DOI 10.1007/s10957-010-9753-7
Semi-Infinite Optimization under Convex Function
Perturbations: Lipschitz Stability
N.Q. Huy· J.-C. Yao
Published online: 19 October 2010
© Springer Science+Business Media, LLC 2010
Abstract This paper is devoted to the study of the stability of the solution map for
the parametric convex semi-infinite optimization problem under convex function per-turbations in short, PCSI. We establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI under perturbations of both objective function and constraint set. The main result obtained is new even when the problems under consideration reduce to linear semi-infinite optimization. Examples are given to il-lustrate the obtained results.
Keywords Convex programming· Semi-infinite optimization · Solution map ·
Lipschitz stability· Slater constraint qualification
1 Introduction
We will denote by CO(Rn)the set of all the finite convex functions onRn. Let T be
nonempty compact subset of a metric space. C1(Rn× T ) be the set of all continuous function g: Rn× T → R, such that gt(·) := g(·, t) is convex for all t ∈ T .
Consider parametric convex semi-infinite optimization problem PCSI under func-tional perturbations of both objective function and constraint set on the parameter
This work was supported by the Grant NSC 99-2221-E-110-038-MY3 and was supported by the Vietnam’s National Foundation for Science and Technology Development (NAFOSTED), respectively.
N.Q. Huy
Department of Mathematics, Hanoi Pedagogical University No. 2, Xuan Hoa, Phuc Yen, Vinh Phuc Province, Vietnam
e-mail:[email protected]
J.-C. Yao (
)Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan e-mail:[email protected]
space
P := CO(Rn)× C1(Rn× T )
formulated as follows: For each the parameter p:= (f, g), we have the convex semi-infinite optimization problem
(CSI)p: min f (x) subject to x ∈ C(p), where
C(p) = {x ∈ | gt(x):= g(x, t) ≤ 0, ∀t ∈ T }
is the set of feasible points of (CSI)pand is a closed convex subset ofRn.
We aim at studying the recent development of parametric convex semi-infinite optimization. Namely, we will investigate the pseudo-Lipschitz property of the solu-tion map of PCSI. It is well known that semi-infinite optimizasolu-tion problems naturally arise in approximation theory, optimal control, and in numerous engineering prob-lems. Many papers are published every year on theory, methods and applications of semi-infinite programming and its extensions; see, e.g., [1–22] and the references therein for more comments and discussions. Especially, solution stability, such as lower semicontinuity, upper semicontinuity, pseudo-Lipschitz property, metric regu-larity, has attracted much attention of researchers in last years (see [1–10,12–17,21,
22]).
In parametric linear semi-infinite optimization, sufficient conditions for the lower and upper semi-continuity of the minimal value functions under perturbations of both objective function and constraint set have been given by B. Brosowski in [2]. M.J. Cánovas et al. [3] derived other characterizations of the same properties of the solution map and optimal value function together with the Lipschitz property of opti-mal value function. Furthermore, the pseudo-Lipschitz property of the solution map was investigated in [8] for parametric linear vector semi-infinite optimization prob-lems under linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraints. In the framework of parameter convex semi-infinite optimization problems, sufficient conditions for the metric regularity of the inverse of the solution map (or, equivalently, for the pseudo-Lipschitz property of so-lution map) under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function was presented in [5].
The main goal of this paper is to establish sufficient conditions for the pseudo-Lipschitz property of the solution map of PCSI
S(p) := {x∗∈ C(p) | f (x∗)≤ f (x), ∀x ∈ C(p)}
under convex functional perturbations of both objective function and constraint set. The main result obtained in the paper is new even when the problems under consider-ation reduce to linear semi-infinite optimizconsider-ation. Our result extends the corresponding result in [5] for convex semi-infinite optimization problems under canonical pertur-bations.
The paper is organized as follows. In Sect.2, we recall some basic definitions and preliminaries from convex analysis and set-valued analysis, and give some auxiliary
results, which will be useful in next section. In Sect.3, we present sufficient condi-tions for the pseudo-Lipschitz property of the solution map of PCSI under convex functional perturbations of both objective function and constraint set. Moreover, ex-amples are given to illustrate the obtained results. An application to parametric linear semi-infinite optimization problems is presented in Sect.4.
2 Preliminaries and Auxiliary Results
Let us first recall some standard notions from convex analysis and set-valued analysis; see, e.g., [19,23–25]. Let (X, d) be a metric space. Given M⊂ X, the closure and the interior of M are denoted by cl M and int M, respectively. We will use N (x) to denote the set of all neighborhoods of x∈ X. The distance from x ∈ X to M is defined by
d(x, M):= inf {dist(x, y) | y ∈ M},
where dist(x, y) denotes the distance between two points x and y, and d(x,∅) :=
+∞.
In particular, when X= Rn (n∈ N), we will denote by co M and cone M the convex hull and the convex conical hull of M, respectively. Write co∅ = ∅ and cone∅ = {0n}, where 0n is the zero vector of Rn. B stands for the open unit ball
inRn, andB(x, ρ) := x + ρB.
Let F : X ⇒ Y be a multifunction between metric spaces. The effective domain and the graph of the multifunction F are given, respectively, by
dom F:= {x ∈ X | F (x) = ∅}, gph F:= {(x, y) ∈ X × Y | y ∈ F (x)}.
Definition 2.1
(i) F is said to be closed at the point x0∈ X iff for all sequences {xi} in X and {yi}
in Y satisfying xi→ x0, yi→ y0, yi∈ F (xi)for all i∈ N, one has y0∈ F (x0). (ii) F is upper semicontinuous (usc for brevity) at x0∈ X iff for every open set V containing F (x0)there exists U0∈ N (x0)such that F (x)⊂ V for all x ∈ U0. (iii) F is said to be lower semicontinuous (lsc for brevity) at x0∈ dom F iff for any
open set V ⊂ Y satisfying V ∩ F (x0)= ∅ there exists U
0∈ N (x0)such that V∩ F (x) = ∅ for all x ∈ U0.
(iv) F is said to be continuous at x0∈ dom F iff it is both upper semicontinuous and lower semicontinuous at x0.
(v) F is pseudo-Lipschitz (also called Aubin continuous or Lipschitz-like) at (x0, y0)∈ gph F iff there exist U ∈ N (x0), V ∈ N (y0)and a constant > 0, such that
d(y2, F (x1))≤ d(x1, x2), for all x1, x2∈ U, and all y2∈ V ∩ F (x2).
Now we define, as in [20], a distance between convex functions in CO(Rn). Let
{Km}∞
m=1be a sequence of compact sets inR
n, such that Km⊂ int Km+1 and Rn= ∞ m=1 Km. (1)
In particular, we could have considered Km= m(cl B). Let f, g ∈ CO(Rn). First, for
each m= 1, 2, . . ., we define
δm(f, g):= sup x∈Km
{|f (x) − g(x)|}.
Then we define a metric σ on CO(Rn)by σ (f, g):= ∞ m=1 1 2m δm(f, g) 1+ δm(f, g) ∀f, g ∈ CO(R n), (2)
which describes the topology of the uniform convergence of convex functions on compact sets.
Lemma 2.1 [20, Proposition 3.1] Let f ∈ CO(Rn)and {fk}∞k=1⊂ CO(Rn). Then, σ (fk, f )→ 0 as k → ∞, if and only if fkconverges uniformly to f on any compact subset ofRn.
Let h: Rn→ R ∪ {+∞} be a proper, lower semicontinuous and convex function. Denote by ∂h(¯x) the subdifferential of f at ¯x ∈ dom f ,
∂h(¯x) := {u∗∈ Rn| h(x) − h( ¯x) ≥ u∗, x− ¯x, ∀x ∈ Rn}. We denote by N(¯x), the normal cone to ⊂ Rnat ¯x ∈ cl ,
N(¯x) := {u∗∈ Rn| u∗, x− ¯x ≤ 0, ∀x ∈ }.
The set of active constraints at x∈ C(p) is defined by Tp(x)= {t ∈ T | gt(x):= g(x, t) = 0}.
Lemma 2.2 The multifunctionT : P × Rn⇒ T , such that T (p, x) = Tp(x), is usc at every (p0, x0)∈ P × Rn.
Proof The proof is straightforward and so is omitted. Let p∈ P . We say, as in [12], that p satisfies the Slater constraint qualification iff there exists ˆx ∈ such that
Lemma 2.3 Let p= (f, g) ∈ P . Then, p satisfies the Slater constraint qualification,
if and only if 0n∈ co({∂gt/ (¯x) | t ∈ Tp(¯x)}) + N(¯x) for all ¯x ∈ C(p) with Tp(¯x) = ∅.
Proof Define the function h: Rn→ R by h(x) := maxt∈Tgt(x). Obviously, h is
convex onRnand
C(p) = {x ∈ | h(x) ≤ 0}.
Since h is a finite-valued convex function, it is continuous. Moreover, for each ¯x ∈ C(p) with Tp(¯x) = ∅, we have
h(¯x) := max
t∈T gt(¯x) = 0.
Then p satisfies the Slater constraint qualification, if and only if there is no¯x ∈ C(p) with Tp(¯x) = ∅ such that it is a minimizer of h on . This is equivalent to
0n∈ ∂h( ¯x) + N/ (¯x). (3)
It follows from [19, Theorem VI.4.4.2] that
∂h(¯x) = co({∂gt(¯x) | t ∈ Tp(¯x)}). (4)
Combining (3) and (4) we obtain 0n∈ co({∂gt/ (¯x) | t ∈ Tp(¯x)}) + N(¯x).
Next, we define the distance between functions in C1(Rn× T ), which is given by σ∞(g,¯g) := sup
t∈T
σ (gt,¯gt) for all g,¯g ∈ C1(Rn× T ), (5)
where gt(x):= g(x, t), ∀x ∈ Rn ∀t ∈ T . Then (σ∞,C1(Rn× T )) is a complete
metrizable space. It is well known that the convergence of a sequence{gk}∞ k=1⊂ C1(Rn× T ) to g ∈ C1(Rn× T ) describes the uniform convergence, on T , of the functions gk(x,·) to g(x, ·) for every x ∈ Rn.
From now on, · denotes the Euclidean norm in Rn. The distance between
two elements p= (f, g) and ¯p = ( ¯f ,¯g) belonging to the parameter space P :=
CO(Rn)× C1(Rn× T ) is formulated by
d∞(p, ¯p) := maxσ (f, ¯f ), σ∞(g,¯g), (6) where σ and σ∞are defined as in (2) and (5), respectively.
Lemma 2.4 Let p0= (f0, g0)∈ P . If p0satisfies the Slater constraint qualification, thenC : P ⇒ Rnis lower semicontinuous at p0.
Proof Let p0= (f0, g0)∈ P . Let V be an open convex set such that V ∩ C(p0)= ∅. Since p0 satisfies the Slater constraint qualification and T is compact, there must exist an element ˆx ∈ C(p0)and ρ > 0, such that
Take x∈ V ∩ C(p0)and choose a number r∈ ]0, 1] such that xr:= x + r( ˆx − x) ∈ V.
By the convexity ofC(p0)and V , xr∈ V ∩ C(p0)for all r∈ [0, r]. It follows from
(7) that
g0(xr, t )≤ (1 − r)g0(x, t )+ rg0(ˆx, t) ≤ −rρ ∀t ∈ T , ∀r∈ [0, r]. (8)
It remains to prove that there exists r1>0 such that, for every ¯p := ( ¯f ,¯g) ∈ P satis-fying d∞(¯p, p0) < r1ρ,
V ∩ C( ¯p) = ∅. (9)
Take an arbitrary ε > 0. Let
B(p0, ρε):= {p ∈ P | d
∞(p, p0) < ρε}. (10) Take any ¯p := ( ¯f ,¯g) ∈ B(p0, ρε). Let{Km} be a sequence of compact subsets of
Rndefined as in (1). Then there exists j∈ N such that xr∈ Kj for all r∈ [0, 1]. Let
t∈ T . Since the function τ → 1+ττ is increasing on[0, +∞[, it follows from (1) that
0≤ 1 2j δj(¯gt, gt0) 1+ δj(¯gt, g0t)≤ ∞ i=1 1 2i δi(¯gt, gt0) 1+ δi(¯gt, g0t)= σ ( ¯gt, g 0 t)≤ σ∞(¯g, g 0). Obviously, σ∞(¯g, g0)≤ d∞(¯p, p0). Hence, 0≤ 1 2j(1+ δj(¯gt, g0 t)) δj(¯gt, gt0) ρ ≤ d∞(¯p, p0) ρ < ε. This impliesδj(¯gt,g0t)
ρ → 0 as ε → 0 for all t ∈ T . Therefore, there must exist r1>0
and r0∈]0, r[ such that δj(¯gt, gt0)≤ r0ρ for all t∈ T whenever d∞(¯p, p0)≤ r1ρ. Hence,
δj(¯gt, gt0)≤ r0ρ < rρ ∀t ∈ T . (11)
Let xs := x + s( ˆx − x), s ∈ [r0, r]. Clearly, xs ∈ V ∩ C(p0). We have xs ∈ C( ¯p).
Indeed, it follows from (11) that, for every t∈ T ,
¯g(xs, t )− g0(xs, t )≤ sup x∈Kj | ¯g(x, t) − g0(x, t )| =: δj(¯gt, g0 t)≤ r0ρ. Then, by (8), ¯g(xs, t )≤ g0(xs, t )+ r0ρ≤ −(s − r0)ρ≤ 0 ∀t ∈ T .
This implies xs∈ C( ¯p) whenever d∞(¯p, p0)≤ r1ρ. ThusC is lower semicontinuous
Lemma 2.5 Let p0= (f0, g0)∈ P . Suppose that the following conditions hold: (i) p0satisfies the Slater constraint qualification;
(ii) S(p0)= {x0}.
ThenS is lower semicontinuous at p0.
Proof We first show that C is closed at p0. Let {pk = (fk, gk)}∞k=1 ⊂ P and
{xk}∞
k=1⊂ Rnbe sequences such that xk∈ C(pk), xk→ x
0and pk→ p0as k→ ∞. It is sufficient to show that x0∈ C(p0). From Lemma2.1and the compactness of T , it follows that, for any ε > 0, there exist a ρ > 0 and a positive integer k0such that
|gk(x, t )− g0(x0, t )| <ε
2 ∀x ∈ B(x
0, ρ),∀k ≥ k
0, ∀t ∈ T . This implies that
g0(x0, t ) <ε 2 + g k(xk, t )≤ε 2 ∀k ≥ k0,∀t ∈ T . Letting ε→ 0, we have g0(x0, t )≤ 0 ∀t ∈ T .
Since xk∈ C(pk)⊂ for all k implies x0∈ , we have x0∈ C(p0). We next prove that there is U (p0)∈ N (p0)such that
S(p) = ∅ for all p ∈ U(p0).
(12)
Indeed, if our claim were false then there would exist{pk= (fk, gk)}∞k=1∈ P con-verging to p0, such that
S(pk)= ∅ for all k ≥ 1. (13)
By the lower semicontinuity of C at p0, there would exist λ > 0, yk ∈ C(pk)∩
clB(x0, λ)such that yk→ x0 as k→ ∞. For each k = 1, 2, . . ., since S(pk)= ∅
it follows that there must exist zk∈ C(pk)\ B(x0, λ)such that
fk(zk)− fk(yk) <0 (14) (otherwise for all z∈ C(pk)\ B(x0, λ), fk(z)≥ fk(yk). This implies that fk has a minimizer onC(pk)∩ cl B(x0, λ)). Consider the following two possible cases:
(a) A subsequence of{zk} converges. We can assume that limk→∞zk= z0. Letting k→ ∞ in (14), we obtain from the closeness ofC at p0that z0∈ C(p0)and
f0(z0)− f0(x0)≤ 0. Hence, z0= x0, which contradicts z0∈ C(p0)\ B(x0, λ).
(b) limk→∞zk = +∞. Without loss of generality, we can assume that
lim
k→∞
zk
By the convexity of fk, we have fk 1 zkz k+ 1− 1 zk yk − fk(yk)≤ 1 zk(f k(zk)− fk(yk)).
It follows from (14) that
fk 1 zkz k+ 1− 1 zk yk − fk(yk) < 0. (15) Obviously, z1kz k+ (1 − 1 zk)y
k∈ C(pk)for all k≥ 1. Letting k → ∞ in (15), we
have
ˆz + x0∈ C(p0) and f0(ˆz + x0)− f0(x0)≤ 0.
Hence ˆz + x0= x0, which is impossible. Combining cases (a) and (b), we have proved (12).
It remains to prove thatS is lower semicontinuous at p0. Let{pk= (fk, gk)}∞k=1⊂ P be a sequence such that pk→ p0as k→ ∞. Then, by (12), there exist k0≥ 1 and xk∈ C(pk)such that
xk∈ S(pk) for all k≥ k0.
Since C is lower semicontinuous at p0, it follows that for each k= 1, 2, . . . there exists wk∈ C(pk)such that lim
k→∞wk= x0. Clearly,
fk(xk)− fk(wk)≤ 0 for all k ≥ k0. (16) Then{xk}∞k=1is bounded. Indeed, if limk→∞xk = +∞ then, we can assume that
lim
k→∞
xk
xk= ˆx, ˆx = 1.
It follows from the convexity of fkthat
fk 1 xkx k+ 1− 1 xk wk − fk(wk)≤ 1 xk(f k(xk)− fk(wk)).
Letting k→ ∞, we can assert from (16) that
ˆx + x0∈ C(p0)
and f0(ˆx + x0)− f0(x0)≤ 0. This implies that ˆx + x0= x0, a contradiction. Hence,{xk}∞k=1is bounded.
Without loss of generality, we can assume that limk→∞xk= ¯x ∈ C(p0). It follows
from (16) that f0(¯x) − f0(x0)≤ 0. Thus, ¯x = x0. The proof is complete. Final in this section, we recall a important result from [24] on the convergence of subdifferentials of convex functions.
Lemma 2.6 [24, Theorem 24.5] Let ϕ∈ CO[Rn] and {ϕk}∞k=1⊂ CO[Rn] such that ϕk converges pointwise to ϕ on an open convex set D⊂ Rn as k→ ∞. Let x ∈ D and{xk}∞k=1⊂ D converging to x. Then, for each ε > 0, there exists an index k0∈ N such that
∂ϕk(xk)⊂ ∂ϕ(x) + εB for all k ≥ k0.
3 Lipschitz Stability of the Solution Map
In this section we present sufficient conditions for the pseudo-Lipschitz property of the solution set mappingS of PCSI at the nominal parameter.
We first recall a optimality condition for PCSI at a given point.
Lemma 3.1 Let p0= (f0, g0)∈ P . Suppose that p0satisfies the Slater constraint qualification. Then x0is a solution of PCSI, if and only if
−∂f (x0)∩ cone t∈T (x0) ∂gt(x0) + N(x0) = ∅.
Proof It is immediate from Theorem 4.1 in [22]. From the above optimality condition we can establish the following result, which is useful in the sequel.|T | denotes the cardinality of T .
Proposition 3.1 Let p0= (f0, g0)∈ P and x0∈ S(p0). Suppose that the following conditions hold:
(i) p0satisfies the Slater constraint qualification; (ii) There is no T0⊂ Tp0(x0)with|T0| < n satisfying
−∂f0(x0)∩ cone t∈T0 ∂g0t(x0) + N(x0) = ∅.
Then, the following statements are valid:
(a) for any {(pk, xk)= (fk, gk, xk)}∞k=1⊂ gph S which converges to (p0, x0)= (f0, g0, x0)in gphS, there exist uk∈ ∂fk(xk), tik∈ Tpk(xk), uki ∈ ∂gk
tik(x k)and
λki >0 for i∈ {1, 2, . . . , n}, such that
−uk− n i=1
λkiuki ∈ N(xk) for k large enough
and{uk1, . . . , ukn} forms a basis of Rn; (b) S(p0)= {x0}.
Proof (a) It is easily seen that p satisfies the Slater constraint qualification in a some neighborhood of p0. Let{(pk, xk)= (fk, gk, xk)}∞k=1be a sequence of gphS such that{(pk, xk)} converges to (p0, x0)= (f0, g0, x0)∈ gph S. Since (pk, xk)→ (p0, x0)as k→ ∞, it follows that pksatisfies the Slater constraint qualification for k large enough. Applying Lemma3.1, we can assert from the Carathéodory’s theorem that, for k large enough, there exist q∈ N, uk∈ ∂fk(xk), tik∈ Tpk(xk), uki ∈ ∂gk
tik(x k)
and λki >0 for i∈ {1, 2, . . . , q}, such that q ≤ n,
−uk− q i=1 λkiuki ∈ N(xk), (17) and{uk
i | i = 1, . . . , q} is a linearly independent system.
It remains to prove that q= n. Conversely, suppose that q < n. By the compact-ness of T , we can assume, by taking a subsequence if necessary, that{tik} converges to some ti∈ T for each i ∈ {1, . . . , q}. Since tik∈ Tpk(tik), it follows from Lemma2.2 that ti∈ Tp0(x0).
We claim that, for each i∈ {1, . . . , q}, there exists λi≥ 0 such that lim
k→∞λ k
i = λi. (18)
Indeed, if our claim were false then, by taking a subsequence if necessary, we can assume that there exists i0∈ {1, . . . , q} such that
lim k→∞λ k i0= +∞. Put μk:=q i=1λ k
i, k≥ 1. Then limk→∞μk = +∞, and there is no loss of
gen-erality in assuming that the sequence{λ k i
μk}k≥k0 converges to some μi ≥ 0 for each
i∈ {1, . . . , q}. Dividing by μkin (17) and letting k→ ∞, we deduce from assump-tion (i), the robustness of the normal cone [10, p. 58], Lemma2.6and the compactness of the subgradient of a convex function that
− q i=1 μiui∈ N(x0) with q i=1 μi= 1 and ui∈ ∂g0ti(x
0). This means that 0
n∈ co({∂g0t(x0)| t ∈ Tp0(x0)})+N(x0), which
contradicts the assertion of Lemma2.3, and (18) follows. Letting k→ ∞ in (17), we get u∈ ∂f0(x0), ui∈ ∂g0ti(x 0) (i= 1, 2, . . . , q), −u − q i=1 μiui∈ N(x0) with{t1, . . . , tq} ⊂ Tp0(x0)and q < n, (19)
(b) Since x0∈ S(p0), it follows from the Carathéodory’s theorem and assump-tion (ii) that there exist u∈ ∂f0(x0), ti ∈ Tp0(x0), ui ∈ ∂gt0i(x
0) and λ
i >0 for
i∈ {1, . . . , n}, such that {u1, . . . , un} is a basis of Rnand
−u − n i=1 λiui∈ N(x0). (20) Take any y∈ S(p0). Then, for each i∈ {1, . . . , n}, gt0i(x0)= 0 and
ui, y− x0 ≤ g0t i(y)− g 0 ti(x 0)= g0 ti(y). Hence, ui, y− x0 ≤ 0. (21) Besides, −u − n i=1 λiui, y− x0 ≤ 0. Therefore, 0= f0(y)− f0(x0)≥ u, y − x0 ≥ − n i=1 λiui, y− x0 ≥ 0.
This implies that ni=1λiui, y− x0 = 0 and so, ui, y− x0 = 0 for every i ∈ {1, . . . , n}. Since {u1, . . . , un} is a basis of Rn, it follows that there exist the real numbers βi, i∈ {1, . . . , n}, such that y − x0=
n i=1βiui. Hence, (y − x0n)2= y − x0, y− x0 = n i=1 βiui, y− x0 = 0.
Thus y= x0. The proof is complete.
Note that in [6] the combination of both conditions (i) and (ii) of Proposition3.1
is referred to as extended Nürnberger condition; for more details and discussions we refer the reader to [5–7].
Let{Km}∞m=1be a sequence of compact sets ofRndefined as in (1). The following lemma is useful in the sequel.
Lemma 3.2 Let be a compact subset ofRn and let{xk}∞
k=1 be a sequence
be-longing to \ {0}. Let {fk}∞k=1⊂ CO(Rn)and f∈ CO(Rn). If limk→∞σ (f
k,f )
xk = 0 then, for each m∈ N such that ⊂ Km, one has limk→∞δm(f
k,f )
Proof Let m∈ N be sufficient large such that ⊂ Km. Since the function γ (r):= r 1+r is increasing onR+:= [0, +∞[, we have δm(fk, f ) 1+ δm(fk, f )≤ δm+k(fk, f ) 1+ δm+k(fk, f ) for every k≥ 1. Obviously, ∞ i=m+1 1 2i δm(fk, f ) 1+ δm(fk, f )≤ ∞ j=1 1 2j δj(fk, f ) 1+ δj(fk, f )=: σ (f k, f ). Hence 0≤ 1 2m δm(fk, f ) 1+ δm(fk, f )≤ σ (f k, f ) and so, 0≤ 1 2m(1+ δm(fk, f )) δm(fk, f ) xk ≤ σ (fk, f ) xk .
Combining this with Lemma2.1we obtain the aimed conclusion. We now state the main result.
Theorem 3.1 Let p0= (f0, g0)∈ P and x0∈ S(p0). Suppose that the following conditions hold:
(i) p0satisfies the Slater constraint qualification; (ii) There is no T0⊂ Tp0(x0)with|T0| < n satisfying
−∂f0(x0)∩ cone t∈T0 ∂g0t(x0) + N(x0) = ∅. ThenS is pseudo-Lipschitz at (p0, x0).
Proof Let p0= (f0, g0)∈ P and x0∈ S(p0). Suppose, in the contrary to our claim, that there exist a sequence {xk}∞k=1⊂ Rn converging to x0, the sequences {pk = (fk, gk)}∞k=1 and{ ¯pk= ( ¯fk,¯gk)}∞
k=1 in P , both converging to p
0, such that xk∈
S(pk)and
d(xk,S( ¯pk)) > kd∞(pk, ¯pk) for all k≥ 1. (22) SinceS is lower semicontinuous at p0by Lemma 2.5, it follows that there exists
¯xk ∈ S( ¯pk)satisfying ¯xk → x0 as k→ +∞. Hence, by (22), for each k≥ 1 and xk= ¯xk, we have max σ (fk, ¯fk) xk− ¯xk, σ∞(gk,¯gk) xk− ¯xk =d∞(pk, ¯pk) xk− ¯xk < 1 k. (23)
From Proposition3.1, it follows that, for k large enough, there exist uk∈ ∂fk(xk), ¯uk ∈ ∂ ¯fk(¯xk), tk i ∈ Tpk(xk), ¯tik ∈ T¯pk(¯xk), uki ∈ ∂gk tik(x k), ¯uk i ∈ ∂ ¯gk¯tk i (¯xk), λki >0, ¯λk
i >0 for all i= 1, 2, . . . , n, such that both {u k
1, . . . , u
k
n} and { ¯uk1, . . . ,¯u
k n} form bases ofRnand −uk− n i=1 λkiuki ∈ N(xk), − ¯uk− n i=1 ¯λk i¯u k i ∈ N(¯x k). (24)
By taking a subsequence if necessary, we can assume that, for each i∈ {1, . . . , n}, the sequences{tik}k≥k0 and{¯tik}k≥k0 converge to some ti and ¯ti, respectively. Then,
by Lemma2.2, we obtain
ti, ¯ti∈ Tp0(x0), i∈ {1, 2, . . . , n}.
As in the proof of Proposition3.1, we can assume that{λki}k≥k0 and{¯λ
k
i}k≥k0
con-verge to some λi and ¯λi, respectively. Letting k→ ∞ in (24), we deduce from
as-sumption (i), the robustness of the normal cone [23, p. 58], Lemma 2.6, and the compactness of the subgradient of any finite-valued convex function that, by taking a subsequence if necessary, limk→∞uk= u ∈ ∂f0(x0), limk→∞¯uk = ¯u ∈ ∂f0(x0),
limk→∞¯uki = ¯ui∈ ∂gt0i(x
0)with i∈ {1, 2, . . . , n}, and −u − n i=1 λiui∈ N(x0), − ¯u − n i=1 ¯λi¯ui∈ N(x0). (25)
From the Carathéodory’s theorem and assumption (ii), it follows that λi>0, ¯λi>0
for all i= 1, . . . , n, and both {u1, . . . , un} and { ¯u1, . . . ,¯un} must be basis of Rn. Let{Km}∞m=1be a sequence of compact subset ofRndefined as in (1). Then there must exist m∈ N such that
{ ¯xk− xk}∞
k=1,{xk}∞k=1,{ ¯xk}∞k=1⊂ Km.
On the hand, since tik∈ Tpk(xk)and ¯xk∈ S( ¯pk)⊂ C( ¯pk)for every i∈ {1, . . . , n}, it follows that gk tik(x k)= 0, ¯gk tik(¯x k)≤ 0. Hence, uk i,¯x k− xk ≤ g tk i(¯x k)− g tk i(x k)≤ g tk i(¯x k)− ¯g tk i(¯x k)≤ δm(gk tk i ,¯gk tk i ). Therefore, uki, ¯x k− xk xk− ¯xk ≤ δm(gtkk i ,¯gk tk i ) xk− ¯xk . (26) Furthermore, σ (gtk,¯gtk)≤ sup t∈T σ (gtk,¯gtk)=: σ∞(gk,¯gk)≤ d∞(pk, ¯pk) ∀t ∈ T .
This and (23) imply lim k→∞ σ (gk tik,¯g k tik) xk− ¯xk = 0 ∀i ∈ {1, 2, . . . , n}. (27)
By taking a subsequence if necessary, we can assume that{x¯xkk− ¯x−xkk
n}k≥k0 converges to some z∈ Rn withz = 1. Letting k → ∞ in (26), we obtain from (27) and Lemma3.2that
ui, z ≤ 0 ∀i ∈ {1, . . . , n}. (28)
Besides, the first inclusion in (24) implies
−uk− n i=1 λkiuki,¯xk− xk ≤ 0. (29)
Dividing both sides of (29) byxk− ¯xk and letting k → ∞, we can assert from (28) that −u, z ≤ n i=1 λiui, z ≤ 0. (30)
On the another hand, since ¯tik ∈ T¯pk(¯xk) and xk ∈ S(pk)⊂ C(pk) for every i∈
{1, . . . , n}, it follows that ¯gk ¯tk i (¯xk)= 0, gk¯tk i (xk)≤ 0. This implies that
¯uk i, xk− ¯xk ¯xk− xk ≤ δm(¯gtkk i , gk tik) ¯xk− xk .
In the same manner we see that, for every i∈ {1, . . . , n},
¯ui,−z ≤ 0, − ¯u, −z ≤ n i=1 ¯λi ¯ui,−z. Hence, ¯ui, z ≥ 0, − ¯u, z ≥ n i=1 λi ¯ui, z ≥ 0. (31)
By the convexity of fk and ¯fk, we have
uk,¯xk− xk ≤ fk(¯xk)− fk(xk) and
Therefore,
uk,¯xk− xk + ¯uk, xk− ¯xk ≤ fk(¯xk)− ¯fk(¯xk)+ ¯fk(xk)− fk(xk)
≤ 2δm(fk, ¯fk). (32)
Dividing both sides of (32) byxk− ¯xk and letting k → ∞, we can assert from (23) and Lemma3.2that
u, z + ¯u, −z ≤ 0,
and so,
− ¯u, z ≤ −u, z. (33)
Combining (28), (30), (31), and (33) we conclude that, for every i∈ {1, . . . , n},
¯ui, z = ui, z = 0.
Since both {uk1, . . . , ukn} and { ¯uk1, . . . ,¯ukn} are basis of Rn, it follows that z= 0, a
contradiction. The proof is complete.
The next example illustrates Theorem3.1.
Example 3.1 For PCSI, let T = [1, 2], P := CO(R) × C1(R × T ) and := R. Let p0= (f0, g0)∈ P be defined by f0(x)= x2+ 3x + 2∀x ∈ R, and
gt0(x)= −x − t ∀t ∈ T , x ∈ R.
ThenC(p0)= [−1, +∞[. Let x0= −1 ∈ S(p0). We now check the assumptions of Theorem 3.1. Clearly, ˆx = 3 is a Slater element for p0 and so, assumption (i) is fulfilled. Let us examine assumption (ii). It is easy to show that ∂f0(x0)= {1}, Tp0(x0)= {1} and N(x0)= {0}. If there exists T0⊂ Tp0(x0)such that|T0| < 1 then T0= ∅. Hence −∂f0(x0)∩ cone t∈T0 ∂gt0(x0) + N(x0) = ∅,
and assumption (ii) is fulfilled. Applying Theorem3.1we conclude thatS is pseudo-Lipschitz at (p0, x0).
The following examples show that the assertion of Theorem3.1may be false, if one of the assumptions (i) and (ii) is violate.
Example 3.2 For PCSI, let T = {1, 2, 3} ∪ [4, 5] ⊂ R, P := CO(R2)× C1(R2× T ) and := R2+. Let p0= (f0, g0)∈ P be formulated by
gt0(x)= ⎧ ⎪ ⎨ ⎪ ⎩ −x1, if t= 1, 2, −x2, if t= 3, x1+ x2− 1, if t ∈ [4, 5], ∀x = (x1, x2)∈ R2.
Let x0:= (0, 0) ∈ S(p0). We now check the assumptions of Theorem3.1. Clearly,
ˆx = (1 4,
1
4)is a Slater element for p
0. However, assumption (ii) is violate. Indeed, we have Tp0(x0)= {1, 2, 3}, T0= {2} ⊂ Tp0(x0)and|T0| = 1 < 2. It is easily seen that ∂f0(x0)= (0, 0), ∂g02(x0)= {(−1, 0)} and N(x0)= −R2+. It follows that cone(t∈T0∂gt0(x0))= −R+× {0} and −∂f0(x0)∩ cone t∈T0 ∂g0t(x0) + N(x0) = {0}.
We next examine the pseudo-Lipschitz property of S at (p0, x0). Let {pk = (f0, gk)}∞k=1⊂ P be a sequence such that
gtk(x)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 2kx2− x1, if t= 1, −x1, if t= 2, −x2, if t= 3, x1+ x2− 1, if t ∈ [4, 5].
We claim that pk= (f0, gk)→ p0= (f0, g0) as k→ ∞. Indeed, it is sufficient to show that gk → g0 as k→ ∞. Let {Km}∞m=1be a sequence of compact subset ofR2such that Km:= m(cl B), where B stands for the open unit ball of R2. Then R2=∞ m=1Km. Clearly, δm(gkt, gt0):= max x∈Km |gk t(x)− gt0(x)| = 1 2km, if t= 1, 0, if t∈ {2, 3} ∪ [4, 5]. Hence, σ (gkt, g0t)= 0 for t ∈ {2, 3} ∪ [4, 5], and for t = 1 we have
σ (g1k, g10):= ∞ m=1 1 2m σm(g1k, g01) 1+ σm(g1k, g10)≤ 1 2k ∞ m=1 1 2m= 1 2k.
Then σ∞(gk, g0):= maxt∈T σ (gkt, g0t)≤ 2k1 and so, gk→ g0as k→ ∞.
We see thatS is not pseudo-Lipschitz at (p0, x0). Indeed, taking ¯pk:= (f0, g0),
¯xk= (1, 0) ∈ S( ¯pk)and xk∈ S(pk)= {(0, 0)}, we have
1= d( ¯xk,S(pk)) >1 2≥ kd(p
k, ¯pk) ∀k ≥ 1.
Example 3.3 For PCSI, let T= [0, 1] ∪ {2}, P := CO(R) × C1(R × T ) and := R. Let p0= (f0, g0)∈ P be defined by f0(x)= x + 1, ∀x ∈ R, gt0(x)= −tx − t, if t ∈ [0, 1], 0, if t= 2, ∀x ∈ R.
Then,C(p0)= [−1, +∞[. Let x0= −1. We now check the assumptions of Theo-rem3.1. Clearly, p0does not satisfy the Slater constraint qualification by g00(x)= 0 for all x∈ R. Hence, assumption (i) is violate. Let us examine assumption (ii). It is easily seen that ∂f0(x0)= {1}, Tp0(x0)= [0, 1] ∪ {2} and N(x0)= {0}. If there
exists T0⊂ Tp0(x0)such that|T0| < 1 then T0= ∅. Hence,
−∂f0(x0)∩ cone t∈T0 ∂gt0(x0) + N(x0) = ∅,
and so, assumption (ii) is fulfilled.
We next check the pseudo-Lipschitz property ofS at (p0, x0). We haveS(p0)=
{−1}. Let pk= (f0, gk)∈ P , gtk(x)= −tx −k+1 k t+ 1 k, if t∈ [ 1 k+1,1], 0, if t∈ [0,k+11 [ ∪ {2}.
We claim that pk → p0 as k→ ∞. Indeed, it is sufficient to show that gk → g0 as k→ ∞. Let {Km}∞m=1be a sequence of compact subset of R such that Km:= m[−1, 1]. Then R =∞m=1Km. Clearly, δm(gtk, gt0):= max x∈Km |gk t(x)− g0t(x)| = ⎧ ⎪ ⎨ ⎪ ⎩ 0, if t∈ {0, 2}, t (m+ 1), if t ∈ ]0,k+11 [, −1 kt+ 1 k, if t∈ [ 1 k+1,1].
Hence, σ (gkt, g0t)= 0 for t ∈ {0, 2}. For t ∈ [k+11 ,1] we have
σ (gkt, g0t)= ∞ m=1 1 2m δm(gtk, g0t) 1+ δm(gtk, gt0)= −1 kt+ 1 k ∞ m=1 1 2m. This implies σ (gtk, gt0)= −1kt+ 1 k for all t∈ [ 1 k+1,1]. For t ∈ ]0, 1 k+1[ we have σ (gtk, gt0)= ∞ m=1 1 2m t (m+ 1) 1+ t(m + 1)= t ∞ m=1 1 2m (m+ 1) 1 t + (m + 1) ≤ t∞ m=1 1 2m= t. Therefore, σ∞(gk, g0):= max t∈T σ (g k t, g 0 t)≤ 1 k+ 1,
and so, gk→ g0as k→ ∞. This implies that pk→ p0 as k→ ∞. It is a simple matter to check thatS(pk)= {0} for every k ≥ 1. Thus, S is not pseudo-Lipschitz at (p0, x0).
We now consider a special case of PCSI which has the form
(CSI)(c,b): min f (x) + cTx subject to x∈ Rn, gt(x)≤ bt, t∈ T ,
where c∈ Rn, cT denotes the transpose of c, f: Rn→ R and gt: Rn→ R (t ∈ T ) are given convex functions such that (x, t)→ gt(x)is continuous on Rn× T , and b∈ C0(T )-the set of all continuous functions on T . The set of feasible points of (CSI)(c,b)is denoted by (c, b).S(c, b) stands for the set of all solutions of (CSI)(c,b).
The set of active constraints at x∈ (c, b) is given by T(c,b)(x):= {t ∈ T | gt(x)= bt}.
The following corollary is an immediate consequence of Theorem3.1by taking
˜
f (x)= f (x) + cTx, ˜g(x, t) = gt(x)− bt, and = Rn.
Corollary 3.1 [5, Theorem 10] For (CSI)(c,b), let c0∈ Rn, b0∈ C0(T ) and x0∈
S(c0, b0). Suppose that the following conditions hold: (i) (c0, b0)satisfies the Slater constraint qualification; (ii) There is no T0⊂ T(c0,b0)(x0)with|T0| < n satisfying
−c0+ ∂f0(x0)∩ cone t∈T0 ∂gt0(x0) = ∅. ThenS is pseudo-Lipschitz at ((c0, b0), x0).
4 Application to Linear Semi-Infinite Optimization Problems
In this section we establish sufficient conditions for pseudo-Lipschitz property of the solution mapping of parametric linear semi-infinite optimization problems at the nominal parameter.
Let T be a nonempty compact metric space and let C0(T ) and C0(T ,Rn)be,
respectively, the set of all continuous mappings
b: T → R and a : T → Rn normed by
b∞:= sup
t∈T
|b(t)| and a∞:= sup
t∈T a(t),
For every triple of parameters p= (c, a, b) ∈ P := Rn× C0(T ,Rn)× C0(T )we consider the linear semi-infinite problem
(LSI)p minc, x subject to x ∈ C(p),
whereC(p) := {x ∈ |a(t), x ≤ b(t) for every t ∈ T } and is a nonempty closed convex subset ofRn.
Theorem 4.1 Let p0= (c0, a0, b0)∈ P and x0∈ S(p0). Suppose that the following conditions hold:
(i) p0satisfies the Slater constraint qualification; (ii) There is no T0⊂ Tp0(x0)with|T0| < n satisfying
−c0∈ cone({a0(t )| t ∈ T0}) + N(x0). ThenS is pseudo-Lipschitz at (p0, x0).
Proof The assertion of the theorem follows by the same method as in the proof of Theorem3.1with d∞(·) replaced by · ∞as well noting thatvt− ¯vt, x ≤√nv −
¯v∞x for every v, ¯v ∈ C0(T ,Rn), x∈ Rnand for every t∈ T . Finally, in the case that = Rn and p just depends on the parameters c and b, i.e., the problems under consideration become the linear semi-infinite optimization problems under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. For this class of problems, the pseudo-Lipschitz property holds, if and only if both conditions (i) and (ii) of Theorem4.1
hold at the nominal parameter, as shown in [5, Theorem 16]. However, we do not known at present how to proceed with the case that p depends on the triple of para-meters c, a and b.
5 Concluding Remarks
We have established sufficient conditions for the pseudo-Lipschitz property of the solution map of parametric convex semi-infinite optimization problem (PCSI) under convex functional perturbations of both objective function and constraint set. More-over, examples are provided to illustrate the obtained results. An application to para-metric linear semi-infinite optimization problems is also given.
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