Journal of Applied Analysis Vol. 13, No. 2 (2007), pp. 183–196
ON STRICT PSEUDOCONVEXITY V. I. IVANOV Received March 29, 2005 and, in revised form, June 7, 2006
Abstract. The present paper provides first and second-order characterizations of a radially lower semicontinuous strictly pseudoconvex function f : X → R defined on a convex set X in the real Euclidean space Rn in terms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived.
1. Introduction Strictly pseudoconvex (in short, s.p.c.) functions play important role in global optimization. They were introduced by Ponstein [21] employing the Mangasarian’s definition of pseudoconvexity. It is well known that a local minimizer of a s.p.c. function over some convex set is the unique global one (see for example, Bazaraa, Shetty [3, Theorem 3.5.9]). Important characterizations of the s.p.c. functions are obtained by Diewert, Avriel, Zang 2000 Mathematics Subject Classification. 26B25, 90C26, 47J20, 47H05. Key words and phrases. Generalized convexity, strict pseudoconvexity, nonsmooth analysis, nonsmooth optimization, Dini-directional derivatives.
ISSN 1425-6908
c Heldermann Verlag.
184
V. I. IVANOV
[6, Theorem 11], Karamardian, Schaible [11, Proposition 4.1], Komlosi [14, Proposition 2.2, Theorem 3.4], Ponstein [21, Theorem 1], Schaible, Zang [23, Section 2]. Some characterizations are derived in terms of generalized directional derivatives (see Komlosi [16, Theorem 4.3 (iii)]). The s.p.c. quadratic functions are investigated by Avriel, Schaible [2] and Schaible [22, 24]. Other results are generalizations of the differentiable case (see for example, Luc [18]). In some other papers only sufficient conditions for strict pseudoconvexity are given (see for example, Crouzeix [5], Ginchev, Ivanov [8], Komlosi [14], Mereau, Paquet [19]). However, there is still too much to be done in this direction. Our aim in this paper is an attempt to fill a part of the existing gap. We derive some characterizations of the strictly pseudoconvex functions in terms of the lower (upper) Dini-directional derivatives. The Dini-directional derivatives are important tool in nonsmooth analysis. It would be worth emphasizing that the first use of the Dini derivatives in quasiconvex analysis goes back to Diewert [7], Crouzeix [5]. Diewert extended Mangasarian’s (strict) pseudoconvexity concept using the lower Dini derivative. It embraces a wider class of functions than the upper Dini pseudoconvexity. An important step in the use of generalized directional derivatives in generalized convexity (in particular pseudoconvexity) is the work of Penot [20]. The upper Dini derivative has a remarkable property for quasiconvex functions. In this case the upper Dini derivative is a quasiconvex positively homogeneous function of the direction. On the basis of this property, Komlosi introduced and investigated the class of generalized upper quasidifferentiable functions [13]. The generalized monotonicity properties of the Dini derivatives were investigated first by Komlosi, including the case of strict pseudoconvexity [15, Theorem 4 and Theorem 6]. The paper is organized as follows. In Section 2 we obtain some necessary and sufficient conditions for strict pseudoconvexity concerning other classes of generalized convex functions. In Section 3 we obtain a characterization which is similar and weaker than the well known second-order one due to Diewert, Avriel, Zang [6]. Moreover, we replace one of the conditions of this characterization with an implication which is easier to be checked. In Section 4 we receive criteria for strict pseudoconvexity which are related to the nonlinear programming problem and variational inequality problems.
ON STRICT PSEUDOCONVEXITY
185
2. Connections of the strictly pseudoconvex functions with other classes of functions In this section we derive necessary and sufficient conditions for strict pseudoconvexity which are connected with other classes of generalized convex functions. Throughout the work we denote by Rn a real finite-dimensional Euclidean space and f : X → R is a finite-valued real function defined on a set X ⊂ Rn . Here R is the set of the reals and R = R ∪ {+∞} ∪ {−∞}. Let f¯: Rn → R be the extension of f such that f¯(x) = +∞ for x ∈ Rn \ X. The lower Dini-directional derivative f−0 (x, u) of f at x ∈ X = dom f¯ in direction u ∈ Rn is defined as an element of R by f−0 (x, u) = lim inf t→+0
f¯(x + tu) − f (x) . t
The difference in the right-hand side is well defined, since only the term f¯(x + tu) eventually takes an infinite value. Applying f¯ instead of f in this definition one gains the convenience to handle easily the possibility x + tu ∈ / X. If the set X is open we meet only with the situation when x + tu ∈ X for all sufficiently small positive t. Obviously in this case the definition of the Dini derivative works with f (x + tu) instead of f¯(x + tu). Recall the following well-known notions: A function f , defined on a set X ⊂ Rn , is said to be lower Dini (strictly) pseudoconvex on X if x, y ∈ X, f (y) < f (x) imply f−0 (x, y − x) < 0 (x, y ∈ X, f (y) ≤ f (x), x 6= y imply f−0 (x, y − x) < 0). For brevity we omit the words “lower Dini” saying simply “pseudoconvex” or “strictly pseudoconvex” function respectively. A function f , defined on a convex set X, is said to be (strictly) quasiconvex on X if for all t ∈ (0, 1) and x, y ∈ X f (y) ≤ f (x) implies f (x + t(y − x)) ≤ f (x). (f (y) ≤ f (x), y 6= x imply f (x + t(y − x)) < f (x)).
A function f : X → R is said to be radially lower (upper) semicontinuous if the function of one variable ϕ : X(x, y) → R, ϕ(t) = f (x + t(y − x)) where X(x, y) = {t ∈ R | x + t(y − x) ∈ X} is lower (upper) semicontinuous for all x, y ∈ X. We use also the abbreviations r.l.s.c. (r.u.s.c.) for words radially lower semicontinuous (radially upper semicontinuous). A function is said to be radially continuous if it is both r.l.s.c. and r.u.s.c. A function f is said to be radially nonconstant on the convex set X [16, 1] if there is no line segment [x, y] in X along which f is constant.
186
V. I. IVANOV
A point x ∈ X is said to be an inf-stationary point of the function f over X with respect to the lower Dini-directional derivative if the condition f−0 (x, d) ≥ 0 is fulfilled at x for every direction d ∈ Rn .
Theorem 1. Let X ⊂ Rn be a convex set and f : X → R r.l.s.c. function. Then f is s.p.c. on X if and only if f is strictly quasiconvex on X and it obey the following property: For any distinct x, y ∈ X such that f (y) ≤ f (x) the number t = 0 is not an inf-stationary point of the function of one variable ϕ : X(x, y) → R, ϕ(t) = f (x + t(y − x)), where X(x, y) = {t ∈ R | x + t(y − x) ∈ X}.
Proof. Let f be s.p.c. We show that f is strictly quasiconvex. Assume in the contrary that there exist x, y ∈ X, x 6= y, f (y) ≤ f (x) and z ∈ (x, y) such that f (x) ≤ f (z). By strict pseudoconvexity f−0 (z, x − z) < 0 and f−0 (z, y − z) < 0. Therefore there are x1 ∈ (x, z) and y1 ∈ (y, z) with f (x1 ) < f (z), f (y1 ) < f (z). On the other hand it is well-known that every r.l.s.c. pseudoconvex function is quasiconvex on each convex set (see Diewert [7]). By quasiconvexity we have f (z) ≤ max{f (x1 ), f (y1 )} which is a contradiction. Let x, y ∈ X, x 6= y, f (y) ≤ f (x). Consider the function ϕ. By strict pseudoconvexity f−0 (x, y − x) < 0 which implies that t = 0 is not an infstationary point of ϕ. Let f be strictly quasiconvex and ϕ possesses the mentioned property. We show that f is s.p.c. Take any x, y ∈ X with f (y) ≤ f (x) and x 6= y. Since X is convex, then the closed interval [0, 1] belongs to the domain of ϕ. If t = 0 is boundary of the domain of ϕ, then ϕ0− (0, −1) = +∞. By the hypothesis of the theorem ϕ0− (0, 1) < 0. Consider the case when t = 0 is an interior point of the domain of ϕ. By strict quasiconvexity ϕ(0) < max{ϕ(1), ϕ(−t)} for all sufficiently small values t > 0. Using that ϕ(1) ≤ ϕ(0) we get the inequality ϕ(−t) > ϕ(0). Therefore ϕ0− (0, −1) ≥ 0. Since t = 0 is not an inf-stationary point, then ϕ0− (0, 1) < 0. Thus, f is s.p.c. We cannot drop the assumption f is r.l.s.c. in Theorem 1 as the following example shows: Example 1. Consider the function f : [−1, 1] → R, defined by f (x) = x2 when x ∈ [−1, 0) ∪ (0, 1], and f (x) = 1 when x = 0. It is s.p.c., but f is not strictly quasiconvex. It is immediately seen that f is not l.s.c. at x = 0. Theorem 2. Let X ⊂ Rn be a convex set and f : X → R pseudoconvex on X r.l.s.c. function. Then f is s.p.c. on X if and only if it is radially nonconstant.
ON STRICT PSEUDOCONVEXITY
187
Proof. Let f be radially nonconstant. We show that it is s.p.c. Assume that x, y ∈ X, y 6= x with f (y) ≤ f (x). If f (y) < f (x), then it follows from pseudoconvexity that f−0 (x, y − x) < 0. By radial lower semicontinuity f is quasiconvex on X. Therefore f (x + t(y − x)) ≤ f (x) for all t ∈ [0, 1]. It follows from radial nonconstancy that there exist z ∈ (x, y) with f (z) < f (x). According to pseudoconvexity and the positive homogeneity of Dini derivatives with respect to the direction we have f−0 (x, y − x) < 0. The inverse claim follows directly from the definition of strict pseudoconvexity. The following example shows that the radial lower semicontinuity plays an essential role in Theorem 2. Example 2. Consider the function f : R → R defined by f (x) = 0 if x is rational, and f (x) = 1 if x is irrational. It is obvious that for every u ∈ R we have f−0 (x, u) = 0 when x is rational, and f−0 (x, u) = −∞ when x is irrational. Therefore f is pseudoconvex, but it is not strictly pseudoconvex. On the other hand f is radially nonconstant. Theorem 2 is similar, but different from a claim due to Aussel [1, Proposition 5.2]. We may compare Theorems 1 and 2 with the following well-known property of the s.p.c. differentiable functions due to Ponstein [21, Theorem 1]: A function f is s.p.c. on an open set if and only if it is both strictly quasiconvex and pseudoconvex. 3. A second-order characterization of the strictly pseudoconvex functions The starting point of this section are the first and second-order characterizations due to Diewert, Avriel, Zang [6, Theorem 11, Corollary 11.1]. Proposition 1. Let f be a directionally differentiable function defined on a set X ⊂ Rn . Then f is s.p.c. on X if and only if x ∈ X, uT u = 1, t¯ > 0, x + t¯u ∈ X, f 0 (x, u) = 0 implies g(t) ≡ f (x + tu) attains a strict local minimum at t = 0. Proposition 2. A twice continuously differentiable function f defined on an open set X ⊂ Rn is s.p.c. if and only if x ∈ X, uT u = 1, uT ∇f (x) = 0 implies (i) uT ∇2 f (x)u > 0 or (ii) uT ∇2 f (x)u = 0 and g(t) ≡ f (x + tu) attains a strict local minimum at t = 0.
188
V. I. IVANOV
We make Proposition 2 weaker and generalize it. Consider the secondorder lower Dini-directional derivative (see for example, Ginchev, Ivanov [8]) of f at x ∈ X in direction u ∈ Rn which is defined as follows f−00 (x, u) = lim inf 2t−2 (f¯(x + tu) − f (x) − tf−0 (x, u)) t→+0
supposing that it exists if f−0 (x, u) is finite. In the case of an infinite f−0 (x, u) the derivative f−00 (x, u) is not considered. Theorem 3. Let X ⊂ Rn be a convex set and f : X → R radially continuous function. (First-Order Conditions) Then f is s.p.c. if and only if the following implications hold for all x, y ∈ X, y 6= x, u ∈ Rn f−0 (x, u) ≤ 0, f−0 (x, −u) ≤ 0 f−0 (x, y − x) = 0
implies
imply
max{f−0 (x, u), f−0 (x, −u)} = 0;
∃τ ∈ (0, 1] : f (x + τ (y − x)) > f (x).
(1)
(Second-Order Conditions) The function f is s.p.c. if and only if the following implications hold for all x, y ∈ X, y 6= x, u ∈ Rn f−0 (x, u) ≤ 0, f−0 (x, −u) ≤ 0 imply max{f−0 (x, u), f−0 (x, −u)} = 0; f−0 (x, y f−0 (x, y
− x) = 0 − x) =
implies
f−00 (x, y
f−00 (x, y
− x) = 0
− x) ≥ 0;
(2) (3)
imply
∃τ ∈ (0, 1] : f (x + τ (y − x)) > f (x).
(4)
Proof. We prove only the second-order conditions because the proof of the first-order ones can be obtained as their simplification. Let f be s.p.c. We show implication (2). Since f is pseudoconvex by radial lower semicontinuity f is quasiconvex. Arguing by contradiction assume that there exist x ∈ X and u ∈ Rn such that f−0 (x, u) < 0 and f−0 (x, −u) < 0. Therefore there are t1 > 0 and t2 > 0 with f (x+t2 u) < f (x) and f (x − t1 u) < f (x). The last conclusions are contrary to the quasiconvexity. We show the implications (3) and (4). Let x, y ∈ X, y 6= x, and f−0 (x, y − x) = 0. Since f−0 (x, t(y − x)) = 0 for all t > 0 by strict pseudoconvexity f (x + t(y − x)) > f (x) for all t ∈ (0, 1]. Therefore implications (3) and (4) have place. Conversely, assume that the implications (2), (3), and (4) hold. First, we show that f is quasiconvex. Assume in the contrary that there exist x, y ∈ X and z in the line segment (x, y) such that f (z) > f (x) and f (z) > f (y). By Weierstrass Theorem we may suppose that the maximal value of f over
ON STRICT PSEUDOCONVEXITY
189
[x, y] is attained at z. Hence f−0 (z, x − z) ≤ 0 and f−0 (z, y − z) ≤ 0. Using the positive homogeneity of the Dini derivative, by implication (2) we have max{f−0 (z, x − z), f−0 (z, y − z)} = 0.
Without loss of generality we assume that f−0 (z, x − z) = 0. By the maximality of z we obtain that f−00 (z, x − z) ≤ 0. Taking into account implication (3) we get that f−00 (z, x − z) = 0. By implication (4) there exists τ ∈ (0, 1] such that f (z + τ (x − z)) > f (z) which contradicts the maximality of z. Consequently f is quasiconvex. Second, we show that f is s.p.c. Assume in the contrary that there exist x, y ∈ X, y 6= x such that f (y) ≤ f (x) and f−0 (x, y − x) ≥ 0. By quasiconvexity f (x + t(y − x)) ≤ f (x) for all t ∈ [0, 1]. Therefore f−0 (x, y − x) = 0. By quasiconvexity we receive that f−00 (x, y − x) ≤ 0. It follows from implication (3) that f−00 (x, y − x) = 0. According to implication (4) there exists τ ∈ (0, 1] with f (x + τ (y − x)) > f (x) which is impossible by quasiconvexity. The following examples show that the assumption f is radially continuous in Theorem 3 cannot be dropped and replaced by f is r.u.s.c. or f is r.l.s.c. Example 3. Consider the function f : [−1, 1] → R defined by f (x) = x3 when 0 ≤ x ≤ 1 and f (x) = x + 1 when −1 ≤ x < 0. f fulfills the first-order conditions and the second-order ones because there does not exist x ∈ [−1, 1] and u ∈ R such that f−0 (x, u) ≤ 0 and f−0 (x, −u) ≤ 0. The implications (1), (3), (4) are satisfied too. This function is not s.p.c. Indeed, if we take x = −1/2 and y = 0, then f (y) < f (x), f−0 (x, y − x) = 1/2 > 0 and f is not u.s.c. at x = 0. Consider the function from Example 1. It is s.p.c., but f does not satisfy the implication (2) at x = 0 because f−0 (0, 1) = −∞ and f−0 (0, −1) = −∞. f is not l.s.c. at x = 0. 4. On nonlinear programming problem and variational inequalities In this section we derive some characterizations connected with the nonlinear programming problem and the variational inequalities. Consider the global minimization problem min f (x)
subject to
x∈Y
(P)
where Y ⊂ Rn . Denote the set of its global minimizers by GM (f, Y ) and the set of its strict global minimizers by SGM (f, Y ). It is obvious that the last set is empty or contains an unique element.
190
V. I. IVANOV
Theorem 4. Let f be a given r.l.s.c. function which is pseudoconvex on a convex set X. Then f is s.p.c. on X if and only if (P) has at most one solution for all convex subsets Y ⊆ X. Proof. Assume that f is s.p.c. on X and Y ⊆ X is convex. We show that (P) has at most one solution. By radial lower semicontinuity, f is quasiconvex. The sublevel sets of a quasiconvex function are convex. Hence the solution set of (P) is convex. Assume that there are distinct global minimizers x1 and x2 . According to the convexity of the solution set f (x1 + t(x2 − x1 )) = f (x1 ) for all t ∈ [0, 1]. Therefore f−0 (x1 , x2 − x1 ) = 0 which contradicts the definition of strict pseudoconvexity since x2 6= x1 . Let (P) have at most one solution for all convex subsets Y ⊆ X. Assume that x, y are arbitrary distinct points from X with f (y) ≤ f (x). We show that f−0 (x, y − x) < 0. If f (y) < f (x), then the claim follows from pseudoconvexity. Let f (y) = f (x). It follows from Weierstrass Theorem that f has an unique global minimizer over the segment [x, y]. Then there exists z ∈ (x, y) such that f (z) < f (x). By pseudoconvexity f−0 (x, z − x) < 0. Using that lower Dini derivative is positively homogeneous function of the direction, we obtain that f−0 (x, y − x) < 0. The following example shows that the assumption f to be pseudoconvex is not superfluous. Example 4. Consider the function f : R → R defined by f (x) = x3 . It has at most one global minimizer over all convex subsets of R, but f is not s.p.c. We see that this function is strictly quasiconvex, but it is not pseudoconvex at the point x = 0. The following similar result does not require the pseudoconvexity of the function. Proposition 3. Let f be a given radially continuous function defined on a convex set X. Then f is strictly quasiconvex on X if and only if (P) has at most one solution for all convex subsets Y ⊆ X. Proof. Let f be strictly quasiconvex on X. It follows directly from the definition of strict quasiconvexity that (P) has at most one solution for all convex subsets Y ⊆ X. Let (P) have at most one solution for all convex subsets Y ⊆ X. We show that f is strictly quasiconvex on X. Assume in the contrary that there exist x, y ∈ X and u ∈ (x, y) such that f (u) ≥ max{f (x), f (y)}. By Weierstrass Theorem, f attains its global minimum over [u, x] and [u, y] at some points x1 and y1 respectively. Without loss of generality f (y1 ) ≤ f (x1 ). Since
ON STRICT PSEUDOCONVEXITY
191
y1 6= u, x1 6= u by radial continuity there exist y2 ∈ (u, y1 ] with f (x1 ) = f (y2 ) and y2 is the nearest such point of (u, y1 ] to u. Therefore the points x1 , y2 are global minimizers of f over [x1 , y2 ] which contradicts the hypothesis. The following examples show that Proposition 3 does not hold when f is only r.l.s.c. or r.u.s.c. Example 5. Consider the function f : [1, 3] → R defined by f (x) = x when x ∈ [1, 2) ∪ (2, 3] and f (2) = 0. It is not continuous, but f is l.s.c. This function has an unique global minimizer over every interval [u, v], u, v ∈ [1, 3], but f is not strictly quasiconvex. Indeed f (3/2) > max{f (1), f (2)}. Example 6. Consider the function f : [1, 3] → R defined by f (x) = x when x ∈ [1, 2) ∪ (2, 3] and f (2) = 4. It is not continuous, but f is u.s.c. This function has at most one global minimizer over every interval [u, v], u, v ∈ [1, 3], but f is not strictly quasiconvex. Indeed f (2) > max{f (1), f (3)}. Consider the following Stampacchia variational inequality problem: Find
x ¯∈X
such that f−0 (¯ x, x − x ¯) ≥ 0,
∀x ∈ X.
Denote its solution set by S(f, X). We consider the case when X is convex. It is worth pointing to the fact that any solution of the Stampacchia variational inequality problem is nothing else than an inf-stationary point of f over X, since f−0 (¯ x, x − x ¯) = +∞ > 0 when x ∈ / X. Theorem 5. Let f be a function defined on a convex set X ⊂ the following claims are equivalent: (i) f is s.p.c. on X; (ii) S(f, Y ) ≡ SGM (f, Y ) for all convex subsets Y ⊆ X; (iii) S(f, [x, y]) ≡ SGM (f, [x, y]) for all x, y ∈ X.
Rn.
Then
Proof. Implication (i) ⇒ (ii). Assume that Y is arbitrary convex subset of X. The inclusion SGM (f, Y ) ⊆ S(f, Y ) is trivial and it holds whatever is the function f : strictly pseudoconvex or not. The converse inclusion S(f, Y ) ⊆ SGM (f, Y ) follows from the strict pseudoconvexity. The implication (ii) ⇒ (iii) is trivial. To show the implication (iii) ⇒ (i) assume that S(f, [x, y]) ≡ SGM (f, [x, y]) for all x, y ∈ X. Let f−0 (x, y − x) ≥ 0 for some distinct x, y ∈ X. We show that f (y) > f (x). By the positive homogeneity of the Dini derivative f−0 (x, z − x) ≥ 0 for all z ∈ (x, y]. Since f−0 (x, x − x) = 0 we have that x ∈ S(f, [x, y]). Therefore x ∈ SGM (f, [x, y]). In particular f (y) > f (x).
192
V. I. IVANOV
Consider the following strict Minty variational inequality problem: Find x ¯∈X
such that
¯ − x) < 0, f−0 (x, x
∀x ∈ X, x 6= x ¯.
Denote its solution set by M (f, X). Any solution of the Minty variational inequality is a global minimizer of f over X. This result is due to Komlosi [17, Theorem 2.2] in the case of a differentiable function f . For nondifferentiable functions it is generalized by Crespi, Ginchev, Rocca [4, Theorem 2.1] with the use of the lower Dini derivative. Consider a function f defined on a set X and a point x ¯ ∈ X. Denote z(t) = x ¯ + tu, x ¯ + tu ∈ X where u is a fixed direction. The function f is said to be strictly increasing along rays starting at x ¯ if f (z(t2 )) > f (z(t1 )) for all t2 > t1 ≥ 0 such that x ¯ + t1 u ∈ X, x ¯ + t2 u ∈ X for any direction u. Diewert’s Mean-Value Theorem ([7]). Let X ⊂ Rn be a convex set and f : X → R r.l.s.c. function. Then for all x, y ∈ X there exist ξ ∈ (x, y] such that f−0 (ξ, x − y) ≥ f (x) − f (y). The following claim slightly differs from Crespi, Ginchev, Rocca [4, Theorems 2.1 and 2.2]. They do not consider the strict case. Our proof follows the scheme given there, hidden behind Diewert’s Mean-Value Theorem. Theorem 6. Let X ⊂ Rn be a convex set and f : X → R r.l.s.c. function. If x ¯ ∈ M (f, X), then f is strictly increasing along rays starting at x ¯ and x ¯ is a strict global minimizer of f over X. Proof. Let x ∈ X be fixed. Denote z(t) = x ¯ + t(x − x ¯), 0 ≤ t ≤ 1. Using that x ¯ is a solution of the strict Minty variational inequality we have f−0 (z(t), x ¯ − z(t)) < 0 for all t ∈ (0, 1]. Assume that 0 ≤ t1 < t2 ≤ 1. We show that f (z(t1 )) < f (z(t2 )). According to Diewert’s Mean-Value Theorem there exists t0 ∈ (t1 , t2 ] such that f (z(t1 )) − f (z(t2 )) ≤ f−0 (z(t0 ), z(t1 ) − z(t2 )) = (t2 − t1 )f−0 (z(t0 ), x ¯ − x) < 0.
Therefore f (z(t1 )) < f (z(t2 )). Taking t1 = 0 and t2 = 1 we receive that f (¯ x) < f (x). Hence x ¯ is a strict global minimizer. Taking into consideration that x, t1 , t2 are arbitrary we obtain that f is strictly increasing along rays starting at x ¯. Example 1 shows that Theorem 6 fails if f is not r.l.s.c. The point x = 0 belongs to M (f, [−1, 1]), but it is not a global minimizer.
ON STRICT PSEUDOCONVEXITY
193
Recall that the lower Dini derivative f−0 of the function f is said to be strictly pseudomonotone (in short, s.p.m.) if the following implication holds x, y ∈ X, x 6= y, f−0 (x, y − x) ≥ 0 imply f−0 (y, x − y) < 0. The following theorem is due to Komlosi [12, 15]. Theorem 7. Let the real-valued function f be defined on the convex set X ⊂ Rn and r.l.s.c. Then f is s.p.c. on X if and only if the lower Dini derivative of f is s.p.m. In the proofs of the following theorems we use Theorems 6 and 7. Theorem 8. Let X ⊂ Rn be a convex set and f : X → R be a r.l.s.c. function. Then the following claims are equivalent: (i) f is s.p.c. on X; (ii) M (f, Y ) ≡ S(f, Y ) for all convex subsets Y ⊆ X; (iii) M (f, [x, y]) ≡ S(f, [x, y]) for all x, y ∈ X; (iv) M (f, [x, y]) contains an unique element for all x, y ∈ X; (v) S(f, [x, y]) contains an unique element for all x, y ∈ X; (vi) M (f, Y ) contains at most one element for all convex Y ⊆ X; (vii) S(f, Y ) contains at most one element for all convex Y ⊆ X. Proof. Implication (i) ⇒ (ii). Assume that f is s.p.c. Suppose that Y is any convex subset of X. We show that S(f, Y ) ⊆ M (f, Y ). Assume in the contrary that there exists x ¯ ∈ S(f, Y ), but x ¯∈ / M (f, Y ). Therefore there is x ∈ X, x 6= x ¯ such that f−0 (¯ x, x − x ¯) ≥ 0,
f−0 (x, x ¯ − x) ≥ 0.
By Theorem 7, f−0 is s.p.m. Then it follows from f−0 (x, x ¯ − x) ≥ 0 that f−0 (¯ x, x − x ¯) < 0 which is a contradiction. To show the inclusion M (f, Y ) ⊆ S(f, Y ) we do not need the strict pseudoconvexity. Let x ¯ be any element from M (f, Y ). By Theorem 6, x ¯ is a strict global minimizer of f over X. It is obvious that any global minimizer is a solution of the Stampacchia variational inequality problem. The implication (ii) ⇒ (iii) is trivial. To show the implication (iii) ⇒ (iv) assume that x and y are arbitrary distinct points from X. According to Weierstrass Theorem, f assumes its global minimal value over the closed segment [x, y] at some point u. According to the minimality u ∈ S(f, [x, y]) which implies that M (f, [x, y]) 6= ∅. Assume in the contrary that M (f, [x, y]) contains at least two points x1 and x2 . It follows from S(f, [x, y]) ≡ M (f, [x, y]) and x1 ∈ S(f, [x, y]) that f−0 (x1 , x2 − x1 ) ≥ 0 which contradicts the inclusion x2 ∈ M (f, [x, y]).
194
V. I. IVANOV
Implication (iv) ⇒ (i). Let x, y ∈ X, y 6= x, and f−0 (x, y − x) ≥ 0. We show that f−0 (y, x − y) < 0. Using the positive homogeneity of Dini derivatives we obtain from f−0 (x, y − x) ≥ 0 that f−0 (x, z − x) ≥ 0 for all z ∈ (x, y]. Therefore any z ∈ (x, y] is not included in M (f, [x, y]). Since M (f, [x, y]) contains an unique element we receive that x ∈ M (f, [x, y]). Hence f−0 (y, x − y) < 0. Thus f−0 is s.p.m. According to Theorem 7, f is s.p.c. The equivalences (i) ⇔ (v), (i) ⇔ (vi), (i) ⇔ (vii) are obvious. Similar to some of the characterizations in Theorem 8 are obtained by John [9] in the case of continuous mapping, and [10] in the case of multivalued one. Theorem 9. Let X ⊂ Rn be a convex set and f : X → R r.l.s.c. function. Then the following claims are equivalent: (i) f is s.p.c. on X; (ii) M (f, Y ) ≡ GM (f, Y ) for all convex subsets Y ⊆ X; (iii) M (f, [x, y]) ≡ GM (f, [x, y]) for all x, y ∈ X; (iv) M (f, Y ) ≡ SGM (f, Y ) for all convex subsets Y ⊆ X; (v) M (f, [x, y]) ≡ SGM (f, [x, y]) for all x, y ∈ X. Proof. Implication (i) ⇒ (ii). Let Y be any convex subset of X. It follows from Theorem 6 that M (f, Y ) ⊆ SGM (f, Y ). By the strict pseudoconvexity S(f, Y ) ⊆ M (f, Y ). Using that SGM (f, Y ) ⊆ GM (f, Y ) ⊆ S(f, Y ) we obtain that M (f, Y ) ≡ GM (f, Y ). The implication (ii) ⇒ (iii) is trivial. To show the implication (iii) ⇒ (i) assume that x, y be any points from X with f (y) ≤ f (x), y 6= x. According to Weierstrass Theorem, f attains its global minimal value on the closed segment [x, y] at least at some point z. By f (y) ≤ f (x) we can suppose that z 6= x. Therefore z ∈ M (f, [x, y]) which implies that f−0 (x, z−x) < 0. By the positive homogeneity of the Dini derivative f−0 (x, y − x) < 0. Hence f is s.p.c. The equivalence (i) ⇔ (iv) follows from (i) ⇔ (ii) and Theorem 8 ((i) ⇔ (vi)). The equivalence (i) ⇔ (v) follows from (i) ⇔ (iii) and Theorem 8 ((i) ⇔ (iv)). The assumption f r.l.s.c. is essential in Theorems 8 and 9 as Example 1 shows. It is easy to see that S(f, [−1, 1]) = GM (f, [−1, 1]) = SGM (f, [−1, 1]) = ∅, but M (f, [−1, 1]) = {0}. It is clear that f is not l.s.c. It is easy to see that all results hold when we use the upper Dinidirectional derivative instead of the lower one.
ON STRICT PSEUDOCONVEXITY
195
Acknowledgements. The author is grateful to the referees for their helpful comments.
References [1] Aussel, D., Subdifferential properties of quasiconvex and pseudoconvex functions: Unified approach, J. Optim. Theory Appl. 97(1) (1998), 29–45. [2] Avriel, M., Schaible, S., Second-order characterizations of pseudoconvex functions, Math. Program. 14 (1978), 170–185. [3] Bazaraa, M. S., Shetty, C. M., Nonlinear Programming. Theory and Algorithms, John Wiley & Sons, New York, 1979. [4] Crespi, G.P., Ginchev, I., Rocca, M., Minty variational inequalities, increase along rays property and optimization, J. Optim. Theory Appl. 123(3) (2004), 479–496. [5] Crouzeix, J.-P., Contributions ` a l´etude des fonctiones quasiconvexes, Th`ese d’Etat, U.E.R. des Sciences Exactes et Naturelles, Universit´e de Clermont-Ferrand II, 1977. [6] Diewert, W. E., Avriel, M., Zang, I., Nine kinds of quasiconcavity and concavity, J. Econom. Theory 25 (1981), 397–420. [7] Diewert, W. E., Alternative characterizations of six kinds of quasiconvexity in the nondifferentiable case with applications to nonsmooth programming, in: “Generalized Concavity in Optimization and Economics”, Academic Press, New York, 1981, 51–95. [8] Ginchev, I., Ivanov, V. I., Second-order characterizations of convex and pseudoconvex functions, J. Appl. Anal. 9(2) (2003), 261–273. [9] John, R., Variational inequalities and pseudomonotone functions: some characterizations, in: “Generalized Convexity, Generalized Monotonicity”, Nonconvex Optim. Appl. 27 (1998), Kluwer Academic Publisher, Dordrecht, 291–301. [10] John, R., A note on Minty variational inequalities and generalized monotonicity, in: “Generalized Convexity and Generalized Monotonicity”, Lecture Notes in Econom., and Math. Systems 502 (2001), Springer, Berlin, 240–246. [11] Karamardian, S., Schaible, S., Seven kinds of monotone maps, J. Optim. Theory Appl. 66(1) (1990), 37–46. [12] Koml´ osi, S., Generalized monotonicity of generalized derivatives, in: “Generalized Concavity for Economic Applications”, Proc. of the Workshop on generalized concavity for economic applications (Pisa, 1992), 1–7. [13] Koml´ osi, S., On generalized upper quasidifferentiability, in: “Nonsmooth Optimization: Methods and Applications”, Gordon and Breach, London, 1992, 263–275. [14] Koml´ osi, S., On pseudoconvex functions, Acta Sci. Math. (Szeged) 57 (1993), 569– 586. [15] Koml´ osi, S., Generalized monotonicity in non-smooth analysis, in: “Generalized Convexity”, Springer Verlag, Heidelberg, 1994, 263–275. [16] Koml´ osi, S., Generalized monotonicity and generalized convexity, J. Optim. Theory Appl. 84(2) (1995), 361–376. [17] Koml´ osi, S., On the Stampacchia and Minty variational inequalities, in: “Generalized Convexity and Optimization for Economics and Financial Decisions”, Pitagora Editrice, Bologna, 1999, 231–260. [18] Luc, D. T., On generalized convex nonsmooth functions, Bull. Austral. Math. Soc. 49(1) (1994), 139–149. [19] Mereau, P., Paquet, J.-G., Second order conditions for pseudoconvex functions, SIAM J. Appl. Math. 27(1) (1974), 131–137.
196
V. I. IVANOV
[20] Penot, J.-P., Generalized convexity in the light of nonsmooth analysis, in: “Recent Developments in Optimization”, Lecture Notes in Econom., and Math. Systems 429 (1995), Springer Verlag, Berlin, 269–290. [21] Ponstein, J., Seven kinds of convexity, SIAM Rev. 9(1) (1967), 115–119. [22] Schaible, S., Second-order characterizations of pseudoconvex quadratic functions, J. Optim. Theory Appl. 21(1) (1977), 15–26. [23] Schaible, S., Zang, I., On the convexifability of pseudoconvex C 2 -functions, Math. Program. 19 (1980), 289–299. [24] Schaible, S., Quasiconvex, pseudoconvex, and strictly pseudoconvex quadratic functions, J. Optim. Theory Appl. 35(3) (1981), 303–338.
Vsevolod I. Ivanov Department of Mathematics Technical University of Varna 9010 Varna, Bulgaria
[email protected] 
