, t ∈ ]0, ∞[}. Obviously, F = ]−∞, 0] and f (0) = −1. This abnormality is a consequence of dom(f ) = F , which implies that f is not continuous at x = 0. Next, we add new properties of W (x; γ) to this pair. Lemma 1. If σ = {at x ≤ bt , t ∈ T } is consistent and W (x; γ0 ) is bounded for a certain γ0 > 0, then inf{bt | t ∈ T and at ∈ W (x; γ0 )} > −∞. Proof. Otherwise, there would exist a sequence {tr }∞ r=1 ⊂ T such that atr ∈ W (x; γ0 ), r = 1, 2, . . . , and lim btr = −∞. Since W (x; γ0 ) is bounded, there exists a subsequence of {atr }∞ r=1 (for simplicity, we denote r
it by the same symbol) converging to a certain a0 . Taking any x0 ∈ F , we obtain a contradiction: (a0 ) x0 = lim atr x0 ≤ lim btr = −∞. r
r
The lemma is proved. Proposition 1. Given the consistent system σ = {at x ≤ bt , t ∈ T }, the sup-function f (x) := sup{at x − bt | t ∈ T }, and the point x ∈ bd(F ), we have x ∈ int(dom(f )) if and only if there exists a positive scalar γ0 such that W (x; γ0 ) is bounded. Proof. First, assume that x ∈ int(dom(f )); this entails the existence of δ > 0 such that x + δ cl(B) ⊂ int(dom(f )). Consider K0 := max{f (x) | x ∈ x + δ cl(B)}. This value K0 is finite since f is continuous on the compact set x + (δ/2) cl(B). Let us see that W (x; δ/2) is bounded. If at0 ∈ W (x; δ/2), then there exists y 0 ∈ x + (δ/2)B such that at0 y 0 = bt0 . If at0 = 0n , then there is nothing to prove. If, alternatively, at0 = 0n , then we take the point y0 +
δ at0 ∈ x + δ cl(B). 2 at0
Now we can write K0 ≥ f (y 0 +
δ at0 δ at0 δ ) ≥ at0 (y 0 + ) − bt0 = at0 2 at0 2 at0 2
and at0 ≤ (2K0 )/δ. In other words, W (x; δ/2) is bounded. Conversely, if W (x; γ0 ) is bounded, we have x + γ0 B ⊂ dom(f ) and, therefore, x ∈ int(dom(f )). According to Lemma 1, β0 := inf{bt | t ∈ T and at ∈ W (x; γ0 )} > −∞. We also use M0 := sup{at | at ∈ W (x; γ0 )}. Take z ∈ x + γ0 B. If at z ≤ bt for all t ∈ T , then, obviously, f (z) ≤ 0 and, therefore, z ∈ dom(f ). Alternatively, if z ∈ / F , then the subset of subscripts T> (z) := {t ∈ T | at z > bt } is nonempty. We take 3361
any t0 ∈ T> (z). Since at0 z > bt0 and simultaneously at0 x ≤ bt0 , the Bolzano theorem yields a point y ∈]z, x] ⊂ x + γ0 B such that at0 y = bt0 . Therefore, f (z) = sup{at z − bt | t ∈ T> (z)} ≤ M0 z − β0 , and, again, z ∈ dom(f ). Now we proceed by studying the relationship between the sets V (x; ρ) and W (x; γ). The following examples show that these sets are, in general, quite different. Example 2. In R, we consider the system σ = {tx ≤ 1, t ∈ ]−1, 1[}. Obviously, −x − 1, if x ≤ 0, f (x) = sup{tx − 1 | t ∈ ]−1, 1[} = x + 1, if x > 0. Thus, F = [−1, 1], and we can take x = 1 ∈ bd(F ). A straightforward calculation yields [1 − ρ, 1[ , if 0 < ρ < 2, V (1; ρ) = ]−1, 1[ , if ρ ≥ 2, and
1 ,1 , if 0 < γ ≤ 2, 1+γ W (1; γ) = 1 1 ∪ , 1 , if γ > 2. −1, 1−γ 1+γ
In this example, we observe that the sets W (1; γ) are even nonconnected and that none of them contains the origin. This excludes the possibility of being contained in any of the sets V (1; ρ), when ρ ≥ 1. The following example shows that V (x; ρ) can even be empty and that its boundedness at x does not imply x ∈ int(dom(f )), as was the case for W (x; γ) (see Proposition 1). Example 3. In R, we consider the system σ = {t2 x ≤ t, t ∈ [1, ∞[}. We have x − 1, if x ≤ 0, f (x) = sup{t2 x − t | t ∈ [1, ∞[} = +∞, if x > 0. Thus, F = ]−∞, 0], and we can take x = 0 ∈ bd(F ). An easy calculation yields ∅, if 0 < ρ < 1, V (1; ρ) = 2 [1, ρ ], if ρ ≥ 1. Obviously, f (0) = −1, and the boundedness of these sets is not sufficient to guarantee a regular behavior of the system at the considered point. The following proposition provides conditions giving rise to a richer relationship between the sets V (x; ρ) and W (x; γ). Proposition 2. Let us consider a consistent system σ = {at x ≤ bt , t ∈ T }, and a point x ∈ bd(F ). Then the following statements hold: (i) if M := sup{at | at ∈ W (x; γ)}, γ > 0, and M is finite, then W (x; γ) ⊂ V (x; γM ); (ii) if, in addition, the SS condition holds, then 0n ∈ / cl W (x; γ) for all γ > 0; 3362
(iii) if 0n ∈ / cl V (x; ρ), then V (x; ρ) ⊂ W (x; where m satisfies 0 < m < inf{at | at ∈ V (x; ρ)}.
ρ ), m
Proof. (i) Given at0 ∈ W (x; γ), let us consider the point y 0 ∈ x + γB such that at0 y 0 = bt0 . If y 0 = x + γu, where u ∈ B, we can write at0 x = at0 (y 0 − γu) = bt0 − γat0 u ≥ bt0 − γM, i.e., at0 ∈ V (x; γM ). (ii) If 0n ∈ cl W (x; γ0 ) for a certain γ0 > 0, then one can find a pair of sequences {tr }∞ r=1 ⊂ T and r ∞ r {y }r=1 ⊂ x + γ0 B such that atr y = btr , r = 1, 2, . . . , and lim atr = 0n . r
Without loss of generality, we can assume that lim y r = y 0 ∈ x + γ0 cl(B). Hence r
lim atr y r r
=
0n y 0
= 0 = lim btr . r
∈ and ε > 0 such that at x0 ≤ bt − ε for all t ∈ T . Since the SS condition holds, there exists Thus, lim atr x0 = 0 ≤ lim btr − ε = −ε; x0
r
Rn r
a contradiction. / cl V (x; ρ), it is obvious that (iii) Since 0n ∈ inf{at | at ∈ V (x; ρ)} > 0. If we take at0 ∈ V (x; ρ), then there exists an associated η ∈ [0, ρ] such that at0 x = bt0 − η. Now we can write
at0 = at0 x + η = bt0 , at0 x + η at0 2 whereas at0 ρ x+η 2 ∈ x + m B, at0 provided that 0 < m < inf{at | at ∈ V (x; ρ)}. In other words, we have already verified that at0 ∈ ρ ). W (x; m 3. Two Alternative Ways of Approaching the Cone of Feasible Directions In this section, we provide two formulas for D(F, x)o involving respectively the families of relaxed active constraint sets studied in the previous section. Proposition 3. Consider the system σ = {at x ≤ bt , t ∈ T } satisfying the SS condition and the point x ∈ bd(F ). Assuming the existence of a positive scalar γ0 such that W (x; γ0 ) is bounded, the following statements hold: (i) (3.1) cl (conv (V (x; ρ))) , D(F, x)o = R+ ρ>0
and dim(F ) = n; (ii) there exists a positive scalar ρ0 such that 0n ∈ / cl V (x; ρ0 ); (iii) there exists a real number ρ0 > 0 such that V (x; ρ0 ) is bounded; 3363
(iv)
D(F, x)o = R+ cl (conv (W (x; γ))) γ>0 = cone cl (W (x; γ)) γ>0 = cone cl (V (x; ρ)) . ρ>0
Proof. (i) According to Proposition 1, the existence of a positive scalar γ0 for which W (x; γ0 ) is bounded is equivalent to the inclusion x ∈ int(dom(f )). Thus, f is continuous at x, and x ∈ bd(F ) implies f (x) = 0. Then, the Valadier formula [12] yields cl (conv (V (x; ρ))) . ∂f (x) = ρ>0
Now, since the SS condition implies the existence of x0 such that f (x0 ) < 0, the function f does not attain its minimum at x. This fact allows us to apply Corollary 23.7.1 in [11] to obtain formula (3.1). Let us prove that F is full-dimensional. Since x ∈ int(dom(f )), the subdifferential ∂f (x) is compact / ∂f (x) since the SS condition holds. As a consequence of the last assertion, ([11, Theorem 23.4]), and 0n ∈ we observe that D(F, x)o = R+ (∂f (x)) is a pointed cone and, therefore, lineality(D(F, x)o ) = 0. Applying Corollary 14.6.1 in [11], we obtain dim(cone(F − x))) = dim(D(F, x)) = dim(cl(D(F, x))) = dim(D(F, x)oo ) = n − lineality(D(F, x)o ) = n, and then dim(F ) = n. (ii) Otherwise, we would have 0n ∈ cl (V (x; ρ)) ⊂ cl (conv (V (x; ρ))) for every ρ > 0, This leads us to the inclusion 0n ∈ ∂f (x), and this possibility is precluded by the SS condition. (iii) is a straightforward consequence of Proposition 2(iii). It is sufficient to select ρ0 small enough / cl V (x; ρ0 ) together with the boundedness of W (x; ρ0 /m0 ), for a certain m0 satisfying to guarantee 0n ∈ 0 < m0 < inf{at | at ∈ V (x; ρ0 )}. (iv) In order to prove the first equality in (iv), it is sufficient to show by applying (i) that the sets cl (conv (W (x; γ))) and B := cl (conv (V (x; ρ))) A := γ>0
ρ>0
{γr }∞ r=1
{ρr }∞ r=1
and are decreasing sequences of positive scalars coincide. It is also obvious that if converging to zero, we can write ∞ cl (conv (W (x; γr ))) = lim conv (W (x; γr )) A := r=1
and B :=
∞ r=1
3364
r
cl (conv (V (x; ρr ))) = lim conv (V (x; ρr )) , r
where lim denotes the limit for the Painlev´e–Kuratowski convergence. r The following bounds are also involved in our reasoning: Mr := sup{at | at ∈ W (x; γr )},
r = 1, 2, . . . ,
and mr satisfying
0 < mr < inf{at | at ∈ V (x; ρr )}, r = 1, 2, . . . . Take a ∈ A. Then a = lim ar , with ar ∈ conv (W (x; γr )), r = 1, 2, . . . . Statement (i) in Proposition 2 r
above ensures that ar ∈ conv (V (x; γr Mr )), r = 1, 2, . . . , and since lim γr Mr = 0 (Mr is finite from certain r
r0 on, and {Mr }∞ r=r0 is a decreasing sequence), we have a ∈ lim conv (V (x; γr Mr )) = r
=
∞
cl (conv (V (x; γr Mr )))
r=1
cl (conv (V (x; ρ))) = B.
ρ>0
We take b ∈ B. Thus, b = lim br , where br ∈ conv (V (x; ρr )), r = 1, 2, . . . . If r0 is large enough to r
guarantee that ρr ≤ ρ0 , r ≥ r0 , and ρ0 satisfies 0n ∈ / cl V (x; ρ0 ) (see (ii)), then Proposition 2(i) yields ρr )), r ≥ r , and since lim (ρ /m ) = 0, we obtain br ∈ conv(W (x; m 0 r r r r
∞ ρr ρr ) = cl conv W (x; ) b ∈ lim conv W (x; r≥r0 mr mr r=r0 = cl (conv (W (x; γ))) = A. γ>0
Next, we proceed by proving the second equality, which holds if we show that cl (conv (W (x; γ))) = conv cl (W (x; γ)) . γ>0
γ>0
For each γ0 > 0, we have cl (W (x; γ)) ⊂ cl (W (x; γ0 )) ⊂ cl (conv (W (x; γ0 ))) . γ>0
Since the set cl (conv (W (x; γ0 ))) is convex, one has conv cl (W (x; γ)) ⊂ cl (conv (W (x; γ0 ))) . γ>0
The last inclusion holds whenever we take γ0 > 0 and, therefore, cl (W (x; γ)) ⊂ cl (conv (W (x; γ))) . conv γ>0
γ>0
To prove the opposite inclusion, it is sufficient to establish the inclusion lim conv (W (x; γr )) ⊂ conv cl (W (x; γ)) r γ>0
for any decreasing sequence of positive scalars converging to zero {γr }∞ r=1 and such that γ1 is small enough to be sure that W (x; γ1 ) is bounded. 3365
Take a ∈ lim conv (W (x; γr )). Then a = limr ar , where ar ∈ conv (W (x; γr )), r = 1, 2, . . . . Applying r
the Carath´eodory theorem, one has, associated with each r, the following sets (some of whose elements might coincide): {ar,1 , ar,2 , ..., ar,n+1 } ⊂ W (x; γr ), {λr,1 , λr,2 , ..., λr,n+1 } ⊂ R+ , such that, for each r ∈ N, a=
n+1
λr,j ar,j and
j=1
n+1
λr,j = 1.
j=1
Since the sequences {ar,j , r = 1, 2, . . . }, j = 1, 2, . . . , n + 1, are contained in the compact set cl(W (x; γ1 )), we can apply a typical recurrent argument to find subsequences {akr ,j , r = 1, 2, . . . } converging to aj , j = 1, 2, . . . , n + 1. Owing to the fact that lim γr = 0, we have r
{aj , j = 1, 2, . . . , n + 1} ⊂ lim W (x; γr ) = r
cl (W (x; γ)) .
γ>0
The same procedure applies to the sequences {λr,j , r = 1, 2, . . . } ⊂ [0, 1]. Without loss of generality, n+1 λj = 1. Actually, we we can assume that lim λkr ,j = λj , j = 1, 2, . . . , n + 1, and we have, in addition, r
j=1
have obtained, by taking limits, a=
n+1
λj aj ∈ conv
cl (W (x; γ))
γ>0
j=1
.
Finally, the last equality in (iv) is proved by means of the same reasoning applied to the equality R+ cl (conv (V (x; ρ))) = cone cl (V (x; ρ)) , ρ>0
ρ>0
which arises as a by-product of previous outputs derived in the proof. Once again, the boundedness of V (x; ρ), for ρ small enough, plays its role in the proof. The formula D(F, x)o = cone
γ>0
cl (W (x; γ))
is also established in [7, Lemma 5.1, Proposition 5.2(iv)], under stronger assumptions on the coefficient vectors at . Also, the full-dimensionality of F concluded in (i) constitutes a stronger statement than Proposition 3.1(i) in that paper. It is also clear that (iv) above leads, as a by-product, to a new Valadier-type formula: cl (conv (W (x; γ))) . ∂f (x) = γ>0
In [1, 2, 4, 6], the stability of the LSIP problem has been investigated in the case where T is an arbitrary (infinite) set with no structure (therefore, the functions t → at and t → bt have no property at all). The approach followed in these papers is based on the use of an (extended) metric between the so-called nominal problem (P ) and the perturbed problem Inf(c1 ) x s.t. 3366
(a1t ) x ≤ b1t , t ∈ T.
(P1 )
This metric, providing the uniform convergence topology on T , is given by the formula 1
at 1 a t d(P1 , P ) = max c − c∞ , supt∈T − , b1t bt ∞
(3.2)
where ·∞ denotes the Chebyshev norm. The metric gives rise to the maximum (supremum) of the perturbations measured componentwise. The space of all the linear SIP problems with the same index set T becomes, by means of this extended metric, a metrizable space, which, in fact, locally behaves as a complete metric space. Since this section only deals with the stability of the solution set, it makes sense to consider, as parameter space, the set Θ of all the linear inequality systems, in Rn , indexed by T . Thus, we neglect the objective function since it has no influence on the feasible set. In this way, if σ = {at x ≤ bt , t ∈ T } and σ1 = {(a1t ) x ≤ b1t , t ∈ T }, the (restricted) distance between these two systems is given by 1
at at . − (3.3) d(σ1 , σ) = sup b1t bt ∞ t∈T (Θ, d) is a complete metrizable space (d (σ, σ1 ) := max{1, d(σ, σ1 )} is a real metric). The convergence in this space, σr → σ, describes the uniform convergence on T of the associated functions (ar(·) ) x − br(·) (considered as functions of t, and for every fixed x) to a(·) x − b(·) , where σr = {(art ) x ≤ brt , t ∈ T }. Next, we consider the subset Ξ := {(σ, x) ∈ Θc × Rn | x ∈ bd(F ), F = solution set of σ} in Θ×Rn , where Θc ⊂ Θ is the subset of all the consistent systems, and we define the mapping W : Ξ ⇒ Rn as follows: cl (conv (W (x; γ))) . W(σ, x) := γ>0
In Ξ, we consider the product topology induced from Θ × Rn , i.e., lim(σr , xr ) = (σ, x) if and only if r
lim d(σr , σ) = 0 and lim xr = x. r r Let Y and Z be topological spaces. We consider a set-valued mapping S : Y ⇒ Z. We say that S is upper semicontinuous in the Berge sense at y ∈ Y if for each open set W ⊂ Z such that S(y) ⊂ W , there exists an open set U ⊂ Y, containing y, such that S(y 1 ) ⊂ W for each y 1 ∈ U . The following result constitutes a stability result, connected with Theorem 2.2 in [8]. Theorem 1. Let us consider a consistent system σ = {at x ≤ bt , t ∈ T } satisfying the SS condition and a point x ∈ bd(F ). Assume the existence of a positive scalar γ0 such that W (x; γ0 ) is bounded. Then the mapping W is upper semicontinuous at (σ, x). Proof. Under these assumptions, and according to the preceding section, we have W(σ, x) = ∂f (x), where f (x) := sup{at x − bt | t ∈ T }. Moreover, this set is bounded. If W fails to be upper semicontinuous at (σ, x), there exists an open set Wo in Rn containing W(σ, x) r and a sequence {(σr , xr )}∞ r=1 in Ξ converging to (σ, x) such that W(σr , x ) Wo for every r. The compactness of W(σ, x) implies the existence of positive scalars ε and δ such that W(σ, x) + εB ⊂Wo and x + δB ⊂ dom(f ). If we consider the sup-functions associated with the systems σr = {(art ) x ≤ brt , t ∈ T }, r = 1, 2, . . . , i.e., fr (x) := sup{(art ) x − brt | t ∈ T }, then we can prove that {fr }∞ r=1 is a sequence of convex functions finite on the open set x + δB. In fact, with x fixed, (art ) x − brt = (at x − bt ) + (art − at ) x + (bt − brt ) ≤ (at x − bt ) + art − at x + |bt − brt | , for all t ∈ T. 3367
Hence fr (x) ≤ f (x) + d(σr , σ)(1 + x), and, assuming that d(σr , σ) is finite, one concludes that fr is finite provided that f is finite. (Recall that lim d(σr , σ) = 0.) r Following the same bounding argument, we obtain the inequalities f (x) ≤ fr (x) + d(σr , σ)(1 + x) and |fr (x) − f (x)| ≤ d(σr , σ)(1 + x). converges pointwise to f on the open convex set x + δB. Thus, the sequence {fr }∞ r=1 According to Theorem 24.5 in [11], there exists r0 such that ∂fr (xr ) ⊂ ∂f (x) + εB ≡W(σ, x) + εB ⊂Wo , for all r ≥ r0 . Theorem 6.1 in [5] states that the SS constraint qualification is equivalent to σ ∈ int(Θc ). Thus, if r0 is chosen large enough, σr also satisfies the SS condition for every r ≥ r0 . Therefore, ∂fr (xr )≡W(σr , xr )⊂Wo , for all r ≥ r0 ; a contradiction. REFERENCES 1. M. J. C´ anovas, M. A. L´ opez, J. Parra, and M. I. Todorov, “Stability and well-posedness in linear semi-infinite programming,” SIAM J. Optimiz., 10, 82–98 (1999). 2. M. J. C´ anovas, M. A. L´ opez, J. Parra, and M. I. Todorov, “Solving strategies and well-posedness in linear semi-infinite programming,” Ann. Oper. Res., 101, 171–190 (2001). 3. M. D. Fajardo and M. A. L´ opez, “Locally Farkas-Minkowski systems in convex semi-infinite programming,” J. Optimization Theory Appl., 103, 313–335 (1999). 4. M. A. Goberna and M. A. Lopez, “Topological stability of linear semi-infinite inequality systems,” J. Optimization Theory Appl., 89, 227–236 (1996). 5. M. A. Goberna and M. A. L´ opez, Linear Semi-Infinite Optimization, Wiley (1998). 6. M. A. Goberna, M. A. L´ opez, and M. Todorov, “Stability theory for linear inequality systems,” SIAM J. Matrix Anal. Appl., 17, 730–743 (1996). 7. M. A. Goberna, M. A. L´ opez, and M. Todorov, “Extended active constraints in linear optimization with applications,” Tech. report, Alicante University (2001). 8. S. Helbig and M. I. Todorov, “Unicity results for general linear semi-infinite optimization problems using a new concept of active constraints,” Appl. Math. Optimiz., 38, 21–43 (1998). 9. W. Li, C. Nahak, and I. Singer, “Constraint qualifications for semi-infinite systems of convex inequalities,” SIAM J. Optimiz., 11, 31–52 (2000). 10. R. Puente and V. Vera de Serio, “Locally Farkas-Minkowski linear semi-infinite systems,” Top, 7, 103–121 (1999). 11. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey (1970). 12. M. M. Valadier, “Sous-diff´erentiels d’une borne sup´erieure et d’une somme continue de fonctions convexes,” Compt. Rend. Acad. Sci., S´er. A, 268, 39–42 (1969).
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