Invariant Pseudolinearity with Applications - Springer Link
J Optim Theory Appl (2012) 153:587–601 DOI 10.1007/s10957-011-9979-z
Invariant Pseudolinearity with Applications Qamrul Hasan Ansari · Mahboubeh Rezaei
Received: 26 April 2011 / Accepted: 1 December 2011 / Published online: 16 December 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we introduce the notion of invariant pseudolinearity for nondifferentiable and nonconvex functions by means of Dini directional derivatives. We present some characterizations of invariant pseudolinear functions. Some characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, pseudolinear program are obtained. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs. Keywords Invex sets · Pseudoinvexity · Invariant pseudolinearity · Dini directional derivatives · Nonsmooth variational-like inequalities · Solution sets of a program 1 Introduction The natural generalization of linearity is the pseudolinearity. It plays an important role in mathematical programming; see, for example, [1–5] and the references therein. The class of pseudolinear functions includes many useful functions such as linear fractional functions [6, 7]. Recently, Lalitha and Mehta [8, 9] generalized the notion of pseudolinearity to nondifferentiable functions and obtained characterizations for such functions. They extended the results given by Chew and Choo [1] and Komlósi [4] for differentiable functions. Under the assumption of pseudolinearity, a characterization for the solution sets of an optimization problem and a variational inQ.H. Ansari () Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India e-mail: [email protected] M. Rezaei Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran e-mail: [email protected]
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equality problem has been obtained. In the last two decades, a large number of results from convex analysis has been extended for invex functions with their applications; See, for example, [10–13] and the references therein. In 1999, Ansari et al. [12] introduced η-pseudolinearity for nonconvex, but differentiable, functions. Such a notion of η-pseudolinearity extends the concept of pseudolinearity for convex and differentiable functions. Several characterizations of η-pseudolinearity are derived. We also investigated some characterizations of the solution set of an η-pseudolinear program. Very recently, the notion of h-pseudoinvexity for a function involving a bifunction h has been introduced in [10]. Several of its properties have also been investigated. Motivated by the work of [8–10], we introduce the concept of invariant h-pseudolinearity by means of a bifunction h and the concept of invariant weakly D-pseudolinearity by means of Dini directional derivatives. We extend the results of Lalitha and Mehta [8] for invariant pseudolinearity and give some characterizations of invariant h-pseudolinearity. The characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, h-pseudolinear program are obtained. We also present some characterizations of the solution set of a nonconvex and nondifferentiable, but h-pseudoinvex, program. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs.
2 Preliminaries Let 0 denote the zero vector of Rn . A mapping f : Rn → R ∪ {±∞} is said to be (a) positively homogeneous iff for all x ∈ Rn and all r > 0, f (rx) = rf (x); (b) subodd iff for all x ∈ Rn \ {0}, f (x) ≥ −f (−x). Definition 2.1 Let f : Rn → R ∪ {±∞} be a function and x ∈ Rn be a point where f is finite. (a) The Dini upper directional derivative at the point x ∈ Rn in the direction d ∈ Rn is defined by f D (x; d) := lim sup t→0+
f (x + td) − f (x) f (x + td) − f (x) = inf sup . s>0 0 f (x) implies h x; η(y, x) > 0, equivalently,
h x; η(y, x) ≤ 0
implies f (y) ≤ f (x);
(c) invariant h-pseudolinear w.r.t. η iff it is both h-pseudoinvex as well as hpseudoincave w.r.t. the same η. Remark 2.1 (a) If h(x; η(y, x)) = f D (x; η(y, x)), then h-pseudoinvexity (respectively, h-pseudoincavity and invariant h-pseudolinearity) w.r.t. η is called D + -pseudoinvexity (respectively, D + -pseudoincavity and invariant D + -pseudolinearity) w.r.t. the same η. (b) If h(x; η(y, x)) = fD (x; η(y, x)), then h-pseudoinvexity (respectively, h-pseudoincavity and invariant h-pseudolinearity) w.r.t. η is called D+ -pseudoinvexity (respectively, D+ -pseudoincavity and invariant D+ -pseudolinearity) w.r.t. the same η. We provide the following example of a function, which is invariant D + -pseudolinear w.r.t. given η. Example 2.1 Let f : R → R be a function defined as 2 x + 2x, if x ≥ 0, f (x) := x, if x < 0, and η : R × R → R be defined as η(x, y) = α(x 3 − y 3 ) such that 0 < α ≤ 13 . Then f is invariant D + -pseudolinear w.r.t. η. Definition 2.5 Let K ⊆ Rn be a nonempty set and η : K × K → Rn be a map. A function f : K → R is said to be invariant weakly D-pseudolinear w.r.t. η iff it is both D + -pseudoinvex as well as D+ -pseudoincave w.r.t. the same η. Lemma 2.1 Let K ⊆ Rn be nonempty set, η : K × K → Rn be map, and h : K × Rn → R be a bifunction. If f : K → R is invariant weakly D-pseudolinear w.r.t. η such that fD (x; ·) ≤ h(x; ·) ≤ f D (x; ·),
for all x ∈ K,
(5)
then f is invariant h-pseudolinear w.r.t. the same η. Proof For all x, y ∈ K, if h(x; η(y, x)) ≥ 0, then from inequality (5), f D (x; η(y, x)) ≥ 0. Since f is D + -pseudoinvex w.r.t. η, we have f (y) ≥ f (x). Thus, f is h-pseudoinvex w.r.t. η.
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If h(x; η(y, x)) ≤ 0, then from inequality (5), fD (x; η(y, x)) ≤ 0. Since f is D+ pseudoincave w.r.t. η, we have f (y) ≤ f (x). Thus, f is h-pseudoincave w.r.t. η. Hence f is invariant h-pseudolinear w.r.t. η. Lemma 2.2 Let K ⊆ Rn be a nonempty set, η : K × K → Rn be a map and h : K × Rn → R be a bifunction. If f : K → R is invariant h-pseudolinear w.r.t. η such that inequality (5) holds, then it is D + -pseudoincave as well as D+ -pseudoinvex w.r.t. the same η. Proof For all x, y ∈ K, if f D (x; η(y, x)) ≤ 0, then from inequality (5), we have h(x; η(y, x)) ≤ 0. By h-pseudoincavity w.r.t. η of f , we have f (y) ≤ f (x). Thus, f is D + -pseudoincave w.r.t. η. If fD (x; η(y, x)) ≥ 0, then from inequality (5), we have h(x; η(y, x)) ≥ 0. By h-pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x). Thus, f is D+ -pseudoinvex w.r.t. η. The following lemma will be used in the sequel. Lemma 2.3 [10, Theorem 27] Let K ⊆ Rn be a nonempty invex set w.r.t. η : K × K → Rn such that the Condition C holds. If f : K → R is D + -pseudoinvex w.r.t. η, then f D x; η(y, x) ≥ 0 implies f D y; η(x, y) ≤ 0.
3 Some Characterizations of Invariant Pseudolinear Functions In this section, we present some characterizations of invariant h-pseudolinear functions and invariant weakly D-pseudolinear functions. Theorem 3.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, and let f : K → R be invariant h-pseudolinear w.r.t. η such that the inequality (5) holds. Then for all x, y ∈ K, h(x; η(y, x)) = 0 if and only if f (x) = f (y). Proof Suppose that h(x; η(y, x)) = 0 for all x, y ∈ K. By h-pseudoinvexity w.r.t. η of f , we have h x; η(y, x) ≥ 0 implies f (y) ≥ f (x). (6) From h-pseudoincavity w.r.t. η of f , we get h x; η(y, x) ≤ 0 implies f (y) ≤ f (x). Combining (6) and (7), we obtain h x; η(y, x) = 0 implies f (x) = f (y).
(7)
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Conversely, suppose that f (x) = f (y) for all x, y ∈ K. To prove that h x; η(y, x) = 0, for all x, y ∈ K, we first show that for any t ∈ (0, 1), f x + tη(y, x) = f (x). If f (x + tη(y, x)) > f (x), then by h-pseudoinvexity w.r.t. η of f , we have h zt , η(x, zt ) < 0,
(8)
(9)
where zt = x + tη(y, x). By Condition C, −t η x, x + tη(y, x) = −tη(y, x) = η y, x + tη(y, x) . 1−t
(10)
Combining (9) and (10), we obtain t η(y, zt ) < 0. h zt ; − 1−t By linearity of h in the second argument, we have h zt ; η(y, zt ) > 0. From h-pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x + tη(y, x)) which contradicts our assumption f (x + tη(y, x)) > f (x) = f (y). Similarly, we can show that f (x + tη(y, x)) < f (x) leads to a contradiction by using h-pseudoincavity w.r.t. η of f . Hence f (x + tη(y, x)) = f (x) for all t ∈ [0, 1]. By the definition of Dini upper and Dini lower directional derivatives, we have fD x; η(y, x) = f D x; η(y, x) = 0. The inequality (5) yields that h(x; η(y, x)) = 0.
Remark 3.1 In Theorem 3.1, we did not use any kind of upper semicontinuity, but we did use the linearity in the second argument of h. Therefore, our result is different from [8, Theorem 3.1]. Theorem 3.1 also extends [12, Proposition 1] for nondifferentiable and invariant h-pseudolinear functions. Theorem 3.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. Let f : K → R be invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds. Then for all x, y ∈ K, h(x; η(y, x)) = 0 if and only if f (x) = f (y). Proof Suppose that h(x; η(y, x)) = 0 for all x, y ∈ K. Then by inequality (5), f D x; η(y, x) ≥ 0.
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By D + -pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x).
(11)
By our supposition, fD (x; η(y, x)) ≤ 0, and so, by using D+ -pseudoincavity w.r.t. η of f , we get f (y) ≤ f (x).
(12)
Combining (11) and (12), we obtain f (x) = f (y). Conversely, suppose that f (x) = f (y) for all x, y ∈ K. To prove that h x; η(y, x) = 0, for all x, y ∈ K, we first show that for any t ∈ (0, 1), f x + tη(y, x) = f (x).
(13)
If f (x + tη(y, x)) > f (x), then by D + -pseudoinvexity w.r.t. η of f , we have (14) f D zt , η(x, zt ) < 0, where zt = x + tη(y, x). By using (10), the above strict inequality becomes t D zt ; − η(y, zt ) < 0. f 1−t Since f D (x; ·) is subodd and positively homogeneous in the second argument, we have f D zt ; η(y, zt ) > 0. From D + -pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x + tη(y, x)) which contradicts our assumption f (x + tη(y, x)) > f (x) = f (y). Similarly, we can show that f (x + tη(y, x)) < f (x) leads to a contradiction by using D+ -pseudoincavity w.r.t. η of f and the inequality (3). Hence f (x + tη(y, x)) = f (x) for all t ∈ [0, 1]. Rest of the proof lies on the lines of the proof of Theorem 3.1. Theorem 3.3 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, and let f : K → R be a function such that the inequality (5) holds. Then f is invariant h-pseudolinear w.r.t. η if and only if there exists a real-valued function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and f (y) = f (x) + p(x, y)h x; η(y, x) . (15) Proof Let f be invariant h-pseudolinear w.r.t. η. We have to construct a function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and (15) holds. For all x, y ∈ K,
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we define
p(x, y) :=
f (y)−f (x) h(x;η(y,x)) ,
1,
if h(x; η(y, x)) = 0, if h(x; η(y, x)) = 0.
(16)
If h(x; η(y, x)) = 0 for all x, y ∈ K, then by Theorem 3.1, we have f (y) = f (x), and thus, (15) holds. If h(x; η(y, x)) = 0, then, we have to show that p(x, y) > 0. If f (y) > f (x), then by h-pseudoincavity w.r.t. η of f , we have h(x; η(y, x)) > 0. From (16), we get p(x, y) > 0. Similarly, if f (y) < f (x), we get p(x, y) > 0 by using h-pseudoinvexity w.r.t. η of f . Conversely, suppose that there exists a real-valued function p : K × K → R such that for any x, y ∈ K, p(x, y) > 0 and (15) holds. If h(x; η(y, x)) ≥ 0, then from (15), we have f (y) − f (x) = p(x, y)h x; η(y, x) ≥ 0. Hence, f is h-pseudoinvex w.r.t. η. Likewise, if h(x; η(y, x)) ≤ 0, we can prove that f is h-pseudoincave w.r.t. the same η. Hence, f is invariant h-pseudolinear w.r.t. η. Remark 3.2 Theorem 3.3 extends [12, Proposition 2], and hence, [21, Proposition 1], and [1, Proposition 2] for nondifferentiable and invariant h-pseudolinear functions. Also, Theorem 3.3 extends and generalizes [8, Theorem 3.3] to invariant h-pseudolinear functions as we have not used any kind of continuity on f . By using the inequality (5) and invariant weakly D-pseudolinearity of f and following the proof of Theorem 3.3, we obtain the following result. Theorem 3.4 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction and f : K → R be a function such that the inequality (5) holds. Then f is invariant weakly D-pseudolinear w.r.t. η on K if and only if there exists a real-valued function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and the equality (15) holds.
4 Characterizations of the Solution Sets of the Optimization Problems Consider the following optimization problem: (OP)
min f (x)
subject to x ∈ K,
where K ⊆ Rn is a nonempty invex set w.r.t. η : K × K → Rn and f : K → R is a function. Throughout this section, we assume that the solution set S = arg minx∈K f (x) of the (OP) is nonempty. It is shown in [12] that the solution set S is invex w.r.t. η if f is preinvex w.r.t. the same η.
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Proposition 4.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear. If f : K → R is invariant h-pseudolinear w.r.t. η such that the inequality (5) holds, then the solution set S of (OP) is invex w.r.t. the same η. Proof Suppose that x1 , x2 ∈ S. Then f (x1 ) ≤ f (y) and f (x2 ) ≤ f (y) for all y ∈ K. Since x1 , x2 ∈ S, we have f (x1 ) = f (x2 ). By Theorem 3.1, we have h(x1 ; η(x2 , x1 )) = 0. The linearity of h in the second argument implies that h x1 ; tη(x2 , x1 ) = 0, for all t ∈ [0, 1]. By (4), we obtain
η x1 + tη x2 , x1 , x1 = tη(x2 , x1 ),
and, therefore, h x1 ; η x1 + tη(x2 , x1 ), x1 = 0,
for all t ∈ [0, 1].
Since f is invariant h-pseudolinear, by Theorem 3.1, f (x1 + tη(x2 , x1 )) = f (x1 ) and, therefore, x1 + tη(x2 , x1 ) is also a solution of (OP). Thus, the solution set of (OP) is invex w.r.t. η. Proposition 4.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. If f : K → R is invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds, then the solution set S of (OP) is invex w.r.t. the same η. Proof Suppose that x1 , x2 ∈ S. Then f (x1 ) ≤ f (y) and f (x2 ) ≤ f (y) for all y ∈ K. Since x1 , x2 ∈ S, we have f (x1 ) = f (x2 ). By Theorem 3.2, we have (17) h x1 ; η(x2 , x1 ) = 0. From the preceding equality and the inequality (5), we obtain f D (x1 ; η(x2 , x1 )) ≥ 0. Since f D (x; ·) is positively homogeneous, we have f D x1 ; tη(x2 , x1 ) ≥ 0, for all t ∈ [0, 1]. By (4), we obtain
η x1 + tη(x2 , x1 ), x1 = tη(x2 , x1 ),
and, therefore, f D x1 ; η x1 + tη(x2 , x1 ), x1 ≥ 0, By D + -pseudoinvexity w.r.t. η of f , we get f x1 + tη(x2 , x1 ) ≥ f (x1 ),
for all t ∈ [0, 1].
for all t ∈ [0, 1].
(18)
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On the other hand, from the equality (17) and the inequality (5), we obtain fD (x1 ; η(x2 , x1 )) ≤ 0. By positive homogeneity of fD (x; ·), we get fD x1 ; tη(x2 , x1 ) ≤ 0, for all t ∈ [0, 1], and, therefore, fD x1 ; η x1 + tη(x2 , x1 ), x1 ≤ 0, By D+ -pseudoincavity w.r.t. η of f , we get f x1 + tη(x2 , x1 ) ≤ f (x1 ),
for all t ∈ [0, 1].
for all t ∈ [0, 1].
(19)
Combining the inequalities (18) and (19), we have f (x1 + tη(x2 , x1 )) = f (x1 ) and, therefore, x1 + tη(x2 , x1 ) is also a solution of (OP). Thus, the solution set of (OP) is invex w.r.t. η. We give some characterizations of the solution set of an invariant h-pseudolinear program in terms of any of its solutions. Theorem 4.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, f : K → R be invariant h-pseudolinear w.r.t. η such that the inequality (5) holds, and x¯ ∈ S. Then S = S1 = S2 = S3 , where S1 := x ∈ K : h x; η(x, ¯ x) = 0 S2 := x ∈ K : h x; ¯ η(x, x) ¯ =0 S3 := x ∈ K : h x; η(x, ¯ x) ≥ 0 . Proof The point x ∈ S if and only if f (x) = f (x). ¯ Then from Theorem 3.1, we have f (x) = f (x) ¯ if and only if h(x; η(x, ¯ x)) = 0. Thus, S = S1 . Similarly, we can show that S = S2 . It is clear that S = S1 ⊆ S3 . We prove that S3 ⊆ S. Assume that x ∈ S3 , that is, x ∈ K such that h(x; η(x, ¯ x)) ≥ 0. By invariant h-pseudoinvexity w.r.t. η of f , we have f (x) ¯ ≥ f (x). This implies that x ∈ S, and hence, S3 ⊆ S. Remark 4.1 Theorem 4.1 partially extends and generalizes [12, Theorem 1], [12, Corollary 1] and [8, Theorems 4.1 and 4.2]. Similarly, by using Theorems 3.2 and 3.4, we obtain the following result whose proof lines on the lines of the proof of above theorem, and therefore, we omit it. Theorem 4.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. Let f : K → R be invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds, and x¯ ∈ S. Then S = S1 = S2 = S3 , where S1 , S2 , and S3 are same as in Theorem 4.1.
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Theorem 4.3 Let K, η, f, and h be the same as in Theorem 4.1. If x¯ ∈ S, then S = S4 , where ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) . S4 := x ∈ K : h x; Proof Let x ∈ S. Then by Theorem 4.1, h x; η(x, ¯ x) ≥ 0 = h x; ¯ η(x, x) ¯ .
(20)
Thus x ∈ S4 , and hence, S ⊆ S4 . We now prove that S4 ⊆ S. Assume that x ∈ S4 . Then h x; ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) .
(21)
Suppose that x ∈ S. Then f (x) ¯ < f (x). By h-pseudoincavity w.r.t. η of f , we have h x; ¯ η(x, x) ¯ > 0. By using (21), we have
h x; η(x, ¯ x) > 0.
By h-pseudoinvexity w.r.t. of η of f , we have f (x) ¯ ≥ f (x). Hence x ∈ S.
Remark 4.2 Theorem 4.3 partially extends [12, Theorem 2] to nondifferentiable and generalized pesudolinear functions and also [8, Theorem 4.4]. Hence, Theorems 4.1 and 4.3 extend the results of Jeyakumar and Yang [3]. Theorem 4.4 Let K, η, f and h be the same as in Theorem 4.2. If x¯ ∈ S, then S = S4 , where S4 is the same as in Theorem 4.3. Proof The proof of S ⊆ S4 is same as in the proof of Theorem 4.3. We now prove that S4 ⊆ S. Assume that x ∈ S4 . Then h x; ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) .
(22)
Suppose that x ∈ S. Then f (x) ¯ < f (x). By D+ -pseudoincavity w.r.t. η of f , we have fD x; ¯ η(x, x) ¯ > 0. By using (5) and (22), we have h x; η(x, ¯ x) > 0. By Theorem 3.4, there exists a function p defined on K × K such that p(x, x) ¯ >0 and f (x) ¯ = f (x) + p(x, x)h ¯ x; η(x, ¯ x) > f (x), which is a contradiction of the fact that f (x) ¯ < f (x). Hence, x ∈ S.
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5 Characterizations of the Solution Sets of Nonsmooth Pseudoinvex Program In this section, we give some characterizations of the solution set of a nonsmooth pseudoinvex program in terms of any of its solutions. Lemma 5.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn . Let f : K → R be D + -pseudoinvex w.r.t. η. If x¯ ∈ K is a solution of (OP), then it is a solution of the following nonsmooth variational-like inequality problem: find x¯ ∈ K such that f D x; ¯ η(y, x) ¯ ≥ 0, for all y ∈ K. (23) Proof Let x¯ ∈ K be a solution of (OP). Then f (x) ¯ ≤ f (y),
for all y ∈ K.
Since K is invex, x¯ + tη(y, x) ¯ ∈ K for all t ∈ (0, 1] and, therefore, f (x + tη(y, x) ¯ − f (x) ¯ f D x; ≥ 0. ¯ η(y, x) ¯ = lim sup t + t→0
Lemma 5.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. If f : K → R is D + -pseudoinvex w.r.t. η and x, ¯ y¯ ∈ S, then f D x; ¯ η(y, ¯ x) ¯ = 0 = f D y; ¯ η(x, ¯ y) ¯ . Proof Since x, ¯ y¯ ∈ S, then by Lemma 5.1, we have f D x; ¯ η(y, ¯ x) ¯ ≥ 0 and f D y; ¯ η(x, ¯ y) ¯ ≥ 0.
(24)
By Lemma 2.3, we get f D y; ¯ η(x, ¯ y) ¯ ≤ 0 and f D x; ¯ η(y, ¯ x) ¯ ≤ 0.
(25)
By combining relations (24) and (25), we get the conclusion.
Theorem 5.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let f : K → R be D + -pseudoinvex w.r.t. η, and x¯ ∈ S, then S = S1 = S1∗ , where S1∗ := x ∈ K : f D (x; η(x, ¯ x)) = 0 , S1∗∗ := x ∈ K : f D (x; ¯ η(x, ¯ x)) ≥ 0 . Proof Let x ∈ S. Since x¯ ∈ S, from Lemma 5.2, we have f D x; η(x, ¯ x) = 0, that is, x ∈ S1∗ . So, S ⊆ S1∗ .
(26) (27)
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Conversely, let x ∈ S1∗ . Then ¯ x) = 0. f D x; η(x, By D + -pseudoinvexity of f , we have f (x) ¯ ≥ f (x). Since x¯ ∈ S, the above inequality implies f (x) ¯ = f (x), that is, S1∗ ⊆ S. Thus. S1∗ = S. It is clear from Lemma 5.1 that S ⊆ S1∗∗ . Assume that x∈K
¯ x) ≥ 0. and f D x; η(x,
¯ ≥ f (x). Since x¯ ∈ S, we have f (x) ¯ = The D + -pseudinvexity of f implies f (x) f (x), that is, S1∗∗ ⊆ S. Thus, S1∗∗ = S. Remark 5.1 Theorem 5.1 extends [22, Theorem 3.1] for nondifferentiable functions. Hence, Theorem 5.1 improves and generalizes [23, Theorem 1] and [3, Theorem 3.1]. Theorem 5.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let f : K → R be D + -pseudoinvex w.r.t. η, and x¯ ∈ S, then S = S5 = S6 , where ¯ η(x, x) ¯ = f D x; η(x, ¯ x) , S5 := x ∈ K : f D x; ¯ η(x, x) ¯ ≤ f D x; η(x, ¯ x) . S6 := x ∈ K : f D x;
(28) (29)
Proof Let x ∈ S. From Lemma 5.2, we have f D (x; η(x, ¯ x)) = f D (x; ¯ η(x, x)) ¯ = 0. Thus, x ∈ S5 , and hence S ⊆ S5 . S5 ⊆ S6 is trivial. Assume that x ∈ S6 . Then, x satisfies ¯ η(x, x) ¯ ≤ f D x; η(x, ¯ x) . f D x;
(30)
¯ η(x, x)) ¯ ≥ 0. By using inequality (30), From x¯ ∈ S and Lemma 5.1, we obtain f D (x; ¯ x)) ≥ 0. By D + -pseudoinvexity w.r.t. η of f , we have f (x) ¯ ≥ we get f D (x; η(x, f (x). Since x¯ ∈ S, we obtain f (x) ¯ = f (x). Thus, x ∈ S, and hence S6 ⊆ S. Remark 5.2 Theorem 5.2 extends [22, Theorem 3.2] for nondifferentiable functions. Therefore, Theorem 5.2 improves and generalizes [3, Theorem 3.2].
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6 Concluding Remarks In this paper, we introduced the concept of invariant h-pseudolinearity by means of a bifunction h and the concept of invariant weakly D-pseudolinearity by means of Dini directional derivatives. We extended the results of Lalitha and Mehta [8] for invariant pseudolinearity and gave some characterizations of invariant h-pseudolinearity. The characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, h-pseudolinear program are obtained. We also presented some characterizations of the solution set of a nonconvex and nondifferentiable, but h-pseudoinvex, program. The results of this paper extended various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs. Acknowledgements In this research, the second author was partially supported by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran. Authors are grateful to the referees for their valuable comments and suggestions to improve the previous draft of this paper.
References 1. Chew, K.L., Choo, E.U.: Pseudolinearity and efficiency. Math. Program. 28, 226–239 (1984) 2. Jeyakumar, V., Mond, B.: On generalised convex mathematical programming. J. Aust. Math. Soc. Ser. B, Appl. Math 34, 43–53 (1992) 3. Jeyakumar, V., Yang, X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995) 4. Komlósi, S.: First and second order characterizations of pseudolinear functions. Eur. J. Oper. Res. 67, 278–286 (1993) 5. Rapcsák, T.: On pseudolinear functions. J. Oper. Res. 50, 353–360 (1991) 6. Martos, B.: Nonlinear Programming; Theory and Methods. North-Holland, Amsterdam (1975) 7. Schaible, S., Ibaraki, T.: Fractional programming. Eur. J. Oper. Res. 12, 325–338 (1983) 8. Lalitha, C.S., Mehta, M.: A note on pseudolinearity in terms of bifunctions. Asia-Pac. J. Oper. Res. 24, 1–9 (2007) 9. Lalitha, C.S., Mehta, M.: Characterization of the solution sets of pseudolinear programs and pseudoaffine variational inequality problems. J. Nonlinear Convex Anal. 8(1), 87–98 (2007) 10. Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: Nonsmooth invexities, invariant monotonicities and nonsmooth vector variational-like inequalities with applications to vector optimization. In: Ansari, Q.H., Yao, J.-C. (eds.) Recent Developments in Vector Optimization, pp. 183–225. Springer, Berlin (2011) 11. Ansari, Q.H., Rezaei, M.: Generalized pseudolinearity. Optim. Lett. (2012) 12. Ansari, Q.H., Schaible, S., Yao, J.C.: η-pseudolinearity. Riv. Mat. Sci. Econ. Soc. 22, 31–39 (1999) 13. Giorgi, G., Guerraggio, A., Thierfelder, J.: Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier, Amsterdam (2004) 14. Diewert, W.E.: Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming. In: Schaible, S., Ziemba, W.T. (eds.) Generalized Concavity in Optimization and Economics, pp. 51–95. Academic Press, New York (1981) 15. Giorgi, G., Komlósi, S.: Dini derivatives in optimization—Part I. Riv. Mat. Sci. Econ. Soc. 15(1), 3–30 (1993) 16. Giorgi, G., Komlósi, S.: Dini derivatives in optimization—Part II. Riv. Mat. Sci. Econ. Soc. 15(2), 3–24 (1993) 17. Giorgi, G., Komlósi, S.: Dini derivatives in optimization—Part III. Riv. Mat. Sci. Econ. Soc. 18(1), 47–63 (1995) 18. Hadjisavvas, N., Komlósi, S., Schaible, S.: Handbook of Generalized Convexity and Generalized Monotonicity. Springer, New York (2005) 19. Jeyakumar, V., Luc, D.T.: Nonsmooth Vector Functions and Continuous Optimization. Springer, New York (2008)
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20. McShane, E.J.: Integration. Princeton University Press, Princeton (1944) 21. Kortanek, K.O., Evans, J.P.: Pseudoconcave programming and Lagrange regularity. Oper. Res. 15, 882–892 (1967) 22. Yang, X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009) 23. Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)
Invariant Pseudolinearity with Applications Qamrul Hasan Ansari · Mahboubeh Rezaei
Received: 26 April 2011 / Accepted: 1 December 2011 / Published online: 16 December 2011 © Springer Science+Business Media, LLC 2011
Abstract In this paper, we introduce the notion of invariant pseudolinearity for nondifferentiable and nonconvex functions by means of Dini directional derivatives. We present some characterizations of invariant pseudolinear functions. Some characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, pseudolinear program are obtained. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs. Keywords Invex sets · Pseudoinvexity · Invariant pseudolinearity · Dini directional derivatives · Nonsmooth variational-like inequalities · Solution sets of a program 1 Introduction The natural generalization of linearity is the pseudolinearity. It plays an important role in mathematical programming; see, for example, [1–5] and the references therein. The class of pseudolinear functions includes many useful functions such as linear fractional functions [6, 7]. Recently, Lalitha and Mehta [8, 9] generalized the notion of pseudolinearity to nondifferentiable functions and obtained characterizations for such functions. They extended the results given by Chew and Choo [1] and Komlósi [4] for differentiable functions. Under the assumption of pseudolinearity, a characterization for the solution sets of an optimization problem and a variational inQ.H. Ansari () Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India e-mail: [email protected] M. Rezaei Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran e-mail: [email protected]
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equality problem has been obtained. In the last two decades, a large number of results from convex analysis has been extended for invex functions with their applications; See, for example, [10–13] and the references therein. In 1999, Ansari et al. [12] introduced η-pseudolinearity for nonconvex, but differentiable, functions. Such a notion of η-pseudolinearity extends the concept of pseudolinearity for convex and differentiable functions. Several characterizations of η-pseudolinearity are derived. We also investigated some characterizations of the solution set of an η-pseudolinear program. Very recently, the notion of h-pseudoinvexity for a function involving a bifunction h has been introduced in [10]. Several of its properties have also been investigated. Motivated by the work of [8–10], we introduce the concept of invariant h-pseudolinearity by means of a bifunction h and the concept of invariant weakly D-pseudolinearity by means of Dini directional derivatives. We extend the results of Lalitha and Mehta [8] for invariant pseudolinearity and give some characterizations of invariant h-pseudolinearity. The characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, h-pseudolinear program are obtained. We also present some characterizations of the solution set of a nonconvex and nondifferentiable, but h-pseudoinvex, program. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs.
2 Preliminaries Let 0 denote the zero vector of Rn . A mapping f : Rn → R ∪ {±∞} is said to be (a) positively homogeneous iff for all x ∈ Rn and all r > 0, f (rx) = rf (x); (b) subodd iff for all x ∈ Rn \ {0}, f (x) ≥ −f (−x). Definition 2.1 Let f : Rn → R ∪ {±∞} be a function and x ∈ Rn be a point where f is finite. (a) The Dini upper directional derivative at the point x ∈ Rn in the direction d ∈ Rn is defined by f D (x; d) := lim sup t→0+
f (x + td) − f (x) f (x + td) − f (x) = inf sup . s>0 0 f (x) implies h x; η(y, x) > 0, equivalently,
h x; η(y, x) ≤ 0
implies f (y) ≤ f (x);
(c) invariant h-pseudolinear w.r.t. η iff it is both h-pseudoinvex as well as hpseudoincave w.r.t. the same η. Remark 2.1 (a) If h(x; η(y, x)) = f D (x; η(y, x)), then h-pseudoinvexity (respectively, h-pseudoincavity and invariant h-pseudolinearity) w.r.t. η is called D + -pseudoinvexity (respectively, D + -pseudoincavity and invariant D + -pseudolinearity) w.r.t. the same η. (b) If h(x; η(y, x)) = fD (x; η(y, x)), then h-pseudoinvexity (respectively, h-pseudoincavity and invariant h-pseudolinearity) w.r.t. η is called D+ -pseudoinvexity (respectively, D+ -pseudoincavity and invariant D+ -pseudolinearity) w.r.t. the same η. We provide the following example of a function, which is invariant D + -pseudolinear w.r.t. given η. Example 2.1 Let f : R → R be a function defined as 2 x + 2x, if x ≥ 0, f (x) := x, if x < 0, and η : R × R → R be defined as η(x, y) = α(x 3 − y 3 ) such that 0 < α ≤ 13 . Then f is invariant D + -pseudolinear w.r.t. η. Definition 2.5 Let K ⊆ Rn be a nonempty set and η : K × K → Rn be a map. A function f : K → R is said to be invariant weakly D-pseudolinear w.r.t. η iff it is both D + -pseudoinvex as well as D+ -pseudoincave w.r.t. the same η. Lemma 2.1 Let K ⊆ Rn be nonempty set, η : K × K → Rn be map, and h : K × Rn → R be a bifunction. If f : K → R is invariant weakly D-pseudolinear w.r.t. η such that fD (x; ·) ≤ h(x; ·) ≤ f D (x; ·),
for all x ∈ K,
(5)
then f is invariant h-pseudolinear w.r.t. the same η. Proof For all x, y ∈ K, if h(x; η(y, x)) ≥ 0, then from inequality (5), f D (x; η(y, x)) ≥ 0. Since f is D + -pseudoinvex w.r.t. η, we have f (y) ≥ f (x). Thus, f is h-pseudoinvex w.r.t. η.
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If h(x; η(y, x)) ≤ 0, then from inequality (5), fD (x; η(y, x)) ≤ 0. Since f is D+ pseudoincave w.r.t. η, we have f (y) ≤ f (x). Thus, f is h-pseudoincave w.r.t. η. Hence f is invariant h-pseudolinear w.r.t. η. Lemma 2.2 Let K ⊆ Rn be a nonempty set, η : K × K → Rn be a map and h : K × Rn → R be a bifunction. If f : K → R is invariant h-pseudolinear w.r.t. η such that inequality (5) holds, then it is D + -pseudoincave as well as D+ -pseudoinvex w.r.t. the same η. Proof For all x, y ∈ K, if f D (x; η(y, x)) ≤ 0, then from inequality (5), we have h(x; η(y, x)) ≤ 0. By h-pseudoincavity w.r.t. η of f , we have f (y) ≤ f (x). Thus, f is D + -pseudoincave w.r.t. η. If fD (x; η(y, x)) ≥ 0, then from inequality (5), we have h(x; η(y, x)) ≥ 0. By h-pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x). Thus, f is D+ -pseudoinvex w.r.t. η. The following lemma will be used in the sequel. Lemma 2.3 [10, Theorem 27] Let K ⊆ Rn be a nonempty invex set w.r.t. η : K × K → Rn such that the Condition C holds. If f : K → R is D + -pseudoinvex w.r.t. η, then f D x; η(y, x) ≥ 0 implies f D y; η(x, y) ≤ 0.
3 Some Characterizations of Invariant Pseudolinear Functions In this section, we present some characterizations of invariant h-pseudolinear functions and invariant weakly D-pseudolinear functions. Theorem 3.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, and let f : K → R be invariant h-pseudolinear w.r.t. η such that the inequality (5) holds. Then for all x, y ∈ K, h(x; η(y, x)) = 0 if and only if f (x) = f (y). Proof Suppose that h(x; η(y, x)) = 0 for all x, y ∈ K. By h-pseudoinvexity w.r.t. η of f , we have h x; η(y, x) ≥ 0 implies f (y) ≥ f (x). (6) From h-pseudoincavity w.r.t. η of f , we get h x; η(y, x) ≤ 0 implies f (y) ≤ f (x). Combining (6) and (7), we obtain h x; η(y, x) = 0 implies f (x) = f (y).
(7)
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Conversely, suppose that f (x) = f (y) for all x, y ∈ K. To prove that h x; η(y, x) = 0, for all x, y ∈ K, we first show that for any t ∈ (0, 1), f x + tη(y, x) = f (x). If f (x + tη(y, x)) > f (x), then by h-pseudoinvexity w.r.t. η of f , we have h zt , η(x, zt ) < 0,
(8)
(9)
where zt = x + tη(y, x). By Condition C, −t η x, x + tη(y, x) = −tη(y, x) = η y, x + tη(y, x) . 1−t
(10)
Combining (9) and (10), we obtain t η(y, zt ) < 0. h zt ; − 1−t By linearity of h in the second argument, we have h zt ; η(y, zt ) > 0. From h-pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x + tη(y, x)) which contradicts our assumption f (x + tη(y, x)) > f (x) = f (y). Similarly, we can show that f (x + tη(y, x)) < f (x) leads to a contradiction by using h-pseudoincavity w.r.t. η of f . Hence f (x + tη(y, x)) = f (x) for all t ∈ [0, 1]. By the definition of Dini upper and Dini lower directional derivatives, we have fD x; η(y, x) = f D x; η(y, x) = 0. The inequality (5) yields that h(x; η(y, x)) = 0.
Remark 3.1 In Theorem 3.1, we did not use any kind of upper semicontinuity, but we did use the linearity in the second argument of h. Therefore, our result is different from [8, Theorem 3.1]. Theorem 3.1 also extends [12, Proposition 1] for nondifferentiable and invariant h-pseudolinear functions. Theorem 3.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. Let f : K → R be invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds. Then for all x, y ∈ K, h(x; η(y, x)) = 0 if and only if f (x) = f (y). Proof Suppose that h(x; η(y, x)) = 0 for all x, y ∈ K. Then by inequality (5), f D x; η(y, x) ≥ 0.
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By D + -pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x).
(11)
By our supposition, fD (x; η(y, x)) ≤ 0, and so, by using D+ -pseudoincavity w.r.t. η of f , we get f (y) ≤ f (x).
(12)
Combining (11) and (12), we obtain f (x) = f (y). Conversely, suppose that f (x) = f (y) for all x, y ∈ K. To prove that h x; η(y, x) = 0, for all x, y ∈ K, we first show that for any t ∈ (0, 1), f x + tη(y, x) = f (x).
(13)
If f (x + tη(y, x)) > f (x), then by D + -pseudoinvexity w.r.t. η of f , we have (14) f D zt , η(x, zt ) < 0, where zt = x + tη(y, x). By using (10), the above strict inequality becomes t D zt ; − η(y, zt ) < 0. f 1−t Since f D (x; ·) is subodd and positively homogeneous in the second argument, we have f D zt ; η(y, zt ) > 0. From D + -pseudoinvexity w.r.t. η of f , we have f (y) ≥ f (x + tη(y, x)) which contradicts our assumption f (x + tη(y, x)) > f (x) = f (y). Similarly, we can show that f (x + tη(y, x)) < f (x) leads to a contradiction by using D+ -pseudoincavity w.r.t. η of f and the inequality (3). Hence f (x + tη(y, x)) = f (x) for all t ∈ [0, 1]. Rest of the proof lies on the lines of the proof of Theorem 3.1. Theorem 3.3 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, and let f : K → R be a function such that the inequality (5) holds. Then f is invariant h-pseudolinear w.r.t. η if and only if there exists a real-valued function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and f (y) = f (x) + p(x, y)h x; η(y, x) . (15) Proof Let f be invariant h-pseudolinear w.r.t. η. We have to construct a function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and (15) holds. For all x, y ∈ K,
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we define
p(x, y) :=
f (y)−f (x) h(x;η(y,x)) ,
1,
if h(x; η(y, x)) = 0, if h(x; η(y, x)) = 0.
(16)
If h(x; η(y, x)) = 0 for all x, y ∈ K, then by Theorem 3.1, we have f (y) = f (x), and thus, (15) holds. If h(x; η(y, x)) = 0, then, we have to show that p(x, y) > 0. If f (y) > f (x), then by h-pseudoincavity w.r.t. η of f , we have h(x; η(y, x)) > 0. From (16), we get p(x, y) > 0. Similarly, if f (y) < f (x), we get p(x, y) > 0 by using h-pseudoinvexity w.r.t. η of f . Conversely, suppose that there exists a real-valued function p : K × K → R such that for any x, y ∈ K, p(x, y) > 0 and (15) holds. If h(x; η(y, x)) ≥ 0, then from (15), we have f (y) − f (x) = p(x, y)h x; η(y, x) ≥ 0. Hence, f is h-pseudoinvex w.r.t. η. Likewise, if h(x; η(y, x)) ≤ 0, we can prove that f is h-pseudoincave w.r.t. the same η. Hence, f is invariant h-pseudolinear w.r.t. η. Remark 3.2 Theorem 3.3 extends [12, Proposition 2], and hence, [21, Proposition 1], and [1, Proposition 2] for nondifferentiable and invariant h-pseudolinear functions. Also, Theorem 3.3 extends and generalizes [8, Theorem 3.3] to invariant h-pseudolinear functions as we have not used any kind of continuity on f . By using the inequality (5) and invariant weakly D-pseudolinearity of f and following the proof of Theorem 3.3, we obtain the following result. Theorem 3.4 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction and f : K → R be a function such that the inequality (5) holds. Then f is invariant weakly D-pseudolinear w.r.t. η on K if and only if there exists a real-valued function p : K × K → R such that for all x, y ∈ K, p(x, y) > 0 and the equality (15) holds.
4 Characterizations of the Solution Sets of the Optimization Problems Consider the following optimization problem: (OP)
min f (x)
subject to x ∈ K,
where K ⊆ Rn is a nonempty invex set w.r.t. η : K × K → Rn and f : K → R is a function. Throughout this section, we assume that the solution set S = arg minx∈K f (x) of the (OP) is nonempty. It is shown in [12] that the solution set S is invex w.r.t. η if f is preinvex w.r.t. the same η.
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Proposition 4.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear. If f : K → R is invariant h-pseudolinear w.r.t. η such that the inequality (5) holds, then the solution set S of (OP) is invex w.r.t. the same η. Proof Suppose that x1 , x2 ∈ S. Then f (x1 ) ≤ f (y) and f (x2 ) ≤ f (y) for all y ∈ K. Since x1 , x2 ∈ S, we have f (x1 ) = f (x2 ). By Theorem 3.1, we have h(x1 ; η(x2 , x1 )) = 0. The linearity of h in the second argument implies that h x1 ; tη(x2 , x1 ) = 0, for all t ∈ [0, 1]. By (4), we obtain
η x1 + tη x2 , x1 , x1 = tη(x2 , x1 ),
and, therefore, h x1 ; η x1 + tη(x2 , x1 ), x1 = 0,
for all t ∈ [0, 1].
Since f is invariant h-pseudolinear, by Theorem 3.1, f (x1 + tη(x2 , x1 )) = f (x1 ) and, therefore, x1 + tη(x2 , x1 ) is also a solution of (OP). Thus, the solution set of (OP) is invex w.r.t. η. Proposition 4.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. If f : K → R is invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds, then the solution set S of (OP) is invex w.r.t. the same η. Proof Suppose that x1 , x2 ∈ S. Then f (x1 ) ≤ f (y) and f (x2 ) ≤ f (y) for all y ∈ K. Since x1 , x2 ∈ S, we have f (x1 ) = f (x2 ). By Theorem 3.2, we have (17) h x1 ; η(x2 , x1 ) = 0. From the preceding equality and the inequality (5), we obtain f D (x1 ; η(x2 , x1 )) ≥ 0. Since f D (x; ·) is positively homogeneous, we have f D x1 ; tη(x2 , x1 ) ≥ 0, for all t ∈ [0, 1]. By (4), we obtain
η x1 + tη(x2 , x1 ), x1 = tη(x2 , x1 ),
and, therefore, f D x1 ; η x1 + tη(x2 , x1 ), x1 ≥ 0, By D + -pseudoinvexity w.r.t. η of f , we get f x1 + tη(x2 , x1 ) ≥ f (x1 ),
for all t ∈ [0, 1].
for all t ∈ [0, 1].
(18)
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On the other hand, from the equality (17) and the inequality (5), we obtain fD (x1 ; η(x2 , x1 )) ≤ 0. By positive homogeneity of fD (x; ·), we get fD x1 ; tη(x2 , x1 ) ≤ 0, for all t ∈ [0, 1], and, therefore, fD x1 ; η x1 + tη(x2 , x1 ), x1 ≤ 0, By D+ -pseudoincavity w.r.t. η of f , we get f x1 + tη(x2 , x1 ) ≤ f (x1 ),
for all t ∈ [0, 1].
for all t ∈ [0, 1].
(19)
Combining the inequalities (18) and (19), we have f (x1 + tη(x2 , x1 )) = f (x1 ) and, therefore, x1 + tη(x2 , x1 ) is also a solution of (OP). Thus, the solution set of (OP) is invex w.r.t. η. We give some characterizations of the solution set of an invariant h-pseudolinear program in terms of any of its solutions. Theorem 4.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let h : K × Rn → R be a bifunction such that for each fixed x ∈ K, h(x; ·) is linear, f : K → R be invariant h-pseudolinear w.r.t. η such that the inequality (5) holds, and x¯ ∈ S. Then S = S1 = S2 = S3 , where S1 := x ∈ K : h x; η(x, ¯ x) = 0 S2 := x ∈ K : h x; ¯ η(x, x) ¯ =0 S3 := x ∈ K : h x; η(x, ¯ x) ≥ 0 . Proof The point x ∈ S if and only if f (x) = f (x). ¯ Then from Theorem 3.1, we have f (x) = f (x) ¯ if and only if h(x; η(x, ¯ x)) = 0. Thus, S = S1 . Similarly, we can show that S = S2 . It is clear that S = S1 ⊆ S3 . We prove that S3 ⊆ S. Assume that x ∈ S3 , that is, x ∈ K such that h(x; η(x, ¯ x)) ≥ 0. By invariant h-pseudoinvexity w.r.t. η of f , we have f (x) ¯ ≥ f (x). This implies that x ∈ S, and hence, S3 ⊆ S. Remark 4.1 Theorem 4.1 partially extends and generalizes [12, Theorem 1], [12, Corollary 1] and [8, Theorems 4.1 and 4.2]. Similarly, by using Theorems 3.2 and 3.4, we obtain the following result whose proof lines on the lines of the proof of above theorem, and therefore, we omit it. Theorem 4.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C, and let h : K × Rn → R be a bifunction. Let f : K → R be invariant weakly D-pseudolinear w.r.t. η such that the inequality (5) holds, and x¯ ∈ S. Then S = S1 = S2 = S3 , where S1 , S2 , and S3 are same as in Theorem 4.1.
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Theorem 4.3 Let K, η, f, and h be the same as in Theorem 4.1. If x¯ ∈ S, then S = S4 , where ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) . S4 := x ∈ K : h x; Proof Let x ∈ S. Then by Theorem 4.1, h x; η(x, ¯ x) ≥ 0 = h x; ¯ η(x, x) ¯ .
(20)
Thus x ∈ S4 , and hence, S ⊆ S4 . We now prove that S4 ⊆ S. Assume that x ∈ S4 . Then h x; ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) .
(21)
Suppose that x ∈ S. Then f (x) ¯ < f (x). By h-pseudoincavity w.r.t. η of f , we have h x; ¯ η(x, x) ¯ > 0. By using (21), we have
h x; η(x, ¯ x) > 0.
By h-pseudoinvexity w.r.t. of η of f , we have f (x) ¯ ≥ f (x). Hence x ∈ S.
Remark 4.2 Theorem 4.3 partially extends [12, Theorem 2] to nondifferentiable and generalized pesudolinear functions and also [8, Theorem 4.4]. Hence, Theorems 4.1 and 4.3 extend the results of Jeyakumar and Yang [3]. Theorem 4.4 Let K, η, f and h be the same as in Theorem 4.2. If x¯ ∈ S, then S = S4 , where S4 is the same as in Theorem 4.3. Proof The proof of S ⊆ S4 is same as in the proof of Theorem 4.3. We now prove that S4 ⊆ S. Assume that x ∈ S4 . Then h x; ¯ η(x, x) ¯ ≤ h x; η(x, ¯ x) .
(22)
Suppose that x ∈ S. Then f (x) ¯ < f (x). By D+ -pseudoincavity w.r.t. η of f , we have fD x; ¯ η(x, x) ¯ > 0. By using (5) and (22), we have h x; η(x, ¯ x) > 0. By Theorem 3.4, there exists a function p defined on K × K such that p(x, x) ¯ >0 and f (x) ¯ = f (x) + p(x, x)h ¯ x; η(x, ¯ x) > f (x), which is a contradiction of the fact that f (x) ¯ < f (x). Hence, x ∈ S.
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5 Characterizations of the Solution Sets of Nonsmooth Pseudoinvex Program In this section, we give some characterizations of the solution set of a nonsmooth pseudoinvex program in terms of any of its solutions. Lemma 5.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn . Let f : K → R be D + -pseudoinvex w.r.t. η. If x¯ ∈ K is a solution of (OP), then it is a solution of the following nonsmooth variational-like inequality problem: find x¯ ∈ K such that f D x; ¯ η(y, x) ¯ ≥ 0, for all y ∈ K. (23) Proof Let x¯ ∈ K be a solution of (OP). Then f (x) ¯ ≤ f (y),
for all y ∈ K.
Since K is invex, x¯ + tη(y, x) ¯ ∈ K for all t ∈ (0, 1] and, therefore, f (x + tη(y, x) ¯ − f (x) ¯ f D x; ≥ 0. ¯ η(y, x) ¯ = lim sup t + t→0
Lemma 5.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. If f : K → R is D + -pseudoinvex w.r.t. η and x, ¯ y¯ ∈ S, then f D x; ¯ η(y, ¯ x) ¯ = 0 = f D y; ¯ η(x, ¯ y) ¯ . Proof Since x, ¯ y¯ ∈ S, then by Lemma 5.1, we have f D x; ¯ η(y, ¯ x) ¯ ≥ 0 and f D y; ¯ η(x, ¯ y) ¯ ≥ 0.
(24)
By Lemma 2.3, we get f D y; ¯ η(x, ¯ y) ¯ ≤ 0 and f D x; ¯ η(y, ¯ x) ¯ ≤ 0.
(25)
By combining relations (24) and (25), we get the conclusion.
Theorem 5.1 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let f : K → R be D + -pseudoinvex w.r.t. η, and x¯ ∈ S, then S = S1 = S1∗ , where S1∗ := x ∈ K : f D (x; η(x, ¯ x)) = 0 , S1∗∗ := x ∈ K : f D (x; ¯ η(x, ¯ x)) ≥ 0 . Proof Let x ∈ S. Since x¯ ∈ S, from Lemma 5.2, we have f D x; η(x, ¯ x) = 0, that is, x ∈ S1∗ . So, S ⊆ S1∗ .
(26) (27)
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Conversely, let x ∈ S1∗ . Then ¯ x) = 0. f D x; η(x, By D + -pseudoinvexity of f , we have f (x) ¯ ≥ f (x). Since x¯ ∈ S, the above inequality implies f (x) ¯ = f (x), that is, S1∗ ⊆ S. Thus. S1∗ = S. It is clear from Lemma 5.1 that S ⊆ S1∗∗ . Assume that x∈K
¯ x) ≥ 0. and f D x; η(x,
¯ ≥ f (x). Since x¯ ∈ S, we have f (x) ¯ = The D + -pseudinvexity of f implies f (x) f (x), that is, S1∗∗ ⊆ S. Thus, S1∗∗ = S. Remark 5.1 Theorem 5.1 extends [22, Theorem 3.1] for nondifferentiable functions. Hence, Theorem 5.1 improves and generalizes [23, Theorem 1] and [3, Theorem 3.1]. Theorem 5.2 Let K ⊆ Rn be an invex set w.r.t. η : K × K → Rn that satisfies Condition C. Let f : K → R be D + -pseudoinvex w.r.t. η, and x¯ ∈ S, then S = S5 = S6 , where ¯ η(x, x) ¯ = f D x; η(x, ¯ x) , S5 := x ∈ K : f D x; ¯ η(x, x) ¯ ≤ f D x; η(x, ¯ x) . S6 := x ∈ K : f D x;
(28) (29)
Proof Let x ∈ S. From Lemma 5.2, we have f D (x; η(x, ¯ x)) = f D (x; ¯ η(x, x)) ¯ = 0. Thus, x ∈ S5 , and hence S ⊆ S5 . S5 ⊆ S6 is trivial. Assume that x ∈ S6 . Then, x satisfies ¯ η(x, x) ¯ ≤ f D x; η(x, ¯ x) . f D x;
(30)
¯ η(x, x)) ¯ ≥ 0. By using inequality (30), From x¯ ∈ S and Lemma 5.1, we obtain f D (x; ¯ x)) ≥ 0. By D + -pseudoinvexity w.r.t. η of f , we have f (x) ¯ ≥ we get f D (x; η(x, f (x). Since x¯ ∈ S, we obtain f (x) ¯ = f (x). Thus, x ∈ S, and hence S6 ⊆ S. Remark 5.2 Theorem 5.2 extends [22, Theorem 3.2] for nondifferentiable functions. Therefore, Theorem 5.2 improves and generalizes [3, Theorem 3.2].
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6 Concluding Remarks In this paper, we introduced the concept of invariant h-pseudolinearity by means of a bifunction h and the concept of invariant weakly D-pseudolinearity by means of Dini directional derivatives. We extended the results of Lalitha and Mehta [8] for invariant pseudolinearity and gave some characterizations of invariant h-pseudolinearity. The characterizations of the solution set of a nonconvex and nondifferentiable, but invariant, h-pseudolinear program are obtained. We also presented some characterizations of the solution set of a nonconvex and nondifferentiable, but h-pseudoinvex, program. The results of this paper extended various results for pseudolinear functions, pseudoinvex functions, and η-pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs, and η-pseudolinear programs. Acknowledgements In this research, the second author was partially supported by the Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran. Authors are grateful to the referees for their valuable comments and suggestions to improve the previous draft of this paper.
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