J Optim Theory Appl (2009) 140: 537–542 DOI 10.1007/s10957-008-9470-7
On Characterizing the Solution Sets of Pseudoinvex Extremum Problems X.M. Yang
Published online: 26 September 2008 © Springer Science+Business Media, LLC 2008
Abstract In this paper, we study the minimization of a pseudoinvex function over an invex subset and provide several new and simple characterizations of the solution set of pseudoinvex extremum problems. By means of the basic properties of pseudoinvex functions, the solution set of a pseudoinvex program is characterized, for instance, by the equality ∇f (x)T η(x, ¯ x) = 0, for each feasible point x, where x¯ is in the solution set. Our study improves naturally and extends some previously known results in Mangasarian (Oper. Res. Lett. 7: 21–26, 1988) and Jeyakumar and Yang (J. Opt. Theory Appl. 87: 747–755, 1995). Keywords Pseudoinvex extremum problems · Solution sets · Characterizations · Invariant pseudomonotone maps
1 Introduction Consider the nonlinear optimization problem (P)
min f (x),
s.t. x ∈ S,
Communicated by F. Giannessi. This research was partially supported by National Natural Science Foundation of China Grants No. 10771228 and 10831009. X.M. Yang () Department of Mathematics, Chongqing Normal University, Chongqing 400047, People’s Republic of China e-mail:
[email protected] X.M. Yang Chongqing Key Laboratory of Operations Research and System Engineering, Chongqing 400047, People’s Republic of China
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where S is an invex subset of Rn and f is a real-valued derivable function, which is defined on an open subset S ⊆ D and is pseudoinvex on S. This is called a pseudoinvex optimization problem. Characterizations and properties of the solution sets are useful for understanding the behavior of solution methods for extremum problems that have multiple optimal solutions. Mangasarian [1] presented several characterizations of the solution sets of convex extremum problems and applied them to study monotone linear complementarity problems. Burke and Ferris in [2] extended the results in [1] to nondifferentiability case. Since then, various extensions of these solution set characterizations to infinite-dimensional convex programs, best approximation problems, vector convex optimization problems and pseudolinear problems have been given in [3–8]. In this paper, we will give some useful and simple characterizations of the solution sets of the nonlinear pseudoinvex optimization problem. We show that, for the pseudoinvex problem (P), the solution set is characterized by the equality ¯ x) = 0, for each feasible point x, where x¯ is in the solution set S. ∇f (x)T η(x,
2 Preliminaries In this section, we recall some properties of pseudoconvex functions that will be used throughout the paper. The derivable function f : D → R is called pseudoconvex on the convex set S ⊂ D if x, y ∈ S, iff x, y ∈ S,
f (x) < f (y)
⇒
∇f (y)T (x − y) < 0;
∇f (y)T (x − y) ≥ 0
⇒
f (x) ≥ f (y).
If −f is a pseudoconvex function on convex set S, then f is said to be a pseudoconcave function on convex set S. If f is both pseudoconvex and pseudoconcave on the convex set S, then f is said to be a pseudolinear function on S. Let D be a subset of Rn and let F be a map from D into Rn . F is pseudomonotone on D if, for every pair of distinct points x ∈ D, y ∈ D, we have F (x)T (y − x) ≥ 0
⇒
F (y)T (x − y) ≤ 0.
Definition 2.1 (See [9]) A set is said to be η-invex if, for any x, y ∈ , λ ∈ [0, 1], y + λη(x, y) ∈ . Definition 2.2 (See [10]) Let ⊂ Rn be an η-invex set. F : → Rn is said to be η-pseudomonotone on of Rn if, for every pair of distinct points x, y ∈ , η(y, x)T ∇F (x) ≥ 0 implies η(x, y)T F (y) ≤ 0. Definition 2.3 (See [10]) A differentiable function f on a subset of Rn is pseudoinvex with respect to η on if, for every pair of distinct points x, y ∈ , η(y, x)T ∇f (x) ≥ 0 implies f (y) ≥ f (x).
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Remark 2.1 If η(y, x) = y −x, then an η-pseudomonotone map reduces a pseudomonotone map and a pseudoinvex function reduces a pseudoconvex function. Thus, a pseudoinvex function is a generalization of a pseudoconvex function and an ηpseudomonotone map is a generalization of a pseudomonotone map. Assumption A (See [10]) Let the set be η-invex and let f : → R. Then, f (y + η(x, y)) ≤ f (x),
for any x, y ∈ .
Assumption C (See [11]) Let η : X × X → Rn . Then, for any x, y ∈ Rn and for any λ ∈ [0, 1], η(y, y + λη(x, y)) = −λη(x, y), η(x, y + λη(x, y)) = (1 − λ)η(x, y). Remark 2.2 If η(y, x) = y − x, Assumptions A and C are obviously true. Lemma 2.1 (See [10]) Let f be differentiable on an open η-invex set of Rn and let f and η satisfy Assumptions A and C, respectively. Then, f is a pseudoinvex with respect to η on if and only if ∇f is η-pseudomonotone on . 3 Characterizations of the Solution Sets We shall assume throughout the paper that the solution set of (P), denoted by S := argmin f (x), x∈S
is nonempty. Now, we state our main results as follows: Lemma 3.1 Let f be differentiable on an open η-invex set of Rn and let f and η satisfy Assumptions A and C, respectively. If f is pseudoinvex on S and x, ¯ y¯ ∈ S, then ¯ x) ¯ = ∇f (y) ¯ T η(x, ¯ y) ¯ = 0. ∇f (x) ¯ T η(y, Proof Since x, ¯ y¯ ∈ S, we obtain ∇f (x) ¯ T η(y, ¯ x) ¯ ≥ 0 and ∇f (y) ¯ T η(x, ¯ y) ¯ ≥ 0.
(1)
Based on the pseudoinvexity of f and Lemma 2.1, the inequalities (1) imply, respectively, ∇f (y) ¯ T η(x, ¯ y) ¯ ≤ 0 and ∇f (x) ¯ T η(y, ¯ x) ¯ ≤ 0. In turn, (1) and (2) imply the thesis.
(2)
The following theorem gives a characterization of the solution set of a pseudoinvex extremum problem.
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Theorem 3.1 Let f be differentiable on an open η-invex set of Rn and let f and η satisfy Assumptions A and C, respectively. If f is pseudoinvex on S and x¯ ∈ S, then S = S˜ = S˜1 , where ¯ x) = 0}, S˜ := {x ∈ S|∇f (x)T η(x,
(3)
¯ x) ≥ 0}. S˜1 := {x ∈ S|∇f (x)T η(x,
(4)
Proof If x ∈ S, then from x¯ ∈ S and Lemma 3.1, we have ∇f (x)T η(x, ¯ x) = 0. ˜ Conversely, if x ∈ S, ˜ then ˜ So, S ⊆ S. That is, x ∈ S. ¯ x) = 0. ∇f (x)T η(x, By the pseudoinvexity of f , we have f (x) ¯ ≥ f (x). Since x¯ ∈ S, the above inequality implies f (x) ¯ = f (x). That is, S˜ ⊆ S. Thus, S˜ = S. It is clear from (3) that the inclusion S ⊆ S˜1 holds. Assume that x∈S
and ∇f (x)T η(x, ¯ x) ≥ 0.
By the pseudoinvexity of f , we have f (x) ¯ ≥ f (x). From x¯ ∈ S and above inequality, we obtain f (x) ¯ = f (x). That is, S˜1 ⊆ S. Thus, S˜1 = S.
If η(x, y) = x − y, based on Theorem 3.1 we have the following results. Corollary 3.1 Let f be pseudoconvex on S; let x¯ ∈ S. Then, S = S˜1 = Sˆ1 , where S˜1 := {x ∈ S|∇f (x)T (x¯ − x) = 0}, S˜11 := {x ∈ S|∇f (x)T (x¯ − x) ≥ 0}. Remark 3.1 Corollary 3.1 improves and generalizes Theorem 1 in [1] and Theorem 3.1 in [2].
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The following theorem gives a characterization of the solution set of a pseudoinvex extremum problem. Theorem 3.2 Let f be differentiable on an open η-invex set of Rn , and let f and η satisfy Assumptions A and C, respectively. If f is pseudoinvex on S and x¯ ∈ S, then S = S ∗ = S1∗ , where ¯ T η(x, x) ¯ = ∇f (x)T η(x, ¯ x)}, S ∗ := {x ∈ S|∇f (x)
(5)
S1∗ := {x ∈ S|∇f (x) ¯ T η(x, x) ¯ ≤ ∇f (x)T η(x, ¯ x)}.
(6)
Proof (i) S ⊆ S ∗ . Let x ∈ S. From Lemma 3.1, we have ∇f (x)T η(x, ¯ x) = ∇f (x) ¯ T η(x, x) ¯ = 0. Thus, x ∈ S ∗ . (ii) S ∗ ⊆ S1∗ . This is trivial. (iii) S1∗ ⊆ S. Assume that x ∈ S1∗ . Then, x satisfies ¯ ≤ ∇f (x)T η(x, ¯ x). ∇f (x) ¯ T η(x, x)
(7)
Since x¯ ∈ S, we know that ∇f (x) ¯ T η(x, x) ¯ ≥ 0. Therefore, using (7), we have ∇f (x)T η(x, ¯ x) ≥ 0. By the pseudoinvexity of f , we have f (x) ¯ ≥ f (x). From x¯ ∈ S and the above inequality, we obtain f (x) ¯ = f (x). That is, x ∈ S. Thus, S1∗ ⊆ S.
Corollary 3.2 Let f be pseudoconvex on S and x¯ ∈ S. Then, S = S 2 = S 3 , where S 2 := {x ∈ S|∇f (x) ¯ T (x − x) ¯ = ∇f (x)T (x¯ − x)},
(8)
S 3 := {x ∈ S|∇f (x) ¯ T (x − x) ¯ ≤ ∇f (x)T (x¯ − x)}.
(9)
Remark 3.2 Corollary 3.3 improves and generalizes Theorem 3.2 in [2]
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