Generalized (ρ,θ)-η-invexity and generalized (ρ,θ)-η-invariant-monotonicity
ABSTRACT In this paper several kinds of (ρ,θ)-η-invariant-monotone functions and generalized (ρ,θ)-η-invariant-monotone functions are considered. For a given Frechet differentiable function f,∇f (gradient of f), (ρ,θ)-η-invariant-monotonicity of ∇f is related to (ρ,θ)-η-invexity of f. It is also shown that (ρ,θ)-η-invariant-monotonicity and generalized (ρ,θ)-η-invariant-monotonicity are proper generalizations of invariant-monotonicity and generalized invariant-monotonicity.
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ABSTRACT: A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.Bulletin of the Australian Mathematical Society 09/1988; 38(02):177 - 189. · 0.48 Impact Factor
Article: Seven kinds of monotone maps[Show abstract] [Hide abstract]
ABSTRACT: Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalized convexity properties of the underlying function. In this way, pure first-order characterizations of various types of generalized convex functions are obtained.Journal of Optimization Theory and Applications 06/1990; 66(1):37-46. · 1.42 Impact Factor
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ABSTRACT: Generalized monotonocity of bifunctions or multifunctions is a rather new concept in optimization and nonsmooth analysis. It is shown in the present paper how quasiconvexity, pseudoconvexity, and strict pseudoconvexity of lower semicontinuous functions can be characterized via the quasimonotonicity, pseudomonotonicity, and strict pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke, and Rockafellar derivatives as well.Journal of Optimization Theory and Applications 01/1995; 84(2):361-376. · 1.42 Impact Factor
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 117, No. 3, pp. 607–625, June 2003 ( 2003)
Generalized Invexity and Generalized
X. M. YANG,2X. Q. YANG,3and K. L. TEO4
Communicated by S. Schaible
generalized invariant monotone maps are introduced. Some examples
are given which show that invariant monotonicity and generalized
invariant monotonicity are proper generalizations of monotonicity and
generalized monotonicity. Relationships between generalized invariant
monotonicity and generalized invexity are established. Our results are
generalizations of those presented by Karamardian and Schaible.
In this paper, several kinds of invariant monotone maps and
tone maps, invex functions, generalized invex functions.
Invariant monotone maps, generalized invariant mono-
Convexity is a common assumption made in mathematical program-
ming. In recent years, there have been increasing attempts to weaken the
convexity condition. Consequently, several classes of (generalized) invex
1This research was partially supported by the National Natural Science Foundation of China,
the key project of the Chinese Ministry of Education (No. 01005), EYTP, and the Research
Committee of Hong Kong Polytechnic University. The authors are thankful to Editor-in-Chief
A. Miele, Associate Editor S. Schaible, an anonymous Associate Editor, and three anonymous
referees for their many valuable comments on an early version of this paper. The authors are
also grateful to Professor B.D. Craven for some discussion on this paper.
2Professor, Department of Mathematics, Chongqing Normal University, Chongqing, China.
Current Address: Department of Applied Mathematics, Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong, China.
3Associate Professor, Department of Applied Mathematics, Hong Kong Polytechnic Univer-
sity, Hung Hom, Kowloon, Hong Kong, China.
4Professor, Department of Applied Mathematics, Hong Kong Polytechnic University, Hung
Hom, Kowloon, Hong Kong, China.
0022-3239?03?0600-0607?0 2003 Plenum Publishing Corporation
JOTA: VOL. 117, NO. 3, JUNE 2003
functions have been introduced in the literature. More specifically, the con-
cept of invexity was introduced in Ref. 1, where it is shown that the Kuhn-
Tucker conditions are sufficient for (global) optimality of nonlinear pro-
gramming problems under invexity conditions. In Ref. 2, Kaul and Kaur
presented strictly pseudoinvex, pseudoinvex, and quasiinvex functions, and
investigated their applications in nonlinear programming. In Refs. 3–4,
Weir and Mond introduced the concept of preinvex functions and applied
it to the establishment of the sufficient optimality conditions and duality
in (multiobjective) nonlinear programming. In Ref. 5, prepseudoinvex and
prequasiinvex functions were introduced and the relationships between
invexity and generalized convexity were established. In Ref. 6, Mohan and
Neogy showed that, under certain conditions, an invex function is preinvex
and a quasiinvex function is prequasiinvex. More recently, characterizations
and applications of preinvex functions, semistrictly preinvex functions, pre-
quasiinvex functions, and semistrictly prequasiinvex functions were studied
in Refs. 7–9.
A concept closely related to the convexity of a real-valued function is
the monotonicity of a vector-valued function. It is well known that the
convexity of a real-valued function is equivalent to the monotonicity of the
corresponding gradient function. It is worth noting that monotonicity has
played a very important role in the study of the existence and solution
methods of variational inequality problems. An important breakthrough
generalization of this relation was given in Ref. 10 for various pseudo?
quasiconvexities and pseudo?quasimonotonicities. Subsequently, there has
been increasing interest in the study of monotonicity and generalized mono-
tonicity and of their relationships to convexity and generalized convexity;
see Refs. 11–13. On the other hand, some relationships between generalized
invexity and generalized invariant monotonicity were given in Refs. 14–
15 under certain conditions. In particular, the existence of solutions to the
variational-like inequality problem was proven under generalized invariant
monotonicity in Ref. 15.
In this paper, we study generalized invariant monotonicity and its
relationships with generalized invexity. We introduce several types of gen-
eralized invariant monotonicities which are generalizations of the (strict)
monotonicity, (strict) pseudomonotonicity, and quasimonotonicity reported
in Ref. 10. The main purpose of this paper is to establish relations among
generalized invariant monotonicities and generalized invexities. Note that
the conditions assumed in this paper are different from those assumed in
Refs. 14–15. Several examples are given to show that these generalized
invariant monotonicities are proper generalization of the corresponding
generalized monotonicities. Moreover, some examples are also presented to
JOTA: VOL. 117, NO. 3, JUNE 2003
illustrate the properly inclusive relations among the generalized invariant
2. Invariant Monotone Maps and Strictly Invariant Monotone Maps
Let Γ be a nonempty subset of ?n, let η be a vector-valued function
from XBX into ?n(X ⊂?n), and let F be a vector-valued function from Γ
to η if there exists an η: ?nB?n→?nsuch that, for any x, y∈Γ and
See Refs. 3–4. A set Γ is said to be invex with respect
pair of points x, y∈Γ,
See Ref. 10. F is said to be monotone on Γ if, for every
invariant monotone on Γ with respect to η if, for every pair of points
Let Γ be an invex set with respect to η. F is said to be
η(x, y)TF( y)Cη(y, x)TF(x)⁄0.
with η(x, y)GyAx, but the converse is not necessarily true.
Every monotone map is an invariant monotone map
Let F and η be maps defined by
xG(x1, x2)∈(0, π?2)B(0, π?2),
η(x, y)G[(sinx1Asiny1)?cosy1, (sinx2Asiny2)?cosy2],
x, y∈(0, π?2)B[0, π?2).
Clearly, F is invariant monotone with respect to η. Let
JOTA: VOL. 117, NO. 3, JUNE 2003
(yAx) [F( y)AF(x)]G−(π?6)(13?2A12?2)F0.
Thus, F is not monotone.
and let η: XBX→? be a vector-valued function. The function f: Γ→? is
said to be preinvex with respect to η if
f(yCλη(x, y))⁄λf(x)C(1Aλ) f(y),
∀x, y∈Γ and λ∈[0, 1].
The function f is said to be strictly preinvex with respect to η if (1) holds
with strict inequality for any pair of distinct points x and y and for
See Refs. 3-4. Let the set Γ be invex with respect to η,
f: Γ→?. Then,
Let the set Γ be invex with respect to η, and let
for any x, y∈Γ.
preinvexity with λG1.
Assumption A is just the inequality of the definition of
and for any λ∈[0, 1],
η(y, yCλη(x, y))G−λη(x, y),
η(x, yCλη(x, y))G(1Aλ)η(x, y).
See Ref.6. Let η: XBX→?n. Then, for any x, y∈?n
We will show that Assumption C holds if
η(x, y)GxAyCo (??xAy??).
In fact, the following two equalities hold:
η(y, yCλη(x, y))
Gη(y, yCλ(xAyCo (??xAy??))
G−λ(xAyCo (??xAy??))Co (λ(??xAyCo (??xAy??))??)
G−λ[xAyCo (??xAy??)Co (??xAyCo (??xAy??)??)]
JOTA: VOL. 117, NO. 3, JUNE 2003
η(x, yCλη(x, y))
Thus, Assumption C holds when we take
if x¤ 0, y¤0,
if xF 0, yF0,
if xH 0, y⁄0,
if x⁄ 0, yH0.
Then, it is easy to verify that f is invex with respect to η on K and that f
and η satisfy Assumptions A and C. However, f is not convex.
The following theorem shows that the preinvexity of a function is
equivalent to the invariant monotone property of its gradient. This is a
generalization of the convexity of a function and the monotonicity of its
gradient obtained in Ref. 16.
is invex with respect to η, then ∇f is an invariant monotone with respect to
Let f be differentiable on an open set containing Γ. If f
The proof follows readily from the definitions of invexity and
be differentiable on Γ. Then, f is a preinvex function with respect to η on Γ
if and only if ∇f is invariant monotone with respect to η on Γ and f satisfies
Let f and η satisfy respectively Assumption C, and let f