INVEX SETS
AND NONSMOOTH INVEX FUNCTIONS
S¸TEFAN MITITELU
We give conditions for invexity of sets and show that the family of some invex subsets ofRnis a nontrivial vector subspace ofRn. Also, a representation form for an invex set is given. We show that if for a nonsmooth function ofRn 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 func- tions 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 mini- mum 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 [gen- eralized] invex functions on open and arbitrary (nonempty) sets. It is divided into four sections. Section 1 is an introduction. In Section 2 we establish nec- essary 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
invex set. Also, it is shown that the family of all subsets ofRn that are invex with respect to vector functions of the formη(x, u) only, and are homogeneous inu, is a nontrivial vector subspace of Rn. In Section 3 we show that if for a nonsmooth function onRn 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 setX⊆Rn is said to be aninvex set at u∈X if there is a vector functionη :X×X→Rn such that
(2.1) ∀x∈X, ∀λ∈[0,1]⇒u+λη(x, u)∈X.
2) A setX is called invex if (2.1) holds for allu∈X.
(There is excluded the situation η = 0 [η is the zero function], whenX 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 toX at the pointu. Therefore,
η(x, u)∈FX(u) ={v ∈Rn| ∀λ∈[0,1] :u+λη(v, u)∈X},
whereFX(u) is the cone of feasible directions toX atu. 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 riFX(u) the relative interior [1] ofFX(u). We remark that riGf =∅. If riX=∅ andFX(u) ={0}, then uis an extremal point [1] of X.
In general, ifFX(u) ⊃ {0} thenFX(u) is not a convex cone. IfX is a convex set, thenFX(u) is a convex cone [1].
Theorem 2.2. Suppose that X is a nonempty set and riFX(u) = ∅. ThenX is an invex set at u.
Proof. Obviously,FX(u)⊃ {0}. Consider the pointsu, x∈X. If [u, x]⊂ X then X is invex at u with respect toη defined byη(x, u) = x−u. Now, assume that [u, x]⊂/ X. Since riFX(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.
Remark2.1. IfX 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.
Remark2.2. 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
X=
u∈X
x∈X
{u+λη(x, u)∈X| ∀λ∈[0,1]}.
Proof. We have
{u+λη(x, u)∈X | ∀λ∈[0,1]}⊆X,
x∈X
{u+λη(x, u)∈X| ∀λ∈[0,1]}⊆X.
Denote
Xu =
x∈X
{u+λη(x, u)∈X | ∀λ∈[0,1]}
to deduce that Xn⊆X and
(2.3)
u∈X
Xu ⊆X.
Let us that the equality sign holds in (2.3). Assume on the contrary that
u∈XXu ⊂ X. Then there exists a point z ∈ X \
u∈XXu. Consequently,
z /∈
u∈XXu. For z ∈ X we have z ∈ Xz ⊆
u∈XXu, that is, a contradiction.
Therefore,X =
u∈XXu.
2.3. The vector subspace of some invex subsets of Rn
Let arbitrary subsetsX and Y ofRn and vector functionsη:X×X → Rn andµ: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 toµthen
∀(x+y)∈X+Y, ∀λ∈[0,1]→(u+v) +λ[η(x, u) +µ(y, v)]∈X+Y.
It follows thatX+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) IfXis invex atu∈Xwith respect toη,η(x, u) is homogeneous with respect touandα∈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 functionsη(x, u) that are homogeneous in the variable u. Then we have
Theorem 2.4. The familyH(Rn)is a vector subspace ofRnwith 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 andλk↓0;f+(x;v) being a upper limit at (x,0), there exists a sequenceδ1
k
↓0 such that ifλk < δ1
k
then
f+ (xk;v)< f(xk+λkv)−f(xk)
λk + 1
k. We have
lim sup
k→∞ f+ (xk;v)≤lim sup
k→∞
f(xk+λkv)−f(xk)
λk =
= lim
k→∞
f(xk+λkv)−f(xk) λk (=l),
where we used the continuity off. There also exists the iterated limit l1= lim
λk↓0 lim
xk→x
f(xk+λkv)−f(xk)
λk = lim
λk↓0
f(x+λkv)−f(x)
λk =
=f(x;v) =f+(x;v). We havel=l1, hence lim sup
k→∞ f+(xk;v) =f+(x;v) [6].
The upper semicontinuity off+ (x;·) at vcan be established in a similar
manner.
In what follows we consider vector functions of the formη:A×A→Rn, whereη= 0.
Definition 3.1 [9]. 1) A function f is said to beinvex at u∈ A if there exists a vector functionη such that
∀x∈A: f(x)−f(u)≥f+ (u;η(x, u)).
2) A functionf is said to be invex on A(invexfor short) if it is invex at each point ofA.
(If f is invex with respect toη, thenf is called η-invex for short).
Definition 3.2 [9]. 1) A function f is said to bepseudoinvex at u ∈A if there exists a vector functionη 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 ofA.
(Iff is pseudoinvex with respect to η, thenf is calledη-pseudoinvex for short.)
Definition 3.3 [9]. 1) A function f is said to be quasiinvex at u ∈ A if there exists a vector functionη 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 ofA.
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 func- tionf+ (u;·) is positively defined. Then, for anyt >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)
λ ≥0.
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 (Koml´osi, [7]). A point x0 ∈ A is said to be astationary point of a functionf iff+ (x0;v)≥0,∀v∈Rn.
The main property that caracterizes the differentiable invex functions is thatevery 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 onA. Then x0 is a stationary point off 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.
Ifη(·, x0) is surjective onA, then it is obvious thatv∈Rnif 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. Thenf is pseudoinvex on the set of its stationary points.
Proof. Ifx0 is a stationary point off, 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 restrictionη(·, 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),
that is f is η-pseudoinvex at x0. Such a function η always exists [K. Kura- towski].
4. NONSMOOTH INVEX FUNCTIONS ON ARBITRARY SETS
Consider now a functionf :X→R, where X is a nonempty set inRn, and let D be an open set in Rn that includes X. Relative to the invexity off on X, comparatively with invexity on open sets, a difference appears on the boundary ofX and it refers toη, which must be a homeomorphism. (We denote the boundary ofX by FrX [1].)
Theorem 4.1. Let Let D be a nonempty open set in Rn, X ⊂ D a nonempty subset and functionsf :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 riX with respect to the restriction η|riX[η : riX×riX→Rn]. Thenf is invex on X with respect to η.
Proof. Asf is η-invex on riX, we have
(4.1) ∀x, u∈riX : f(x)−f(u)≥f+(u, η(x, u)).
Let y, b∈ FrX. 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
(4.2) f(y)−f(b)≥ lim sup
η(x,u)→η(y,b)u→b
f+ (u;η(x, u)) (=l). 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
u→b f+ (u;η(y, b)) =f+ (b;η(y, b)), where we took into account thatf+(x, v) is upper semicontinuous with respect to each variable (see Theorem 3.1). Thenl=l1 and (4.2) becomes
(4.3) f(y)−f(b)≥f+ (b, η(y, b)).
Finally, considering all the variants ofx→y,u→b, we can derive (4.3) for∀y, b∈X.
Theorem 4.2. LetDbe a nonempty open set inRn,X⊂Da 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 the that f is pseudoinvex on riX with respect to restriction η|riX. Thenf is pseudoinvex onX with respect toη.
Theorem 4.3. LetDbe a nonempty open set inRn,X⊂Da nonempty subset and functions f :D→R and η:X×X →Rn. Assume conditions 1) and2)of Theorem 4.1 are satisfied and also that f is quasiinvex on riX with respect to restrictionη|riX. Thenf 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
