LNA

ISSN 2188-8167

2023

104

H. YASUNAKA AND S. SUZUKI

where f is an extended real-valued quasiconvex function on Rn , and A is a convex
subset of Rn . We study dual problems in terms of three types of quasiconjugate
functions, H-quasiconjugate, O-quasiconjugate, and R-quasiconjugate. We introduce three dual problems, and show strong duality theorems. Additionally, we
compare our results with previous ones.
The remainder of the present paper is organized as follows. In Section 2, we
introduce some preliminaries and previous results. In Section 3, we show duality
theorems for quasiconvex programming in terms of quasiconjugate functions. In
Section 4, we discuss about our results.
2. Preliminaries
Let Rn denote the n-dimensional Euclidean space. The inner product of two
vectors v and x in Rn is denoted by ⟨v, x⟩. The indicator function δA is deﬁned by{
0 x ∈ A,δA (x) =∞ otherwise.
Let f be a function from Rn to R = [−∞, ∞]. A function f is said to be convex iffor each x, y ∈ Rn and α ∈ [0, 1],f ((1 − α)x + αy) ≤ (1 − α)f (x) + αf (y).
The epigraph of f is deﬁned as
epif = {(x, α) ∈ Rn × R : f (x) ≤ α}.
A function f is convex if and only if epif is convex. The Fenchel conjugate [3, 4] off , f ∗ : Rn → R, is deﬁned asf ∗ (v) = sup {⟨v, x⟩ − f (x)}.
x∈Rn

A function f is said to be quasiconvex if for each x, y ∈ Rn and α ∈ [0, 1],f ((1 − α)x + αy) ≤ max{f (x), f (y)}.
Deﬁne the level sets of f with respect to a binary relation ⋄ on R aslev(f, ⋄, α) = {x ∈ Rn : f (x) ⋄ α}for each α ∈ R. It is well known that f is quasiconvex if and only if lev(f, ≤, α) isconvex for all α ∈ R.
We deﬁne the following families of open half spaces:
H = {lev(v, <, α) : v ∈ Rn , α ∈ R},
H + = {lev(v, <, α) : v ∈ Rn , α > 0},
H 0 = {lev(v, <, 0) : v ∈ Rn },H − = {lev(v, <, α) : v ∈ Rn , α < 0}.
A subset A of Rn is said to be evenly (H-evenly, O-evenly, and R-evenly) convex

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if it is the intersection of a subfamily of H (H + , H 0 , H − , respectively). We deﬁne
the whole space and the empty set is evenly (H-evenly, O-evenly, and R-evenly,
respectively) convex by convention. A function f is said to be evenly (H-evenly,
O-evenly, and R-evenly) quasiconvex if lev(f, ≤, α) is evenly (H-evenly, O-evenly,
and R-evenly, respectively) convex for all α ∈ R.
We show the following two propositions without proof.
Proposition 2.1. Let A be a subset of Rn . Then, the following statements hold:(i) if A is evenly convex, then A is convex,(ii) if A is open or closed convex, then A is evenly convex,(iii) if A is non-empty, then A is H-evenly convex if and only if A is evenly
convex and contains 0,(iv) if A is O-evenly convex, then for each x ∈ A and t > 0, tx ∈ A,(v) if A is R-evenly convex, then for each x ∈ A and t ≥ 1, tx ∈ A.
Proposition 2.2. Let f be a function from Rn to R. Then, the following statements
hold:(i) if f is evenly quasiconvex, then f is quasiconvex,(ii) if f is lower semicontinuous (lsc) or upper semicontinuous (usc) quasiconvex, then f is evenly quasiconvex,(iii) f is H-evenly quasiconvex if and only if f is evenly quasiconvex and 0 ∈ Rn
is a global minimizer of f in Rn ,(iv) if f is O-evenly quasiconvex or R-evenly quasiconvex, then 0 ∈ Rn is a global
maximizer of f in Rn ,(v) if f is O-evenly quasiconvex, then for each x ∈ Rn and t > 0, f (x) = f (tx),(vi) if f is R-evenly quasiconvex, then for each x ∈ Rn and t ≥ 1, f (x) ≥ f (tx).
Proposition 2.1 and Proposition 2.2 are elementary but important. Various researchers investigate evenly convex sets and evenly quasiconvex functions precisely,
see [1, 2, 4, 9, 17, 18, 20, 21, 23] and references therein.
In quasiconvex programming, various types of quasiconjugate functions have been
investigated. In [6], Martı́nez-Legaz introduces Q-conjugate of f as follows:f Q (v, t) = − inf{f (x) : ⟨v, x⟩ ≥ t}.
In [22], Thach introduces H-quasiconjugate of f as follows:{− inf{f (x) : ⟨v, x⟩ ≥ 1}, v ̸= 0,
H
f (v) =− sup{f (x) : x ∈ Rn },v = 0.
In [11], Passy and Prisman introduce O-quasiconjugate of f as follows:f O (v) = − inf{f (x) : ⟨v, x⟩ ≥ 0}.
In [22], Thach introduces R-quasiconjugate as follows:f R (v) = − inf{f (x) : ⟨v, x⟩ ≥ −1}.
Although Q-conjugate and O-quasiconjugate are deﬁned by other names, we denotef Q and f O for the sake of distinction.

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In this paper, we consider the following quasiconvex programming problem (P ):{minimize f (x),(P )subject to x ∈ A,
where f is an extended real-valued quasiconvex function on Rn , and A is a convex
subset of Rn . In [19], we deﬁne the following dual problem (D) in terms of Qconjugate for the primal problem (P ):{minimize f Q (v, t),(D)∗.
subject to (v, t) ∈ −epiδA
We denote
val(P ) = inf f (x), and val(D) =
x∈A

inf

∗
(v,t)∈−epiδA

f Q (v, t).

In [19], we show the following strong duality theorem.
Theorem 2.3 ([19]). Let f be an usc quasiconvex function from Rn to R, and A anonempty convex subset of Rn . Thenval(P ) = −val(D).
Additionally, we introduce the following three dual problems:{
minimize f Q (v, 1),(D)1∗,
subject to (v, 1) ∈ −epiδA
{
(D)0
{
(D)−1

minimize f Q (v, 0),∗,subject to (v, 0) ∈ −epiδAminimize f Q (v, −1),∗.
subject to (v, −1) ∈ −epiδA

We show the following corollary.
Corollary 2.4 ([19]). Let f be an usc quasiconvex function from Rn to R, and A
a nonempty convex subset of Rn . Thenval(P ) = − min{val(D)1 , val(D)0 , val(D)−1 }.
3. Quasiconjugate duality theorems
In this section, we study dual problems in terms of three types of quasiconjugate
functions, By Corollary 2.4, val(P ) is equal to at least one of val(D)1 , val(D)0 , andval(D)−1 . We show suﬃcient conditions for strong duality theorems between the
primal problem and dual problems.
At ﬁrst, we show a strong duality theorem for (D)1 under H-evenly quasiconvexity.

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Theorem 3.1. Let f be an usc H-evenly quasiconvex function from Rn to R, A aconvex subset of Rn , and x0 ∈ A. Assume that inf x∈A f (x) > inf x∈Rn f (x). Thenval(P ) = −val(D)1 .
Proof. By Corollary 2.4,val(P ) = − min{val(D)1 , val(D)0 , val(D)−1 } ≥ −val(D)1 .
Assume that val(P ) > −val(D)1 . Since val(P ) = inf x∈A f (x) > inf x∈Rn f (x), there
exists α ∈ R such that val(P ) > α > −val(D)1 and val(P ) > α > inf x∈Rn f (x).
This shows that lev(f, <, α) is open convex and 0 ∈ lev(f, <, α) since f is usc Hevenly quasiconvex. By the separation theorem between A and lev(f, <, α), there
exists (v0 , t0 ) ∈ Rn+1 such that for each x ∈ A and z ∈ lev(f, <, α),⟨v0 , x⟩ ≥ t0 > ⟨v0 , z⟩ .
Since 0 ∈ lev(f, <, α), t0 > 0. Hence,
⟨
⟩
⟨
⟩
v0
v0
,x ≥ 1 >
,z
t0
t0
∗ and
Let v ∗ = vt00 , then (v ∗ , 1) ∈ −epiδA−f Q (v ∗ , 1) = inf{f (y) : ⟨v ∗ , z⟩ ≥ 1} ≥ α > −val(D)1 ≥ −f Q (v ∗ , 1).
□

This is a contradiction. This completes the proof.

Next, we show a strong duality theorem for (D)0 under O-evenly quasiconvexity.
Theorem 3.2. Let f be an usc O-evenly quasiconvex function from Rn to R, A aconvex subset of Rn , and x0 ∈ A. Thenval(P ) = −val(D)0 .
Proof. By Corollary 2.4,val(P ) = − min{val(D)1 , val(D)0 , val(D)−1 } ≥ −val(D)0 .
Assume that val(P ) > −val(D)0 . Then, there exists α ∈ R such that val(P ) > α >∗,−val(D)0 . Since (0, 0) ∈ −epiδA
α > −val(D)0 ≥ −f Q (0, 0) = infn f (x).
x∈RThis shows that lev(f, <, α) is nonempty. Furthermore, lev(f, <, α) is open convex
since f is usc O-evenly convex. By the separation theorem between A and lev(f, <, α), there exists (v0 , t0 ) ∈ Rn+1 such that for each x ∈ A and z ∈ lev(f, <, α),⟨v0 , x⟩ ≥ t0 > ⟨v0 , z⟩
For x0 ∈ A, assume that ⟨v0 , x0 ⟩ < 0. This shows that for each z ∈ lev(f, <, α),
0 > ⟨v0 , x0 ⟩ ≥ t0 > ⟨v0 , z⟩ .
0 ,x0 ⟩
Let t = ⟨v
2⟨v0 ,z⟩ , then t > 0 and tz ∈ lev(f, <, α) by Proposition 2.1 (iv). However,⟨v0 , tz⟩ =⟨v0 , x0 ⟩> ⟨v0 , x0 ⟩ .
2

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This is a contradiction. Hence for each x ∈ A, ⟨v0 , x⟩ ≥ 0.
Next, assume that there exists z0 ∈ lev(f, <, α) such that ⟨v0 , z0 ⟩ ≥ 0. If⟨v0 , z0 ⟩ > 0, then t0 > 0. Since f is O-evenly quasiconvex, tz0 ∈ lev(f, <, α) for
each t > 0. However, for suﬃciently large t, ⟨v0 , tz0 ⟩ > t0 . This is a contradiction.
If ⟨v0 , z0 ⟩ = 0, there exists z̄ ∈ lev(f, <, α) such that ⟨v0 , z̄⟩ > 0 since lev(f, <, α)is open. By the similar way, we can prove that ⟨v0 , tz̄⟩ > t0 for suﬃciently large t.
Hence, for each z ∈ lev(f, <, α), ⟨v0 , z⟩ < 0.
∗ andThis shows that (v0 , 0) ∈ −epiδA−f Q (v0 , 0) = inf{f (y) : ⟨v0 , z⟩ ≥ 0} ≥ α > −val(D)0 ≥ −f Q (v0 , 0).
□This is a contradiction. Hence val(P ) = val(D)0 .

Finally, we show a strong duality theorem for R-evenly quasiconvex objectivefunctions.
Theorem 3.3. Let f be an usc R-evenly quasiconvex function from Rn to R, A aconvex subset of Rn , and x0 ∈ A. Thenval(P ) = − min{val(D)0 , val(D)−1 }.
Proof. By Corollary 2.4,val(P ) = − min{val(D)1 , val(D)0 , val(D)−1 } ≥ − min{val(D)0 , val(D)−1 }.
Assume that val(P ) > − min{val(D)0 , val(D)−1 }. Then, there exists α ∈ R such∗,that val(P ) > α > − min{val(D)0 , val(D)−1 }. Since (0, 0) ∈ −epiδA
α > − min{val(D)0 , val(D)−1 } ≥ −f Q (0, 0) = infn f (x).
x∈RThis shows that lev(f, <, α) is nonempty. Furthermore, lev(f, <, α) is open convex
since f is usc R-evenly convex. By the separation theorem between A and lev(f, <, α), there exists (v0 , t0 ) ∈ Rn+1 such that for each x ∈ A and z ∈ lev(f, <, α),⟨v0 , x⟩ ≥ t0 > ⟨v0 , z⟩
If t0 = 0, we can check that val(P ) = −val(D)0 . Additionally, if t0 < 0, thenval(P ) = −val(D)−1 .
Assume that t0 > 0 and there exists z0 ∈ L(f, <, α) such that ⟨v0 , z0 ⟩ ≥ 0. If⟨v0 , z0 ⟩ = 0, then there exists z̄ ∈ L(f, <, α) such that ⟨v0 , z̄⟩ > 0 since f is usc.
Without loss of generality, we can assume that ⟨v0 , z0 ⟩ > 0. Since f is R-evenly
quasiconvex, tz0 ∈ L(f, <, α) for each t ≥ 1 by Proposition 2.1 (v). However, forsuﬃciently large t, ⟨tv0 , z0 ⟩ > t0 . This is a contradiction. Hence, for each x ∈ A
and z ∈ lev(f, <, α),⟨v0 , x⟩ ≥ t0 > 0 > ⟨v0 , z⟩ .
∗ andThis shows that (v0 , 0) ∈ −epiδA−f Q (v0 , 0) = inf{f (y) : ⟨v0 , z⟩ ≥ 0} ≥ α > −val(D)0 ≥ −f Q (v0 , 0).
This is a contradiction. This completes the proof.

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4. Discussion
In this section, we discuss about our results. In particular, we compare our resultswith duality theorems in [22, 23].
4.1. H-quasiconjugate duality. In [22], Thach studies duality theorems in termsof H-quasiconjugate functions. We consider the following maximization problem:{
maximize f (x),subject to x ∈ A,
where f is an usc quasiconvex function and A is a compact convex subset of Rn .
Thach introduces the following dual problem:{
minimize f H (v),subject to v ∈ Rn \ intA0 ,where A0 = {v ∈ Rn : ∀x ∈ A, ⟨v, x⟩ ≤ 1}, and intA0 is the interior of A0 . Thach
investigates a maximization problem of a quasiconvex function and shows duality
theorems in terms of H-quasiconjugate. On the other hand, in this paper, we study
a minimization problem of a quasiconvex function. In Theorem 3.1, we show strong
duality theorem in terms of H-quasiconjugate.
4.2. R-quasiconjugate duality. In [23], Thach introduces the following dual problem:{
minimize f R (v),(D)Rsubject to v ∈ −A0 .
Clearly, (D)R = (D)−1 . Thach shows the following result, see Theorem 3.2 in [23].
Theorem 4.1 ([23]). val(P ) = −val(D)R and v ∗ ∈ Rn is a global minimizer of(D)R if and only if for each x ∈ A and y ∈ lev(f, <, val(P )),(4.1)⟨v ∗ , x⟩ ≥ −1 > ⟨v ∗ , y⟩ .

Thach shows a characterization of a global minimizer of (D)R . However, even if
f is R-evenly quasiconvex, the separation inequality (4.1) does not necessarily holdby Theorem 3.3. We show the following example.
Example 4.2. Let f be the following function on R2 : 0, x ∈ S = {(y1 , y2 ) : y1 > 0, y1 y2 > 1},/ S, x1 + x2 > 21 ,f (x) = 1, x ∈2, otherwise.
We can check easily that f is usc R-evenly quasiconvex.

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Consider a minimization problem of f on A = {(x1 , x2 ) : x2 ≤ 0}. Then val(P ) =∗ , and1, ((0, −1), 0) ∈ −epiδA
f O ((0, −1)) = −1,
that is, val(P ) = −val(D)0 . However, val(P ) ̸= −val(D)−1 . Actually, let v =∗ , then v = 0 and v ≤ 0. This shows(v1 , v2 ) ∈ Rn such that ((v1 , v2 ), −1) ∈ −epiδA
1
2
that
f R (v) = − inf{f (x) : ⟨v, x⟩ ≥ −1} = 0.
Hence,
val(P ) = 1 > 0 = −val(D)−1 .
In Theorem 3.1 and Theorem 3.2, we show strong duality theorems for H-evenly
quasiconvex and O-evenly quasiconvex functions. On the other hand, in Theorem 3.3, we need not only R-quasiconjugate but also O-quasiconjugate for R-evenly
quasiconvex functions. In the following theorem, we show a suﬃcient condition forval(P ) = −val(D)−1 .
Theorem 4.3. Let f be an usc quasiconvex function from Rn to R, A a convex
subset of Rn , and x0 ∈ A. Assume that 0 ∈ intA and inf x∈A f (x) > inf x∈Rn f (x).
Thenval(P ) = −val(D)−1 .
Proof. Let α = val(P ). By the assumption, lev(f, <, α) is nonempty. By theseparation theorem for A and lev(f, <, α), there exists (v0 , t0 ) ∈ Rn+1 such that foreach x ∈ A and z ∈ lev(f, <, α),⟨v0 , x⟩ ≥ t0 > ⟨v0 , z⟩ .
v0
Since 0 ∈ intA, t0 < 0. Let v̄ = −t
, then for each x ∈ A and z ∈ lev(f, <, α),0⟨v̄, x⟩ ≥ −1 > ⟨v̄, z⟩ .
∗ . By Theorem 2.3,
This shows that f Q (v̄, −1) ≤ −α and (v̄, −1) ∈ −epiδAval(P ) = −val(D) ≥ −val(D)−1 ≥ −f Q (v̄, −1) ≥ α.
This completes the proof.

□

In Theorem 4.3, we do not assume that f is R-evenly quasiconvex. For Revenly quasiconvex objective functions, we may need additional assumptions forval(P ) = −val(D)−1 .
4.3. Conjugate functions in convex and quasiconvex programming. We
show the following table of conjugate functions in convex and quasiconvex programming:QUASICONJUGATE DUAL PROBLEMS FOR QUASICONVEX PROGRAMMING

111

convexity of function
conjugate
domain of conjugate∗
convexfRnQ
evenly quasiconvex
fRn+1
H-evenly quasiconvex f H = f Q (·, 1)RnO
Q
O-evenly quasiconvex f = f (·, 0)Rn
R-evenly quasiconvex f R = f Q (·, −1)
Rn

In convex programming, Fenchel conjugate plays an essential role. In quasiconvex
programming, we need extra parameters to obtain duality results, for example t
in f Q (v, t). In this paper, we show duality results for quasiconvex programming
without extra parameters by using f H , f O , and f R . Similar to Fenchel conjugate,these three quasiconjugate functions are deﬁned on Rn . Additionally, these quasiconjugates are closely related to f Q .

Acknowledgements
We would like to express our deepest appreciation to Professor Kazimierz Goebel.
The authors are grateful to the anonymous referee for many comments and suggestions improved the quality of the paper.

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Manuscript received 20 January 2023
revised 20 October 2023

QUASICONJUGATE DUAL PROBLEMS FOR QUASICONVEX PROGRAMMING

Hiroki Yasunaka
Graduate School of Natural Science and Technology, Shimane University, Japan
E-mail address: n21m009@matsu.shimane-u.ac.jp
Satoshi Suzuki
Department of Mathematical Sciences, Shimane University, Japan
E-mail address: suzuki@riko.shimane-u.ac.jp113