J Optim Theory Appl (2012) 154:108–119DOI 10.1007/s10957-012-9998-4

Calmness and Exact Penalization in Constrained ScalarSet-Valued Optimization

X.X. Huang

Received: 7 May 2011 / Accepted: 25 January 2012 / Published online: 16 February 2012© Springer Science+Business Media, LLC 2012

Abstract In this paper, we study a class of constrained scalar set-valued optimiza-tion problems, which includes scalar optimization problems with cone constraints asspecial cases. We introduce (local) calmness of order α for this class of constrainedscalar set-valued optimization problems. We show that the (local) calmness of order α

is equivalent to the existence of a (local) exact set-valued penalty map.

Keywords Constrained scalar set-valued optimization · Calmness · Exactpenalization

1 Introduction

It is well-known that penalty function methods are important and popular in con-strained optimization (see, e.g., [1–4]). The basic idea of these methods is to solvea constrained optimization problem by solving one or a sequence of unconstrainedoptimization problems.

In constrained single-valued scalar optimization, order one exact penalty functionwas usually used (see, e.g., [2, 4, 5]). It was shown in [6] that the existence of a (lo-cal) order one exact penalization is equivalent to the (local) calmness of the originalconstrained optimization problem. Lower order penalty functions were used in [7]to study mathematical programs with equilibrium constraints. It was demonstrated

Communicated by J. Zafarani.

This work is supported by the National Science Foundation of China and a research grant fromChongqing University. The author would like to thank two anonymous referees for their detailed andconstructive comments, which help improve the presentation of the paper.

X.X. Huang (�)School of Economics and Business Administration, Chongqing University, Chongqing 400030,Chinae-mail: huangxuexiang@cqu.edu.cn

J Optim Theory Appl (2012) 154:108–119 109

in [8] that (local) lower order exact penalization is equivalent to (local) calmnessof the original constrained optimization problem. These results further imply that(local) lower order exact penalization requires weaker condition than the usual or-der exact penalization. Subsequently, lower penalization technique was applied tomathematical programs with complementarity constraints [9] and nonlinear semidef-inite programs [10]. It is worth mentioning that the study of general penalization andlower penalization technique has been extended to constrained vector optimizationproblems (see, e.g., [11–14]). However, to the best of our knowledge, exact penal-ization for constrained set-valued optimization problems has not been investigatedin the literature although set-valued optimization has attracted increasing attention inthe optimization community (see, e.g., [15–23] and the references therein).

In this paper, we study a class of (local) exact penalty functions, which includeslower order (when the order is less than one) exact penalty functions as special casesfor a constrained scalar set-valued optimization problem. The main motivation for ourinvestigation of scalar set-valued optimization problems is that scalar optimizationis the simplest set-valued optimization problems (compared with vector set-valuedoptimization problems) and scalarization is an important technique for dealing withvector set-valued optimization problems (see, e.g., [17, 18, 22]).

The paper is organized as follows. In Sect. 2, we present the problems, definitions,notations. In Sect. 3, we derive equivalence between local calmness and local exactpenalization property. In Sect. 4, we obtain equivalence between calmness and globalex1act penalization property.

2 Definitions and Notations

In this section, we present the problems, definitions and notations, which will be usedin the sequel.

Let X and Z be normed spaces, X1 ⊂ X be a nonempty and closed set. Let F :X1 ⇒ R ∪ {+∞} be a strict (F(x) �= ∅,∀x ∈ X1), proper (there exist x0 ∈ X1 andt0 ∈ F(x0) such that t0 < +∞) and closed-valued (F(x) is a closed set for any x ∈X1) map. Let G : X1 ⇒ Z be a nonempty-compact-valued map (G(x) is a nonemptyand compact set for any x ∈ X1). Let D ⊂ Z be a closed and convex cone. Considerthe following constrained scalar set-valued optimization problem:

infF(x) s.t. x ∈ X1, G(x) ∩ −D �= ∅. (SSVO)

Denote by

X0 := {x ∈ X1 : G(x) ∩ −D �= ∅}

the feasible set of (SSVO).Throughout the paper, we make the following assumption.

Assumption A There exist x0 ∈ X0 and t0 ∈ F(x0) such that t0 < +∞.

Denote by v the optimal value of (SSVO). That is,

v := inf{t ∈ R1 : t ∈ F(x), x ∈ X0

}.

110 J Optim Theory Appl (2012) 154:108–119

Definition 2.1 (i) x ∈ X0 is said to be a local optimal solution of (SSVO) iff thereexist a neighbourhood U of x and t ∈ F(x) such that

t ≥ t , ∀t ∈ F(x), ∀x ∈ X0 ∩ U.

It is obvious that t = infF(x).(ii) x ∈ X0 is said to be an optimal solution of (SSVO) iff there exists t ∈ F(x)

such that

t ≥ t , ∀t ∈ F(x), ∀x ∈ X0.

Let z ∈ Z. Consider also the following perturbed problem of (SSVO):

infF(x) s.t. x ∈ X1, G(x) ∩ (−D + z) �= ∅. (SSVOz)

Denote by Xz and vz the feasible set and optimal value of (SSVOz), respectively.We assume that vz = +∞ if Xz = ∅. Clearly, v = v0.

Definition 2.2 Let x ∈ X0 be a local solution of (SSVO) with t = inft∈F(x) t . Letα > 0. (SSVO) is said to be locally calm at x of order α iff there exists M > 0such that for any {zn} ⊂ Z with zn → 0, any {xn} ⊂ Xzn satisfying xn → x and anytn ∈ F(xn), there holds

tn − t

‖zn‖α≥ −M

when n is sufficiently large.

Definition 2.3 Let α > 0. (SSVO) is said to be calm of order α iff either

(i) v = −∞;

or

(ii) when v > −∞, there exist M > 0 and a neighbourhood W of 0 ∈ Z such that

vz − v

‖z‖α≥ −M, ∀z ∈ W\{0}.

Remark 2.1 (i) When F is single-valued and finite, Z = Rm, G is single-valued and

D = {0p} × Rm−p+ , where 0p is the origin of R

p , the local calmness of order 1 at alocal solution x of (SSVO) reduces to the local calmness at a local solution definedin [5] and [2]; the calmness of (SSVO) reduces to the stability defined in [5] and thecalmness studied in [6].

(ii) It is routine to check that local calmness of order α > 0 at a local solution x of(SSVO) is equivalent to the fact that there exist a neighbourhoods W of 0 ∈ Z, U ofx ∈ X and M > 0 such that

t − t

‖z‖α≥ −M, ∀t ∈ F(x), ∀x ∈ Xz ∩ U, ∀z ∈ W\{0},

where t = infF(x0).

J Optim Theory Appl (2012) 154:108–119 111

(iii) The following example shows that, even if x ∈ X0 is a (global) optimal solu-tion of (SSVO) and (SSVO) is locally calm of order α > 0 at x, (SSVO) is not calmof order α > 0.

Example 2.1 Let X = Z = R, X1 = X and

F(x) = x2, if x ≤ 1, F (x) = 2 − x, if x ≥ 1,

G(x) = x/2, if x ≤ 1, G(x) = 1

2x, if x ≥ 1.

Consider the problem (SSVO). It is easily checked that X0 = ]−∞,0], x = 0 isan optimal solution to (SSVO), v = 0 and vz = −∞ if 0 < z < 1/2. Moreover,F(x)−F(x)

‖z‖α = x2

‖z‖α ≥ 0 for any x ∈ R with |x| sufficiently small and any z ∈ R with|z| > 0 sufficiently small. Namely, (SSVO) is locally calm of order α > 0 at x. How-ever, vz−v

‖z‖α = −∞ for 0 < z < 1/2. That is, (SSVO) is not calm of order α > 0.Let A1,A2 be two subsets of a metric space (W,d). The distance between A1 and

A2 is defined by

d(A1,A2) := inf{d(a1, a2) : a1 ∈ A1, a2 ∈ A2

}.

Let α > 0. Consider the penalty problem for (SSVO):

infx∈X1

F(x) + rdα(G(x),−D

), (PPα

r )

where r > 0 is the penalty parameter.Denote by vα(r) the optimal value of (PPα

r ). Clearly, vα(r) is nondecreasing in r .

3 Local Calmness and Local Exact Penalization

In this section, we present results concerning the relationship between the local calm-ness of order α > 0 (e.g., at a local solution x) of (SSVO) and local exact penalizationof (PPα

r ) (i.e., x is a local solution of (PPαr )).

The following theorem establishes the equivalence between local calmness of(SSVO) and local exact penalization of (PPα

r ).

Theorem 3.1 Let Assumption A hold. Let α > 0 and x ∈ X0 be a local solution to(SSVO) with t ∈ F(x) and t = infF(x). Assume that there exist r > 0, m0 ∈ R and aneighbourhood U of x such that

t + rdα(G(x),−D

) ≥ m0, ∀x ∈ X1 ∩ U, ∀t ∈ F(x). (1)

Then the following two statements are equivalent.

(i) (SSVO) is locally calm of order α at x;(ii) there exists r ′ > r such that x is a local solution to (PPα

r ) whenever r ≥ r ′.

112 J Optim Theory Appl (2012) 154:108–119

Proof (i) ⇒ (ii). Suppose to the contrary that (ii) does not hold. Then, there existr < rn ↑ +∞, {xn} ⊂ X1 with xn → x and tn ∈ F(xn) such that

tn + rndα(G(xn),−D

)< t. (2)

We assert that d(G(xn,−D) �= 0 when n is sufficiently large. Otherwise,d(G(xn),−D) = 0. By the nonemptiness and compactness of G(xn), we haveG(xn)∩−D �= ∅. That is, xn ∈ X0. This fact combined with (2) yields tn < t , contra-dicting the fact that x is a local optimal solution to (SSVO) and t = infF(x). From(2) and assumption (1), we deduce that

m0 + (rn − r)dα(G(xn),−D

)

≤ tn + rdα(G(xn),−D

) + (rn − r)dα(G(xn),−D

)< t.

Consequently,

d(G(xn),−D

) ≤[t − m0

rn − r

]1/α

.

Passing to the limit as n → +∞, we have

0 < sn = d(G(xn),−D

) → 0. (3)

By the definition of d(G(xn),−D), there exist zn ∈ G(xn) and dn ∈ −D such that

0 ← ‖zn − dn‖ ≤ 2sn.

Let z′n = zn − dn. Then

0 ← 2sn ≥ ‖z′n‖. (4)

Clearly, zn ∈ z′n − D. That is, zn ∈ G(xn) ∩ (z′

n − D), namely, zn ∈ Xz′n. It follows

from (2) that

tn − t

dα(G(xn),−D)< −rn

when n is sufficiently large. This together with (3) and (4) gives us

tn − t

‖z′n‖α

≤ tn − t

[2sn]α = tn − t

2αdα(G(xn),−D)< −rn/2α

when n is sufficiently large, contradicting the local order α calmness of (SSVO) at x.(ii) ⇒ (i). Suppose to the contrary that (SSVO) is not locally calm of order α at x.

Then, there exist 0 < Mn → +∞, 0 �= zn ∈ Z with zn → 0, xn ∈ Xzn and tn ∈ F(xn)

satisfying xn → x such that

tn − t

‖zn‖α< −Mn.

That is,

tn + Mn‖zn‖α < t. (5)

J Optim Theory Appl (2012) 154:108–119 113

From xn ∈ Xzn , we have G(xn) ∩ (zn − D) �= ∅. That is, zn ∈ G(xn) + D. It followsthat d(G(xn),−D) ≤ ‖zn‖ → 0. This combined with (5) yields

tn + Mndα(G(xn),−D

)< t. (6)

We assert that this implies that there exists no r ′ > 0 such that x is a local solution to(PPα

r ) when r ≥ r ′. Indeed, suppose to the contrary that there exists r ′ > 0 such thatx is a local solution to (PPα

r ′). Then, there exists a neighbourhood U ′ of x such that

t = t + r ′dα(G(x),−D

) ≤ t + r ′dα(G(x),−D

),

∀x ∈ X1 ∩ U ′, ∀t ∈ F(x). (7)

Note that xn → x. It follows from (7) that

t ≤ t ′n + r ′dα(G(xn),−D

), ∀t ′n ∈ F(xn)

when n is sufficiently large. This together with (6) yields

t ≤ tn + Mndα(G(xn),−D

)< t

when n is sufficiently large, which is impossible. The proof is complete. �

Remark 3.1 (i) It is easily seen that assumption (1) holds if there exist a neighbour-hood U ′ of x and m′

0 ∈ R such that t ≥ m′0,∀t ∈ F(x),∀x ∈ U ′ ∩ X1.

(ii) Consider a special case of (SSVO)—scalar single-valued optimization withequality and inequality constraints studied in [24]:

minf (x)

s.t. gi(x) = 0, i = 1, . . . , p, gi(x) ≥ 0, i = p + 1, . . . ,m, x ∈ X1, (P)

where X1 ⊂ X is nonempty and closed, X is a Banach space, f,gi : X1 → R arefunctions. Suppose that x be a local solution of (P). It is easily seen that (SSVO)reduces to problem (P) if we set F = f , G = −g and D = 0p × R

m−p+ . First we note

that all the results of [24] hold true if X (in [24]) is replaced by X1 ∩ B(x, ε) andKx is replaced by Kε

x . In particular, the local regularity condition (8) of [24] (withKx replaced by Kε

x ), which is equivalent to (11) of [24], is equivalent to the existence

of λ ∈ 0p × Rm−p+ such that L(x, λ) ≤ L(x, λ) ≤ L(x, λ),∀x ∈ X1 ∩ B(x, ε),∀λ ∈

0p × Rm−p+ . It follows that the local regularity condition (8) or (11) of [24] implies

f (x) ≤ f (x) − ⟨λ, g(x)

⟩

≤ f (x) + max1≤i≤p

{|λi |}(

m∑

i=1

∣∣gi(x)∣∣ +

m∑

i=p+1

(−gi(x))+

(x)

)

,

∀x ∈ X1 ∩ B(x, ε),

114 J Optim Theory Appl (2012) 154:108–119

where λ := (λ1, . . . , λm), (−gi(x))+ := max{−gi(x),0}. Namely, x is a local solu-tion of the penalty problem:

minx∈X1

f (x) + r

(m∑

i=1

∣∣gi(x)∣∣ +

m∑

i=p+1

(−gi(x))+

(x)

)

(PPr )

with r ≥ max1≤i≤p{|λi |}. In other words, the local regularity condition (8) or (11)of [24] implies the local exact penalization property of the penalty problem (PPr ). Itis obvious that the penalty problem (PPr ) is just problem (PP1

r ) with F = f , G = −g

and D = 0p ×Rm−p+ . Moreover, by Theorem 3.1 of [24], if f is continuous at x, then

the local regularity condition (8) or (11) of [24] implies calmness of problem (P) at x

(which is local calmness of (SSVO) at x with F = f , G = −g and D = 0p ×Rm−p+ ).

As mentioned in [24] (see also [25]), calmness of problem (P) at x is equivalent tocondition (14) in [24]. Recall that condition (14) in [24] is

Hu ∩ cl cone(

Kεx − clH

) = ∅, (8)

where nations T C, conv, cl, cone mean the same as those in [24] and

H = int R+ × {0p} × Rm−p+ ,

Hu = intR+ × {0m},Kε

x = {(f (x) − f (x), g(x)

) : x ∈ X1 ∩ B(x, ε)}.

We show that condition (8) is equivalent to the following condition:

Hu ∩ T C(conv

(Kε

x − clH)) = ∅, (9)

where

H = intR+ × {0m},Hu = H,

Kεx = {(

f (x) − f (x),∣∣g1(x)

∣∣, . . . ,∣∣gp(x)

∣∣,(−gp+1(x)

)+, . . . ,

(−gm(x))+) : x ∈ X1 ∩ B(x, ε)

}

corresponding to H, Hu, Kεx (for problem (P)), respectively, are for the problem:

minf (x)

s.t. |gi(x)| = 0, i = 1, . . . , p,(−gi(x)

)+ = 0,

i = p + 1, . . . ,m, x ∈ X1, (P)

which is equivalent to problem (P) in the sense that both of them have the same set ofoptimal solutions and the same optimal value. First, we show by contradiction that (8)implies (9). Suppose that (9) does not hold. Then, it is known from convex analysis

J Optim Theory Appl (2012) 154:108–119 115

(see, e.g., [26]) that there exist t > 0, tn > 0, sjn ≥ 0, j = 1, . . . ,m+ 2,

∑m+2j=1 s

jn = 1,

αjn ≥ 0, j = 1, . . . ,m + 2 and x

jn ∈ X1 ∩ B(x, ε), j = 1, . . . ,m + 2 such that

tn

m+2∑

j=1

sjn

(f (x) − f

(x

jn

) − αjn

) → t, (10)

tn

m+2∑

j=1

sjngi(xn) → 0, i = 1, . . . , p, (11)

tn

m+2∑

j=1

sjn

(−gi(xn))+ → 0, i = p + 1, . . . ,m. (12)

It is obvious that we can assume without loss of generality that sjn → sj ≥ 0, j =

1, . . . ,m + 2,∑m+2

j=1 sj = 1. Let βjn = f (x) − f (x

jn) − α

jn, j = 1, . . . ,m + 2. Then,

from (10), we have

tn

m+2∑

j=1

sjnβ

jn → t > 0.

Thus, we assume without loss of generality that

tnsj∗n β

j∗n ≥ 1

2(m + 2)

for some j∗ ∈ {1, . . . ,m+ 2} when n is sufficiently large. For n sufficiently large, wetake γn ∈ R+ such that

tnsj∗n β

j∗n − t

4(m + 2)≤ γn ≤ tns

j∗n β

j∗n − t

8(m + 2).

Then,

t

8(m + 2)≤ tns

j∗n β

j∗n − γn ≤ t

4(m + 2).

Thus, we can assume without loss of generality that tnsj∗n β

j∗n − γn → t ′ > 0. That is,

tnsj∗n (β

j∗n − γn/(tns

j∗n )) → t ′ > 0. Hence,

tnsj∗n

(f (x) − f

(x

j∗n

) − αj∗n − γn/

(tns

j∗n

)) → t ′ > 0. (13)

Moreover, from (11) and (12), we deduce

tnsj∗n gi

(x

j∗n

) → 0, i = 1, . . . , p, (14)

tnsj∗n

(−gi

(x

j∗n

))+ → 0, j = p + 1, . . . ,m. (15)

116 J Optim Theory Appl (2012) 154:108–119

Take τn = gi(xj∗n ) if gi(x

j∗n ) ≥ 0 and τn = 0 if gi(x

j∗n ) < 0. Then,

−tnsj∗n

(−gi

(x

j∗n

))+ ≤ tnsj∗n

(gi

(x

j∗n

) − τn

) ≤ 0.

This together with (15) yields

tnsj∗n

(gi

(x

j∗n

) − τn

) → 0. (16)

The combination of (13), (14) and (16) implies

Hu ∩ clcone(

Kεx − clH

) �= ∅,

contradicting (8). Next, we show by contradiction that (9) implies (8). Suppose that(8) does not hold. Then, there exist t > 0, tn > 0, αn ≥ 0, αi

n ≥ 0, i = p + 1, . . . ,m

and xn ∈ X1 ∩ B(x, ε) such that

tn(f (x) − f (xn) − αn

) → t, (17)

tngi(xn) → 0, (18)

tn(gi(xn) − αn

) → 0, i = p + 1, . . . ,m. (19)

From (18), we have

tn∣∣gi(xn)

∣∣ → 0. (20)

From (19), we deduce

tn(−g(xn)

) + tnαin → 0, i = p + 1, . . . ,m.

It follows that there exists a sequence 0 < εn → 0 such that

tn(−g(xn)

) + tnαin ≤ εn, i = p + 1, . . . ,m.

Thus,

tn(−g(xn)

) ≤ εn, i = p + 1, . . . ,m.

Consequently,

0 ≤ tn(−g(xn)

)+ ≤ εn, i = p + 1, . . . ,m.

Hence,

tn(−g(xn)

)+ → 0, i = p + 1, . . . ,m. (21)

The combination of (17), (20) and (21) yields Hu ∩ cl cone(Kεx − clH) �= ∅, contra-

dicting (9). Observe that (9) is just the local regularity (at x) of problem (P), which,by the observation we have made above, is equivalent to the existence of λ′ ∈ R

m

such that

L(x, λ) ≤ L(x, λ′) ≤ L

(x, λ′), ∀x ∈ X1 ∩ B(x, ε) ∩ X1, ∀λ ∈ R

m, (22)

J Optim Theory Appl (2012) 154:108–119 117

where

L(x,λ) = f (x) − ⟨λ, g(x)

⟩, ∀x ∈ X1, ∀λ ∈ R

m.

It is easily checked that (22) is equivalent to the fact that there exists r > 0 such that x

a local solution of (PPr ) whenever r ≥ r . Now we summarize the above observationsas follows:

(a) local regularity of (P) at x implies local exact penalization property of (PPr ) whenr > 0 is sufficiently large;

(b) calmness of (P) at x (called local calmness at x of (P) in this paper) is equivalentto local regularity of (P) at x, which is equivalent to local exact penalizationproperty of (PPr ) when r > 0 is sufficiently large;

(c) local regularity of (P) at x implies calmness of (P) at x if f is continuous at x.

Hence, the combination of (a), (b) and (c) concludes that, if f is continuous at x,calmness of (P) at x is equivalent to local regularity of (P) at x, which is, in turn,equivalent to local exact penalization property of (PPr ) when r > 0 is sufficientlylarge.

4 Calmness and Exact Penalization

In this section, we establish the equivalence between the calmness of order α of(SSVO) and exact penalization of (PPα

r ).

Theorem 4.1 Let Assumption A hold. Let α > 0. Assume that there exist r > 0, m0 ∈R such that

t + rdα(G(x),−D

) ≥ m0, ∀x ∈ X1, ∀t ∈ F(x). (23)

Then the following two statements are equivalent.(i) (SSVO) is calm of order α at x;(ii) there exists r ′ > r such that v = vα(r) whenever r ≥ r ′ ≥ r + 1.

Proof (i) ⇒ (ii). Note that Assumption A implies that vα(r) ≤ v,∀r > 0. Conse-quently, if v = −∞, then (ii) holds automatically. Now we assume that +∞ >

v > −∞ and (i) holds. Suppose to the contrary that there exist r + 1 < rn ↑ +∞,{xn} ⊂ X1 and tn ∈ F(xn) such that

tn + rndα(G(xn),−D

)< v. (24)

By (23), we have

m0 + (rn − r)dα(G(xn),−D

)

≤ tn + rdα(G(xn),−D

) + (rn − r)dα(G(xn),−D

)< v.

Consequently,

d(G(xn),−D

) ≤[v − m0

rn − r

]1/α

.

118 J Optim Theory Appl (2012) 154:108–119

Showing as in the proof of Theorem 3.1 (i) ⇒ (ii), we can prove(a) d(G(xn),−D) �= 0 when n is sufficiently large;(b) there exist zn ∈ G(xn) and dn ∈ −D such that 0 �= z′

n = zn − dn → 0, zn ∈G(xn) ∩ (z′

n − D) and

‖z′n‖ ≤ 2d

(G(xn),−D

). (25)

From (24) and (25), we deduce that

tn − v

‖zn‖α< −rn,

contradicting the order α calmness of (SSVO).(ii) ⇒ (i). Suppose to the contrary that there exist 0 < Mn → +∞, 0 �= zn ∈ Z

with zn → 0, xn ∈ X1 satisfying

G(xn) ∩ (zn − D) �= ∅ (26)

and tn ∈ F(xn) such that

tn − v

‖zn‖α< −Mn.

That is,

tn + Mn‖zn‖α < v. (27)

From (26), we have

d(G(xn),−D

) ≤ ‖zn‖. (28)

The combination of (27) and (28) yields

tn + Mndα(G(xn),−D

)< v. (29)

On the other hand, by the order α exact penalization assumption, there exists r ′ > 0such that

v = infx∈X1

F(x) + r ′dα(G(x),−D

). (30)

From (29) and (30), we have

v ≤ tn + r ′dα(G(xn),−D

)

≤ tn + Mndα(G(xn),−D

)

< v

which is impossible. The proof is complete. �

Remark 4.1 Assumption (23) is fulfilled if there exists m′0 ∈ R such that t ≥ m′

0,∀t ∈F(x),∀x ∈ X1.

J Optim Theory Appl (2012) 154:108–119 119

5 Conclusion

We established equivalent relations between (local) calmness of order α > 0 (at alocal solution) and (local) exact penalization property of (order α > 0) of penaltymaps for a constrained scalar set-valued optimization problem.

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