CUBO, A Mathematical Journal
Vol.22, No¯ 01, (137–154). April 2020

http: // doi. org/ 10. 4067/ S0719-06462020000100137

Level sets regularization with application to optimization
problems

Moussa Barro and Sado Traore

Université Nazi BONI,

Burkina Faso.

mousbarro@yahoo.fr, traore.sado@yahoo.fr

ABSTRACT

Given a coupling function c and a non empty subset of R, we define a closure operator.

We are interested in extended real-valued functions whose sub-level sets are closed

for this operator. Since this class of functions is closed under pointwise suprema, we

introduce a regularization for extended real-valued functions. By decomposition of the

closure operator using polarity scheme, we recover the regularization by bi-conjugation.

We apply our results to derive a strong duality for a minimization problem.

RESUMEN

Dada una función de acoplamiento c y un subconjunto no vaćıo de R, definimos un

operador clausura. Estamos interesados en funciones extendidas a valores reales cuyos

conjuntos de sub-nivel son cerrados para este operador. Dado que esta clase de funciones

es cerrada bajo supremos puntuales, introducimos una regularización para funciones

extendidas a valores reales. Gracias a la descomposición del operador clausura usando

el esquema de polaridad, recuperamos la regularización por bi-conjugación. Aplicamos

nuestros resultados para derivar una dualidad fuerte para un problema de minimización.

Keywords and Phrases: Duality, regularization, level sets, c-elementary functions, polarity,

conjugacy.

2010 AMS Mathematics Subject Classification: 49N15.

http://doi.org/10.4067/S0719-06462020000100137


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1 Introduction

Regularization and conjugation of extended real-valued functions play an important role in opti-

mization theory since it is a base of duality theory. Until the Fenchel’s work([4]), many authors

have introduced and studied different kinds of regularization and conjugation among which we can

cite Moreau ([8]), Crouzeix ([1]), Rockafellar ([13]), Mart́ınez-Legaz ([7]), Singer ([15]), Penot-Volle

([11]), Volle ([16, 17]). In [16], M. Volle used a dual pair of polarities to introduce and study level

set regularization and conjugacy.

In this work, we introduce and study level set regularization and conjugacy by means of a

coupling function and a nonvoid subset of the real numbers. The outline of the paper is as follows.

In Section 2, we recall Moreau conjugation scheme. Section 3 is devoted to the study of the

Γ−regularization of extended real-valued functions and hull of sets. We introduce these notions

and give some properties (Proposition 2, 4 and Theorem 3.8). In Section 4, we introduce the

level set regularization of extended real-valued functions. By decomposition of a closure operator

via a couple of dual polarities, we show that this regularization coincides with the bi-conjugation

relative to the polarity couple (Proposition 8 and Theorem 4.6). We derive an analytic expression

of level set regularization of extended real-valued functions (Proposition 10). Section 5 is devoted

to an application of our theory to a minimization problem. A perturbational dual of this problem

is defined and a necessary and sufficient condition is given to ensure a strong duality property for

this problem (Theorem 5.1, Corollary 5.2 and Corollary 5.3).

2 Preliminaries

Let us start this section by recalling the Moreau conjugation ([8]). Let U, V two nonvoid sets

and c : U × V → R, a coupling function. Given an extended real-valued function h : U → R :=

R ∪ {−∞, +∞}, we define the c−conjugate of h by hc(v) := sup
u∈U

{c(u, v) − h(u)}, for any v ∈ V .

By exchanging the role of U and V , we define the c−conjugate of a given function k : V → R by

kc(u) := sup
v∈V

{c(u, v) − k(v)} for any u ∈ U. The c−bi-conjugate of a given h : U → R is then

defined by hcc(u) := sup
v∈V

{c(u, v) − hc(v)} for any u ∈ U.

Example 2.1. The usual Legendre-Fenchel conjugacy is obtained by taking U := X, a topological

vector space with topological dual V := X∗ and c the standard bilinear coupling function.

Example 2.2. Other examples of coupling functions have been considered in the literature among

which, we can cite:

(1) U = V = {x ∈ Rn|x1 > 0, . . . , xn > 0} and c(u, v) = min
1≤i≤n

viui, ([14]),



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Level sets regularization with application to optimization problems 139

(2) (U, d) a metric space, α > 0, V = U and c(u, v) = −αd(u, v), ([6]),

(3) (U, d) a metric space, V = U×]0, +∞[ and c(u, (v, α)) = −αd(u, v), ([6]),

(4) U = V = Rn, 0 < α ≤ 1, β > 0 and c(u, v) = −β‖u − v‖α, ([7]),

(5) U a topological space, V = C (U,R), space of countinuous real functions on U and c(u, v) =

v(u), ([5],[10]).

Given a function h : U → R, the following notation and definitions will be needed: dom h =

{u ∈ U | h(u) < +∞}, the effective domain of h, [h ≤ t] := {u ∈ U | h(u) ≤ t}, the t-sub-level set

of h(or level set of h in short).

Given a subset A of U, we define its indicator function iA by iA(u) = 0 if u ∈ A and iA(u) =

+∞ if u ∈ U \ A. Following the terminology introduced in [9] we will also use the valley function

vA of A defined by vA(u) = −∞ if u ∈ A and vA(u) = +∞ if u ∈ U \ A.

3 Γ−regularization of functions and hull of sets

3.1 Γ−regularization of functions

The notion of continuous affine functions can be generalized by those of c−elementary functions.

In this work, we call c−elementary function on U (resp. on V ), the function of the form c(., v) − r

(resp. c(u, .) − r) with v ∈ V (resp. u ∈ U) and r ∈ R. The upper hull (i.e., the supremum) of

a family of c−elementary functions is called c-regular. We denote by Γc(U), the set of c−regular

functions defined on U. We call c-hull or Γc−regularization of h : U → R, the greatest c−regular

minorant of h. This function is denoted by hΓc. It is well known ([8]) that

hcc = hΓc, for each h : U → R. (3.1)

Remark 3.1. The equality (3.1) is still valid if the coupling function is an extended real-valued

function. In this case, one must interpret the conjugate hc as follows

hc(v) = − inf
u∈U

{h(u) − c(u, v)},

with the usual conventions (+∞) − (+∞) = (−∞) − (−∞) = +∞.

There exists an equivalent approach to generalized convex duality in terms of Φ -convexity

[2], which consists of working with a set U and a class of functions Φ ⊂ R
U
.



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3.2 Hull of sets

Let P be a nonvoid subset of R. The following definition generalizes the notion of half space.

Definition 1. We call (c,P)−elementary subset of U any subset of Uof the form {u ∈ U | r −

c(u, v) ∈ P}, where (v, r) ∈ V × R. We note it by EPv,r.

Note that, if P = R, then EPv,r = U for any (v, r) ∈ V × R. In this case, the only

(c,P)−elementary subset of U is U itself. The (c,P)−elementary subsets of U allow us to de-

fine a notion of hull of a subset A of U.

Definition 2. The (c,P)−hull of A ⊂ U is the intersection of all (c,P)−elementary subsets of U

containing A. The (c,P)−hull of A is denoted by 〈A〉c,P.

Remark 3.2. If there is not (c,P)−elementary subset of U containing A, then 〈A〉c,P = U by

convention.

Proposition 1. If P 6= R, then 〈∅〉c,P = ∅.

Proof Let s ∈ R \ P. Assume 〈∅〉c,P 6= ∅. Let a ∈ 〈∅〉c,P. Then r − c(a, v) ∈ P for any

(v, r) ∈ V × R. In particular s = (s + c(a, v)) − c(a, v) ∈ P, absurd.

It follows from the definition of 〈.〉c,P that, for each A ⊂ U, and for each u ∈ U, one has

u /∈ 〈A〉c,P ⇐⇒ ∃(v, r) ∈ V × R : A ⊂ E
P

v,r and r − c(u, v) /∈ P. (3.2)

By definition 1, one has A ⊂ 〈A〉c,P , for any A ⊂ U. Moreover, if A ⊂ B then 〈A〉c,P ⊂ 〈B〉c,P.

Therefore, 〈〈A〉c,P〉c,P = 〈A〉c,P, ∀A ⊂ U. We deduce that 〈.〉c,P is an algebraic closure operator.

Definition 3. A subset A of U is said to be (c,P)−regular if A = 〈A〉c,P. We denote Rc,P(U),

the set of all (c,P)-regular subsets of U.

Observe that (c,P)−elementary sets are (c,P)−regular. More generally, any intersection of

(c,P)−regular subsets is (c,P)−regular and the (c,P)−regular hull of A ⊂ U coincides with the

intersection of all (c,P)−regular subsets of U containing A.

In what follows, we will use the following values for P : P1 = R+ := [0, +∞[, P2 = R
∗
+ :=

]0, +∞[, P3 = R
∗ := R \ {0} and P4 = {0}. For i = 1, 2, 3, 4, 〈.〉c,i := 〈.〉c,Pi for short. For

i = 1, 2, 3, the set 〈A〉c,i can be explained as follows:

Proposition 2. For any A ⊂ U, one has:

〈A〉c,1 = {u ∈ U | c(u, v) ≤ sup
a∈A

c(a, v), ∀v ∈ V }, (3.3)

〈A〉c,2 = {u ∈ U | ∀v ∈ V, ∃a ∈ A | c(u, v) ≤ c(a, v)}, (3.4)

〈A〉c,3 = {u ∈ U | ∀v ∈ V, ∃a ∈ A | c(u, v) = c(a, v)}. (3.5)



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Level sets regularization with application to optimization problems 141

Proof By (3.2), one has:

a /∈ 〈A〉c,1 ⇐⇒ ∃(v, r) ∈ V × R : A ⊂ [c(., v) ≤ r] and r < c(a, v)

⇐⇒ ∃(v, r) ∈ V × R : sup
u∈A

c(u, v) ≤ r < c(a, v)

⇐⇒ ∃v ∈ V : sup
u∈A

c(u, v) < c(a, v).

Thus, a ∈ 〈A〉c,1 ⇐⇒ ∀v ∈ V, c(a, v) ≤ sup
u∈A

c(u, v), and (3.3) holds.

a /∈ 〈A〉c,2 ⇐⇒ ∃(v, r) ∈ V × R : A ⊂ [c(., v) < r] and r ≤ c(a, v)

⇐⇒ ∃(v, r) ∈ V × R : c(u, v) < r ≤ c(a, v), ∀u ∈ A

⇐⇒ ∃v ∈ V : c(u, v) < c(a, v), ∀u ∈ A.

Thus, a ∈ 〈A〉c,2 ⇐⇒ ∀v ∈ V, ∃u ∈ A : c(a, v) ≤ c(u, v), and (3.4) holds.

a /∈ 〈A〉c,3 ⇐⇒ ∃(v, r) ∈ V × R : A ⊂ [c(., v) 6= r] and r = c(a, v)

⇐⇒ ∃v ∈ V : c(u, v) 6= c(a, v), ∀u ∈ A.

Thus a ∈ 〈A〉c,3 ⇐⇒ ∀v ∈ V, ∃u ∈ A : c(a, v) = c(u, v), and (3.5) holds.

Remark 3.3. Observe that one cannot remove the real parameter r in the definition of 〈A〉c,4.

Example 3.4. We observe the situation in topological vector case. Assume U is a topological vector

space with topological dual V and c the standard coupling function. The c−elementary functions

are affine continuous functions, and we have:

1. (c,P1)−elementary sets are ∅, U and closed half spaces. Moreover, if U is locally convex,

then by Hahn-Banach separation theorem and (3.3), 〈A〉c,1 = convA, the closed convex hull

of A.

2. (c,P2)−elementary sets are ∅, U and half open spaces. The (c,P2)−hull of a subset of U is

its evenly convex hull ([4],[7],. . . ).

3. (c,P3)−elementary sets are ∅, U and complementary set of closed hyperplane. The (c,P3)−hull

of a subset of U is its evenly co-affine hull ([15]). Observe that ([15], corollary 2.2) A is

evenly convex if and only if A is evenly co-affine and convex.

4. (c,P4)−elementary sets are ∅, U and closed hyperplane. Moreover, if U is locally convex,

then by the Hahn-Banach separation theorem and (3.2), the (c,P4)−hull of a non empty

subset of U is its closed affine hull.

Proposition 3. Let P and Q be two nonvoid subsets of R. Assume that any (c, P)-elementary

set is (c, Q)-regular. Then, 〈A〉c,Q ⊂ 〈A〉c,P , ∀A ⊂ U.



142 Moussa Barro & Sado Traore CUBO
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Proof Let a /∈ 〈A〉c,P . By definition, there exists an (c, P)-elementary set E such that A ⊂ E

and a /∈ E. Since E is also (c, Q)-regular, it follows from (3.2) that a /∈ 〈A〉c,Q, and we are

done.

Corollary 3.5. For any A ⊂ U, one has:

〈A〉c,1 ⊃ 〈A〉c,2 ⊃ 〈A〉c,3 and 〈A〉c,4 ⊃ 〈A〉c,3.

Proof Let v ∈ V and r ∈ R, it is obvious that

{u ∈ U | c(u, v) ≤ r} =
⋂

s>r

{u ∈ U | c(u, v) < s}.

Consequently, any (c,P1)−elementary subset is (c,P2)−regular, and by Proposition 3, one has

〈A〉c,1 ⊃ 〈A〉c,2 for any A ⊂ U.

It is easy to verify that:

{u ∈ U | c(u, v) < r} =
⋂

s≥r

{u ∈ U | c(u, v) 6= s},

{u ∈ U | c(u, v) = r} =
⋂

s6=r

{u ∈ U | c(u, v) 6= s},

therefore, 〈A〉c,2 ⊃ 〈A〉c,3 ⊂ 〈A〉c,4.

We derive from Corollary 3.5, the following comparison between the sets Rc,Pi(U):

Rc,P1(U) ⊂ Rc,P2(U) ⊂ Rc,P3(U) and Rc,P4(U) ⊂ Rc,P3(U). (3.6)

Remark 3.6. Observe that

1. P1 ⊃ P2 and Rc,P1(U) ⊂ Rc,P2(U).

2. P3 ⊃ P2 and Rc,P3(U) ⊃ Rc,P2(U).

3. Rc,P1(U) ⊂ Rc,P3(U) whereas P1 and P3 are not comparable in the sense of inclusion.

4. In particular, in the case of locally convex vector space, we recover the fact that every closed

convex subset is evenly convex.

Proposition 4. Assume that the coupling function c satisfies the property:

∀v ∈ V, ∃w ∈ V | − c(., v) = c(., w). (3.7)

Then, one has:

Rc,P4(U) ⊂ Rc,P1(U) ⊂ Rc,P2(U) ⊂ Rc,P3(U). (3.8)



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Level sets regularization with application to optimization problems 143

Proof By (3.6), we only need to show that Rc,P4(U) ⊂ Rc,P1(U). Let (v, r) ∈ V × R. We

have

[c(., v) = r] = [c(., v) ≤ r]
⋂

[−c(., v) ≤ −r].

By assumption on the coupling function, there exists w ∈ V such that −c(., v) = c(., w). Conse-

quently,

[c(., v) = r] = [c(., v) ≤ r]
⋂

[c(., w) ≤ −r].

We conclude with Proposition 3.

Example 3.7. Assume that U = V = Rn, and coupling function c is defined by c(u, v) = ‖u − v‖,

where ‖.‖ is the euclidean norm. The non trivial (c,P4)−elementary sets are spheres (not convex)

whereas the non trivial (c,P1)−elementary sets are closed balls (closed convex). In this case,

Rc,P1(U) and Rc,P4(U) are not comparable. Observe that in this case assumption (3.7) does not

hold.

Proposition 5. For any A ⊂ U, one has 〈A〉c,1 = [i
Γc
A ≤ 0].

Proof By (3.2), one has

a /∈ 〈A〉c,1 ⇐⇒∃v ∈ V : i
c
A(v) < c(a, v)

⇐⇒0 < sup
v∈V

{c(a, v) − icA(v)}

⇐⇒0 < iΓcA (a)

⇐⇒0 /∈
[

iΓcA ≤ 0
]

.

Thus 〈A〉c,1 =
[

iΓcA ≤ 0
]

.

The following result makes the link between hull of set and Γ-regularization of function by

means of indicator function.

Theorem 3.8. Assume that the coupling function c satisfies the condition:

∀ (v, β) ∈ V × R∗+, ∃v̄ ∈ V | βc(., v) = c(., v̄). (3.9)

Then for each A ⊂ U such that dom icA 6= ∅, one has: i
Γc
A = i〈A〉c,1.

Proof Let b ∈ U.

(1) Assume that b /∈ 〈A〉c,1. By (3.3), there exists (v, ǫ) ∈ V ×R
∗
+ such that c(b, v)−sup

a∈A
c(a, v) ≥

ǫ. From (3.9), one has:

∀n ≥ 1, ∃vn ∈ V : nc(., v) = c(., vn).



144 Moussa Barro & Sado Traore CUBO
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Consequently,

nǫ ≤ c(b, vn) − sup
a∈A

c(a, vn) = c(b, vn) − i
c
A(vn) ≤ i

Γc
A (b), ∀n ≥ 1.

Therefore iΓcA (b) = +∞.

(2) Assume that b ∈ 〈A〉c,1. By (3.3), one has

c(b, w) − sup
a∈A

c(a, w) ≤ 0, ∀v ∈ V.

Thus

iΓcA (b) = sup
v∈V

{

c(b, v) − sup
a∈A

c(a, v)

}

≤ 0.

Let v ∈ dom icA. By (3.9), one gets

∀n ≥ 1, ∃ vn ∈ V :
1

n
c(., v) = c(., vn).

Consequently,

1

n

(

c(b, v) − sup
a∈A

c(a, v)

)

= c(b, vn) − sup
a∈A

c(a, vn) ≤ i
Γc
A (b), ∀n ≥ 1.

Therefore,

0 = lim
n→+∞

1

n

(

c(b, v) − sup
a∈A

c(a, v)

)

≤ iΓcA (b),

and finally, iΓcA (b) = 0.

Remark 3.9. Assumption (3.9) is satisfied by coupling functions (1), (3) and (5) of Example 2.2.

Coupling functions (2) and (4) of the same example do not satisfy assumption (3.9).

4 Level set regularization of functions

In this section, we introduce a notion of (c,P)−level set regularization of extended real-valued

functions. We show that this level set regularization can be interpreted as bi-conjugacy relative

to a couple of dual polarities by decomposition of the closure operator. We then give some other

expressions of these regularizations.

4.1 Definitions and properties

Definition 4. A function h : U → R is said to be (c,P)−level regular if all of its sub-level sets

are (c,P)−regular, i.e 〈[h ≤ r]〉c,P = [h ≤ r], ∀r ∈ R.



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Level sets regularization with application to optimization problems 145

We denote Nc,P(U), the set of (c,P)−level regular functions defined on U to R. Observe that

this set contains the constant function −∞.

Proposition 6. The set Nc,P(U) is closed under pointwise suprema, i.e given (hi)i∈I a family of

(c,P)−level regular functions,then h := sup
i∈I

hi is (c,P)−level regular.

Proof Let r ∈ R. Since [h ≤ r] = ∩i∈I[hi ≤ r], the conclusion follows from the fact that any

intersection (c,P)−regular sets is (c,P)−regular.

We define the (c,P)−level set regularization of an extended real-valued function as follows.

Definition 5. The (c,P)−level set regularization of a function h : U → R is the greatest

(c,P)−level regular minorant of h. This function is denoted by h〈〉c,P.

Example 4.1. Assume U is topological vector space with topological dual V and c the standard

coupling function. (c,P2)−level regular functions are evenly quasi-convex functions. Moreover,

if U is locally convex then (c,P1)−level regular functions are lower semi-continuous quasi-convex

functions.

Example 4.2. Assume that U is a metric space, V = C (U,R) a space of continuous functions

from U to R, and c : U × V → R defined by c(u, v) = v(u). A function h : U → R ∪ {+∞} is

(c,P1)−level regular if and only if h is lower semi-continuous ([3], corollary 11).

Proposition 7. Any c−elementary function is (c,Pi)−level regular for i = 1, 2, 3. More precisely,

one has

Γc(U) ⊂ Nc,P1(U) ⊂ Nc,P2(U) ⊂ Nc,P3(U) and Nc,P4(U) ⊂ Nc,P3(U).

Proof Let h := c(., v) − r an c−elementary function. For any t ∈ R, we have

[h ≤ t] = {u ∈ U | t + r − c(u, v) ≥ 0},

which is obviously (c,P1)−elementary set. Therefore Γc(U) ⊂ Nc,P1(U). The other inclusions

follow from (3.6).

Remark 4.3. c−elementary functions are not necessary (c,P4)−level set regular functions. For

example, in the topological case, one cannot write a half space as an intersection of affine hyper-

planes.

Example 4.4. Let n ≥ 1, an integer number. Assume that U = V = Rn and c a standard scalar

product of Rn. Let h1, h2 : R
n → J0; nK two functions defined by

h1(x) =

{

0 if x = 0

max{i ∈ J1, nK | xi 6= 0} if x 6= 0,



146 Moussa Barro & Sado Traore CUBO
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h2(x) =

{

0 if xi 6= 0, ∀i ∈ J1, nK

max{i ∈ J1, nK | xi = 0} else .

For any r ∈ R, we have

[h1 ≤ r] =











∅ if r < 0

{x ∈ Rn | xi+1 = . . . = xn = 0} if i ≤ r < i + 1, i = 0, 1, . . . , n − 1

R

n if n ≤ r,

[h2 ≤ r] =











∅ if r < 0

{x ∈ Rn | xi+1 6= 0, . . . , xn 6= 0} if i ≤ r < i + 1, i = 0, 1, . . . , n − 1

R

n if n ≤ r.

It is clear that:

(1) h1 is (c,P4)−level regular. In particular, h1 ∈ Nc,Pi(U), for i = 1, 2, 3, 4.

(2) h2 is (c,P3)−level regular but not (c,P2)−level regular since [h2 ≤ n−1] = {x ∈ R
n | xn 6= 0}

is not convex.

Example 4.5. Let U = V = R, c the standard product of R. The indicator function of R∗,i
R

∗ is

(c,P3)−level regular but not quasi-convex.

4.2 Decomposition of 〈〉c,P

Let us consider a map ∆c,P : 2
U → 2V ×R defined by:

∆c,P(A) := {(v, r) ∈ V × R | A ⊂ E
P

v,r}, (4.1)

which, we simply denote ∆ in the sequel. Given (Ai)i∈I a family of subsets of U, we have

∆
⋃

i∈I

Ai :=

{

(v, r) ∈ V × R |
⋃

i∈I

Ai ⊂ E
P

v,r

}

=
{

(v, r) ∈ V × R | Ai ⊂ E
P

v,r, ∀i ∈ I
}

=
⋂

i∈I

{

(v, r) ∈ V × R | Ai ⊂ E
P

v,r

}

=
⋂

i∈I

∆Ai.

Therefore ∆ is said to be a polarity ([16]). We associate to ∆, its dual polarity ∆∗ : 2V ×R → 2U

defined by

∆∗(B) =
⋃

{A ∈ 2U | B ⊂ ∆(A)}. (4.2)

Observe that for each (v, r) ∈ V × R and for each u ∈ U, one has

u ∈ ∆∗(v, r) ⇐⇒ (v, r) ∈ ∆(u) ⇐⇒ u ∈ EPv,r, (4.3)



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Level sets regularization with application to optimization problems 147

therefore ∆∗(v, r) = EPv,r. Since ∆
∗ is a polarity, then we have for each B ⊂ V × R,

∆∗(B) = ∆∗
(

⋃

(v,r)∈B

{(v, r)}
)

=
⋂

(v,r)∈B

∆∗(v, r) =
⋂

(v,r)∈B

EPv,r. (4.4)

The operator 〈〉c,P can be decomposed as follows.

Proposition 8. For any A ⊂ U, we have (∆∗ ◦ ∆)(A) = 〈A〉c,P.

Proof Given A ⊂ U, one has

(∆∗ ◦ ∆)(A) = ∆∗({(v, r) ∈ V × R | A ⊂ EPv,r}) =
⋂

A⊂EPv,r

EPv,r = 〈A〉c,P.

4.3 Conjugacy associated to polarities ∆ and ∆∗ ([16, 17])

The conjugate of a function h : U → R relative to the polarity ∆ is the function h∆ : V × R → R

given by

h∆(v, r) := sup
u/∈∆∗(v,r)

−h(u) = sup
u/∈EP

v,r

−h(u). (4.5)

Analogously, the conjugate of a function k : V × R → R relative to the polarity ∆∗ is defined by

k∆
∗

(u) := sup
(v,r)/∈∆(u)

−k(v, r) = sup
u/∈EPv,r

−k(v, r). (4.6)

Thus, the bi-conjugacy relative to polarities ∆, ∆∗ of a function h : U → R is the function

h∆∆
∗

: U → R given by

h∆∆
∗

(a) := sup
(v,r)/∈∆(a)

inf
u/∈∆∗(v,r)

h(u) = sup
a/∈EP

v,r

inf
u/∈EPv,r

h(u). (4.7)

It is well known ([16]) that this conjugacy can be interpreted by means of coupling function

δ : U × (V × R) → R defined by

δ(u, (v, r)) =

{

0 si u /∈ EPv,r
−∞ si u ∈ EPv,r.

More precisely, given a function h : U → R, we have h∆ = hδ and h∆∆
∗

= hδδ.

Theorem 4.6 ([16]). The (c,P)−level regularization of a function h : U → R coincides with

bi-conjugacy relative to polarities ∆, ∆∗: h〈〉c,P = h∆∆
∗

.

Corollary 4.7. For any subset A of U, we have

i
〈〉c,P
A = i〈A〉c,P and v

〈〉c,P
A = v〈A〉c,P.



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Proof Let A ⊂ U. Let a ∈ A. It follows from Theorem 4.6 that

i
〈〉c,P
A (a) = sup

a/∈EP
v,r

inf
u/∈EP

v,r

iA(u).

We distinguish two cases:

• We first assume that a /∈ 〈A〉c,P. There exists (v, r) ∈ V ×R such that a /∈ E
P

v,r and A ⊂ E
P

v,r.

Consequently,

inf
u/∈EP

v,r

iA(u) = +∞.

• Secondly, assume that a ∈ 〈A〉c,P. For all (v, r) ∈ V × R such that a /∈ E
P

v,r, there exists

u ∈ A such that u /∈ EPv,r. Consequently,

i
〈〉c,P
A (a) = sup

a/∈EP
v,r

inf
u/∈EP

v,r

iA(u)

=

{

0 if a ∈ 〈A〉c,P

+∞ if a /∈ 〈A〉c,P

=i〈A〉c,P(a).

Analogously, we have

v
〈〉c,P
A (a) = sup

a/∈EPv,r

inf
u/∈EP

v,r

vA(u)

=

{

−∞ if a ∈ 〈A〉c,P

+∞ if a /∈ 〈A〉c,P

=v〈A〉c,P(a).

Proposition 9. For any function h : U → R and for any real number t, one has

[h〈〉c,P ≤ t] =
⋂

s>t

〈[h ≤ s]〉c,P.

Proof Let s > t and a /∈ 〈[h ≤ s]〉c,P. There exists (v̄, r̄) ∈ V × R such that a /∈ E
P

v̄,r̄ and

[h ≤ s] ⊂ EPv̄,r̄. We deduce that

h〈〉c,P(a) = sup
a/∈EP

v,r

inf
u/∈EP

v,r

h(u) ≥ inf
u/∈EPv̄,r̄

h(u) ≥ s > t.

Let a /∈
〈

[h〈〉c,P ≤ s]
〉

c,P
. There exists s ∈ R such that h〈〉c,P(a) > s > t. By Theorem 4.6, there

exists (v, r) ∈ V × R such that

a /∈ EPv,r and inf
u/∈EP

v,r

h(u) > s,

thus [h ≤ s] ⊂ EPv,r, and finally a /∈ 〈[h ≤ s]〉c,P.



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Level sets regularization with application to optimization problems 149

4.4 Other expressions of (c,P)−level regularizations

We now give another expression of the (c,P)−level regularization of an extended real-valued func-

tion h. These expressions give the value of the (c,P)−level regularization of h at a given point.

Given h : U → R and a ∈ U, we define sets Ih(a) and Jh(a) by:

Ih(a) := {t ∈ R | a /∈ 〈[h ≤ t]〉c,P} and Jh(a) = {t ∈ R | a ∈ 〈[h ≤ t]〉c,P}. (4.8)

Sets Ih(a) and Jh(a) are two intervals of R such that Ih(a) ∩ Jh(a) = ∅ and Ih(a) ∪ Jh(a) = R.

Moreover, for any (r, s) ∈ Ih(a) × Jh(a), we have r < s. We deduce that sup Ih(a) = inf Jh(a).

Proposition 10.

h〈〉c,P(a) = sup
{

t ∈ R : a /∈ 〈[h ≤ t]〉c,P

}

= inf
{

t ∈ R : a ∈ 〈[h ≤ t]〉c,P

}

.

Proof Let t ∈ Ih(a). There exists (v, r) ∈ V × R such that a /∈ E
P

v,r and [h ≤ t] ⊂ E
P

v,r.

Therefore

inf
u/∈EP

v,r

h(u) ≥ t and so sup
a/∈EP

v,r

inf
u/∈EP

v,r

h(u) ≥ t.

By Theorem 4.6, one gets h〈〉c,P(a) ≥ t. Hence h〈〉c,P(a) ≥ sup Ih(a). Conversely, let t < h
〈〉c,P(a),

then a /∈ [h〈〉c,P ≤ t] and by Proposition 9, there exists s > t such that a /∈ 〈[h ≤ s]〉c,P. Conse-

quently, sup Ih(a) ≥ s > t. Hence sup Ih(a) ≥ h
〈〉c,P(a).

5 Applications to an optimization problem: sub-level set
duality

Let us consider the following minimization problem:

min
x

f(x), s.t x ∈ X, (P)

where X is a nonempty set and f : X → R ∪ {+∞} is an extended real-valued function.

5.1 Level set perturbational duality

We consider a perturbation function F : X × U → R satisfying

∃a ∈ U : F(., a) = f(.). (5.1)

We associate to F a valued function h : U → R defined by

h(u) := inf
x∈X

F(x, u). (5.2)



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We denote by α the optimal value of (P). It is obvious that α = h(a).

The perturbational dual of (P) ([16]) is given by

max
(v,r)

−h∆(v, r) s.t a /∈ EPv,r. (D)

We denote by β the optimal value of (D). By definition, one has

− ∞ ≤ β := sup (D) = h∆∆
∗

(a) ≤ h(a) =: α = inf (P) ≤ +∞. (5.3)

Thus, the weak duality holds. The following theorem gives a necessary and sufficient condition to

assure the strong duality.

Theorem 5.1. The following statements are equivalent:

(1) The strong duality holds for (P) i.e inf (P) = max (D),

(2) a /∈ 〈[h < α]〉c,P.

Proof. Assume that (1) holds. There exists (v̄, r̄) ∈ V × R such that

a /∈ EPv̄,r̄ and α = h(a) = −h
∆(v̄, r̄) := inf

u/∈EPv̄,r̄

h(u).

We deduce that [h < α] ⊂ EPv̄,r̄. Thus a /∈ 〈[h < α]〉c,P. Conversely, assume that (2) holds. There

exists (v̄, r̄) ∈ V × R such that a /∈ EPv̄,r̄ and [h < α] ⊂ E
P

v̄,r̄ . Therefore

inf
u/∈EPv̄,r̄

h(u) ≥ α ≥ β.

Remember that

β := sup
a/∈EP

v,r

−h∆(v, r) = sup
a/∈EP

v,r

inf
u/∈EP

v,r

h(u).

Thus

β ≥ inf
u/∈EPv̄,r̄

h(u) = −h∆(v̄, r̄) ≥ α ≥ β.

Hence β = −h∆(v̄, r̄) = α.

Theorem 5.1 is interesting in evenly convex case which is used in economic theory.

5.2 Evenly quasi-convex duality ([1],[7],[11],[12],[17])

We assume X and U are topological vector spaces, V = U∗ the topological dual of U, c = 〈, 〉 the

standard coupling function between U and U∗.



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Level sets regularization with application to optimization problems 151

Corollary 5.2. Assume that function F : X × U → R is quasi-convex and for each x ∈ X,

F(x, .) : U → R is upper semi-continuous. One has:

inf (P) = max
u∗∈U∗

inf
(x,u)∈X×U
〈u−a,u∗〉≥0

F(x, u).

Proof Since F is quasi-convex and for each x ∈ X, F(x, .) is upper semi-continuous then h

is quasi-convex and upper semi-continuous. Consequently, [h < α] is open convex set and so it is

evenly convex. As a /∈ [h < α], it results from Theorem 5.1 that

inf (P) = max (D) = max
r−〈a,u∗〉≤0

inf
r−〈u,u∗〉≤0

h(u) = max
u∗∈U∗

inf
〈u,u∗〉≥〈a,u∗〉

h(u),

where the last equality follows from the fact that for each u∗ ∈ U∗, function ku∗ : R → R defined

by ku∗(r) = inf
r−〈u,u∗〉≤0

h(u) is not decreasing.

Corollary 5.3. Assume that function F : X × U → R is quasi-convex and for each x ∈ X,

F(x, .) : U → R is upper semi-continuous. One has:

inf (P) = max
u∗∈U∗

inf
(x,u)∈X×U
〈u−a,u∗〉=0

F(x, u).

Proof We know that under these assumptions on F , [h < α] is convex open set, therefore it is

(〈, 〉,R∗)−regular. Since a /∈ [h < α], it results from Theorem 5.1 that

inf (P) = max (D)

= max
(u∗,r)∈U∗×R
〈a,u∗〉 = r

inf
u∈U

〈u,u∗〉=r

h(u)

= max
u∗∈U∗

max
r∈R

〈a,u∗〉=r

inf
u∈U

〈u,u∗〉=r

h(u)

= max
u∗∈U∗

inf
u∈U

〈u−a,u∗〉=0

h(u)

= max
u∗∈U∗

inf
(x,u)∈X×U
〈u−a,u∗〉=0

F(x, u) by definition of h.

6 Conclusion

In this work, we introduced a closure operator by means of coupling function and a subset of R.

This operator allowed us to define a hull of sets and level set regularization of extended real-valued

functions. By decomposition of closure operator, we showed that a level set regularization of any



152 Moussa Barro & Sado Traore CUBO
22, 1 (2020)

extended real-valued function coincides with its bi-conjugacy relative to a couple of dual polarities.

We derive an analytic expression of a level set regularization of extended real-valued function. Our

results are applied to derive a strong duality for a minimization problem.

Acknowledgements The authors are grateful to the anonymous referees and the editor for their

constructive comments which have contributed to the final presentation of the paper.



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Level sets regularization with application to optimization problems 153

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	Introduction
	Preliminaries
	-regularization of functions and hull of sets
	-regularization of functions
	 Hull of sets

	Level set regularization of functions
	Definitions and properties
	 Decomposition of "426830A "526930B c,P
	Conjugacy associated to polarities  and * (Volle-1985,Volle-1987)
	Other expressions of (c,P)-level regularizations

	Applications to an optimization problem: sub-level set duality
	Level set perturbational duality
	Evenly quasi-convex duality (Crouzeix-1977,Legaz-book,Penot-Volle-1990,Penot-Volle-2004,Volle-1987)

	Conclusion