INVEX SETS
AND NONSMOOTH INVEX FUNCTIONS
ŞTEFAN MITITELU
We give conditions for invexity of sets and show that the family of some invex
subsets of Rn is a nontrivial vector subspace of Rn . Also, a representation form
for an invex set is given. We show that if for a nonsmooth function of Rn every
stationary point is a global minimum, then this function is pseudoinvex on the set
of its stationary points. The domains of the invex, pseudoinvex and quasiinvex
functions are invex sets. We extend the notions of invexity, pseudoinvexity and
quasiinvexity for real functions defined on arbitrary sets.
AMS 2000 Subject Classification: 26B25, 49J52, 90C25.
Key words: invexity, preinvexity, generalized convexity, nonsmooth analysis.
1. INTRODUCTION
Pini [13] introduced in 1991 the notion of η-invex set and Mititelu [8]
defined in 1994 the invex sets. The invex, pseudoinvex and quasiinvex functions were introduced by Hanson [5] in 1981 in the differentiable framework. A
differentiable function is invex if and only if every stationary point is a global
minimum point [2]. If a function is pseudoinvex, then every stationary point
of it is a global minimum [9]. For a quasiinvex function, every local minimum point is a global one [9]. For enjoying these properties, the functions
mentioned are used in optimization theory. We recall that the domains of
differentiable invex functions are invex sets [8].
Craven [3] defined in 1986 the nonsmooth invex functions. All types
of nonsmooth invex and generalized invex functions at a point, grouped in
ρ-classes, were introduced by Giorgi and Mititelu [4], using the upper Dini
directional derivative, and by Mititelu and Stancu-Minasian [12], using the
Clarke directional derivative.
This paper establishes some properties of invex sets and nonsmooth [generalized] invex functions on open and arbitrary (nonempty) sets. It is divided
into four sections. Section 1 is an introduction. In Section 2 we establish necessary and sufficient conditions for the invexity of the sets using the cone of
feasible directions [1]. In addition, there is given a representation form of an
REV. ROUMAINE MATH. PURES APPL., 52 (2007), 6, 665–672
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Ştefan Mititelu
2
invex set. Also, it is shown that the family of all subsets of Rn that are invex
with respect to vector functions of the form η(x, u) only, and are homogeneous
in u, is a nontrivial vector subspace of Rn . In Section 3 we show that if for a
nonsmooth function on Rn every stationary point is a global minimum, then
this function is pseudoinvex on the set of its stationary points. Also, it is
shown that the domains of the invex, pseudoinvex and quasiinvex nonsmooth
functions are invex sets. In Section 4 we consider invex functions defined on
arbitrary (nonempty) sets.
2. INVEX SETS
Definition 2.1 [8]. 1) A nonempty set X ⊆ Rn is said to be an invex set
at u ∈ X if there is a vector function η : X × X → Rn such that
(2.1)
∀x ∈ X, ∀λ ∈ [0, 1] ⇒ u + λη(x, u) ∈ X.
2) A set X is called invex if (2.1) holds for all u ∈ X.
(There is excluded the situation η = 0 [η is the zero function], when X
is a trivial invex set.)
2.1. Invexity conditions for sets
We recall that the invex sets generalize the convex sets (property that
results for η(x, u) = x − u). If X is invex with respect to η then it is called
η-invex for short. From (2.1) we get that η(x, u) is a feasible direction to X
at the point u. Therefore,
η(x, u) ∈ FX (u) = {v ∈ Rn | ∀λ ∈ [0, 1] : u + λη(v, u) ∈ X},
where FX (u) is the cone of feasible directions to X at u. Then the invexity of
X at u amounts to
(2.2)
η(x, u) ∈ FX (u) ⇒ u + λη(x, u) ∈ X, ∀λ ∈ [0, 1] (η = 0).
Example 2.1. The graph of the function f : [−1, 1] → R, f (x) = x3 ,
namely Gf = {(x, y) ∈ R2 | y = x3 , x ∈ [−1, 1]}, is an invex set at no point
of it. We remark that at the current point (u, u3 ) ∈ Gf we have FGf (u, u3 ) =
{(0, 0)}. For this reason (it results η = 0) the set Gf is some invex at no of
its points.
Theorem 2.1. If a nonempty set X is an η-invex set at u, then
FX (u) ⊃ {0}.
We denote by ri FX (u) the relative interior [1] of FX (u). We remark that
ri Gf = ∅. If ri X = ∅ and FX (u) = {0}, then u is an extremal point [1] of X.
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Invex sets and nonsmooth invex functions
667
In general, if FX (u) ⊃ {0} then FX (u) is not a convex cone. If X is a convex
set, then FX (u) is a convex cone [1].
Theorem 2.2. Suppose that X is a nonempty set and ri FX (u) = ∅.
Then X is an invex set at u.
Proof. Obviously, FX (u) ⊃ {0}. Consider the points u, x ∈ X. If [u, x] ⊂
X then X is invex at u with respect to η defined by η(x, u) = x − u. Now,
assume that [u, x] ⊂
/ X. Since ri FX (u) = ∅, we can consider a point x ∈ X
such that [u, x ] ⊂ X and x − u ∈ FX (u). In this case, the correspondence
(u, x) → x defines an application η by x = u + η(x, u), for which we have
u + η(x, u) ∈ X and u + λη(x, u) ∈ X, ∀λ ∈ [0, 1]. With this η, X is an
η-invex set. Remark 2.1. If X is η-invex at u then, according to (2.2), it is sufficient
that η : X × X → FX (u) (η = 0). If X is invex (at each u ∈ X), then
η : X × X → Rn .
Remark 2.2. Any nonempty open set is invex.
2.2. A representation of an invex set
We have the following result.
Theorem 2.3. If a set X is invex with respect to a vector function
η : X × X → Rn , then it admits the representation
{u + λη(x, u) ∈ X | ∀λ ∈ [0, 1]}.
X=
u∈X x∈X
Proof. We have
{u + λη(x, u) ∈ X | ∀λ ∈ [0, 1]} ⊆ X,
{u + λη(x, u) ∈ X | ∀λ ∈ [0, 1]} ⊆ X.
x∈X
Denote
Xu =
{u + λη(x, u) ∈ X | ∀λ ∈ [0, 1]}
x∈X
to deduce that Xn ⊆ X and
(2.3)
Xu ⊆ X.
u∈X
Let
us that the equality sign holds in (2.3). Assumeon the contrary that
Xu ⊂ X. Then there exists a point z ∈ X \
Xu . Consequently,
u∈X
u∈X
668
Ştefan Mititelu
z ∈
/
Xu . For z ∈ X we have z ∈ Xz ⊆
u∈X
Xu . Therefore, X =
4
u∈X
Xu , that is, a contradiction.
u∈X
2.3. The vector subspace of some invex subsets of Rn
Let arbitrary subsets X and Y of Rn and vector functions η : X × X →
and µ : Y × Y → Rn be given. We define the operations below.
(O1) If X is invex at u ∈ X with respect to η and Y is invex at v ∈ Y
with respect to µ then
Rn
∀(x + y) ∈ X + Y, ∀λ ∈ [0, 1] → (u + v) + λ[η(x, u) + µ(y, v)] ∈ X + Y.
It follows that X + Y is an invex set with respect to the vector function
σ : (X + Y ) × (X + Y ) → Rn ,
σ(x + y, u + v) = η(x, u) + µ(y, v).
(O2) If X is invex at u ∈ X with respect to η, η(x, u) is homogeneous with
respect to u and α ∈ R, then ∀α ∈ R, ∀x ∈ X, ∀λ ∈ [0, 1] → αu + λη(x, αu) ∈
αX. Therefore, αX is an invex set with respect to this η.
We denote by H(Rn ) the family of all subsets of Rn which are invex
with respect to vector functions η(x, u) that are homogeneous in the variable
u. Then we have
Theorem 2.4. The family H(Rn ) is a vector subspace of Rn with respect
to the operations (O1) and (O2).
3. NONSMOOTH INVEX FUNCTIONS ON OPEN SETS
Let A ⊆ Rn be a nonempty open set and a function f : A → R. The
upper Dini directional derivative of f at the point x ∈ A in the direction
v ∈ Rn is defined by
f+ (x; v) = lim sup
λ↓0
f (x + λv) − f (x)
.
λ
Theorem 3.1. (a) If f+ is finite near (x, v), then the function f+ (x; ·)
is positively homogeneous [9]. (b) Assume that f is continuous near x and f+
is finite near (x; v). Then the function f+ (·; v) is upper semicontinuous at x
while the function f+ (x; ·) is upper semicontinuous at v.
Proof. (b) Let a sequence (xk ) ⊂ A, xk →x and λk ↓ 0; f+ (x; v) being
a
upper limit at (x, 0), there exists a sequence δ k1 ↓ 0 such that if λk < δ k1
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Invex sets and nonsmooth invex functions
then
f+ (xk ; v) <
f (xk + λk v) − f (xk ) 1
+ .
λk
k
We have
lim sup f+ (xk ; v) ≤ lim sup
k→∞
669
k→∞
f (xk + λk v) − f (xk )
=
λk
f (xk + λk v) − f (xk )
(= l),
= lim
k→∞
λk
where we used the continuity of f . There also exists the iterated limit
l1 = lim lim
λk ↓0 xk →x
f (xk + λk v) − f (xk )
f (x + λk v) − f (x)
= lim
=
λ
↓0
λk
λk
k
= f (x; v) = f+ (x; v).
We have l = l1 , hence lim sup f+ (xk ; v) = f+ (x; v) [6].
k→∞
The upper semicontinuity of f+ (x; ·) at v can be established in a similar
manner. In what follows we consider vector functions of the form η : A × A → Rn ,
where η = 0.
Definition 3.1 [9]. 1) A function f is said to be invex at u ∈ A if there
exists a vector function η such that
∀x ∈ A :
f (x) − f (u) ≥ f+ (u; η(x, u)).
2) A function f is said to be invex on A (invex for short) if it is invex at
each point of A.
(If f is invex with respect to η, then f is called η-invex for short).
Definition 3.2 [9]. 1) A function f is said to be pseudoinvex at u ∈ A if
there exists a vector function η such that
(3.1)
∀x ∈ A :
f+ (u; η(x, u)) ≥ 0 ⇒ f (x) ≥ f (u).
2) A function f is said to be pseudoinvex on A if it is pseudoinvex at
each of A.
(If f is pseudoinvex with respect to η, then f is called η-pseudoinvex for
short.)
Definition 3.3 [9]. 1) A function f is said to be quasiinvex at u ∈ A if
there exists a vector function η such that
∀x ∈ A :
f (x) ≤ f (u) ⇒ f+ (u; η(x, u)) ≤ 0.
2) A function f is said to be quasiinvex on A if it is quasiinvex at each
point of A.
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Ştefan Mititelu
6
Theorem 3.2. The domain of an invex, preinvex or quasiinvex function
is an invex sets.
Proof. We establish the pseudoinvex variant of the theorem. The function f+ (u; ·) is positively defined. Then, for any t > 0, it follows from (3.1) that
f+ (u; tη(x, u)) = lim sup
λ↓0
f (u + λ t η(x, u)) − f (u)
≥ 0.
λ
Therefore, for all x ∈ A, t > 0 for which 0 ≤ f+ (u; t η(x, u) < ∞ we have
u + λ t η(x, u) ∈ A. Hence u + λ t η(x, u) ∈ A, ∀λ t > 0. Putting µ = λ t,
we obtain in particular that u + µη(x, u) ∈ A, ∀µ ∈ [0, 1], that is, A is an
η-invex set. Definition 3.4 (Komlósi, [7]). A point x0 ∈ A is said to be a stationary
point of a function f if f+ (x0 ; v) ≥ 0, ∀v ∈ Rn .
The main property that caracterizes the differentiable invex functions is
that every stationary point of it is a global minimum and conversely [2].
In a nonsmooth framework, the necessity of this property only holds,
even for pseudoinvex functions (see Theorem 2.9 of [11]). In what follows,
we show that the sufficiency of this property also holds under appropriate
restricted conditions.
Lemma 3.3. Let x0 ∈ A and functions f : A → R and η : A × A → Rn .
Assume that the function η(·, x0 ) is surjective on A. Then x0 is a stationary
point of f if and only if f+ (x0 ; η(x, x0 )) ≥ 0, ∀x ∈ A.
Proof. Equivalently, we should establish the relation
(3.2)
f+ (x0 ; v) ≥ 0, ∀v ∈ Rn ⇔ f+ (x0 ; η(x, x0 )) ≥ 0, ∀x ∈ A.
If η(·, x0 ) is surjective on A, then it is obvious that v ∈ Rn if and only if
there exists η(x, x0 ) ∈ Rn (where x ∈ A) such that v = η(x, x0 ) and so, (3.2)
holds. Theorem 3.4. Assume that for a nonsmooth function f : A → R every
stationary point is a global minimum. Then f is pseudoinvex on the set of its
stationary points.
Proof. If x0 is a stationary point of f , then we have
(3.3)
f+ (x0 , v) ≥ 0, ∀v ∈ Rn ⇒ f (x) ≥ f (x0 ), ∀x ∈ A.
According to Lemma 3.3, if for a vector function η : A × A → Rn the
restriction η(·, x0 ) is a surjective function, then relation (3.2) holds. Moreover,
it follows from (3.2) and (3.3) that
(3.4)
∀x ∈ A :
f+ (x0 ; η(x, x0 )) ≥ 0 ⇒ f (x) ≥ f (x0 ),
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Invex sets and nonsmooth invex functions
671
that is f is η-pseudoinvex at x0 . Such a function η always exists [K. Kuratowski]. 4. NONSMOOTH INVEX FUNCTIONS
ON ARBITRARY SETS
Consider now a function f : X → R, where X is a nonempty set in Rn ,
and let D be an open set in Rn that includes X. Relative to the invexity
of f on X, comparatively with invexity on open sets, a difference appears on
the boundary of X and it refers to η, which must be a homeomorphism. (We
denote the boundary of X by Fr X [1].)
Theorem 4.1. Let Let D be a nonempty open set in Rn , X ⊂ D a
nonempty subset and functions f : D → R and η : X × X → Rn . Moreover,
assume that
1) f is continuous and f+ is finite on D × Rn ;
2) η is a homeomorphism;
3) f is invex on ri X with respect to the restriction
η|ri X [η : ri X × ri X → Rn ].
Then f is invex on X with respect to η.
Proof. As f is η-invex on ri X, we have
∀x, u ∈ ri X :
(4.1)
f (x) − f (u) ≥ f+ (u, η(x, u)).
Let y, b ∈ Fr X. Because η is a homeomorphism we have (x → y, u →
b) ⇔ (u → b, η(x, u) → η(y, b)). Taking lim sup as x → y, u → b in (4.1),
we obtain
f (y) − f (b) ≥
(4.2)
lim sup f+ (u; η(x, u)) (= l).
u→b
η(x,u)→η(y,b)
Also, there exists the iterated limit
l1 = lim sup
u→b
lim sup
η(x,u)→η(y,b)
f+ (u; η(x, u)) = lim sup f+ (u; η(y, b)) = f+ (b; η(y, b)),
u→b
f+ (x, v)
is upper semicontinuous with respect
where we took into account that
to each variable (see Theorem 3.1). Then l = l1 and (4.2) becomes
(4.3)
f (y) − f (b) ≥ f+ (b, η(y, b)).
Finally, considering all the variants of x → y, u → b, we can derive (4.3)
for ∀y, b ∈ X. Theorem 4.2. Let D be a nonempty open set in Rn , X ⊂ D a nonempty
subset and functions f : D → R and η : X × X → Rn . Assume conditions
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Ştefan Mititelu
8
1) and 2) of Theorem 4.1 are satisfied and also the that f is pseudoinvex on
riX with respect to restriction η|ri X . Then f is pseudoinvex on X with respect
to η.
Theorem 4.3. Let D be a nonempty open set in Rn , X ⊂ D a nonempty
subset and functions f : D → R and η : X × X → Rn . Assume conditions 1)
and 2) of Theorem 4.1 are satisfied and also that f is quasiinvex on ri X with
respect to restriction η|ri X . Then f is quasiinvex on X with respect to η.
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Received 1st March 2006
Technical University of Civil Engineering
Department of Mathematics and Computer Science
Bd. Lacul Tei 124
020396 Bucharest, Romania