Generalized parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions Page: 187
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Math Sci (2016) 10:185-199 187
(x, x*; #(x, x*)Vf(x*)) + (z, V2f(x*)z)
> - p(x,x*)0(x,x*)'
- (f(x) -f(x*))(>) >0.
The function f is said to be (strictly) (F, /3, 0, p, 0, m)-
pseudosounivex on X if it is (strictly) (F, /3, 4, p, 0, m) -
pseudosounivex at each x* E X.
Definition 1.3 The function f is said to be (prestrictly)
(F, /3, , p, 0, m)-quasisounivex at x* if there exist func-
tions /3: Xx X - R+, : - R, p : X x X R,0:
X x X - R", a sublinear function .F(x, x*;.) : " -n R,
and a positive integer m, such that for each x E X and
Z E R",
0(f(x) -f(x*))(<) <o0
z c
4 r(x, x*;/3(x,x*)Vf(x*)) +(z, V2f(x*)z)
< - p(x,x*)0(x,x*)'.
The function f is said to be (prestrictly) (F, 3, , p, 0, m)-
quasisounivex on X if it is (prestrictly) (F, /, , p, 0, m)-
quasisounivex at each x* E X.
From the above definitions, it is clear that if f is
(F, /3, 4), p, 8, m)-sounivex at x*, then it is both
(F, /3, 4, p, 8, m)-pseudosounivex and (.F, /3, 4, p, 8, m)-
quasisounivex at x*, if f is (.F, /3, 0, p, 0, m)-quasisounivex
at x*, then it is prestrictly (F, /3, 0, p, 0, m)-quasisounivex
at x*, and iff is strictly (F, /3, , p, 0, m)-pseudosounivex at
x*, then it is (F, /3, 4, p, 0, m)-quasisounivex at x*.
In the proofs of the duality theorems, sometimes, it may
be more convenient to use certain alternative but equivalent
forms of the above definitions. These are obtained by
considering the contrapositive statements. For example,
(, /, <, p, 0, m)-quasisounivexity can be defined in the
following equivalent way:
The function f is said to be (F, /3, 0, p, 0, m)-qua-
sisounivex at x* if there exist functions /3 : X x X -*
R+, 0 : R ->,p:XxX-R,0:XxX -> , a sub-
linear function F(x, x*;.-) : R -> R, and a positive integer
m, such that for each x E X and z E R,
F (x, x*; /(x,x*)Vf(x*)) + (z, V2f(x*)z)
> - p (x, x* )118(x, x*) I '
- q5(f(x) -f(x*))> 0.
Needless to say that the new classes of generalized convex
functions specified in Definitions 1.1-1.3 contain a variety
of special cases that can easily be identified by appropriate
choices of JF, /3, 0, p, 0, and m. For example, if let
r (x, x*; Vf(x*)) (Vf(x*),u(x,x*)) and /3(x, x*) =-1,
then we obtain the definitions of (strictly) (, , p, 0, m)-sonvex, (strictly) (0, , p, 0, m)-pseudosonvex, and (pre-
strictly) (0, , p, 0, m)-quasisonvex functions introduced
recently in [10], where the "second-order invexity" is
compactly abbreviated as "sonvexity." The notion of the
sonvexity/generalized sonvexity has been applied in
developing a new optimality-duality theory in nonlinear
programming based on second-order necessary and suffi-
cient optimality conditions [1, 8-10, 12, 22].
Definition 1.4 The function f is said to be (strictly)
(s, t, p, 0, m)-sonvex at x* if there exist functions
S: -R,:XxX - R",p: X x X - R, and 0:Xx
X - D", and a positive integer m, such that for each x E
X(x f x*) and z E[f,
0b(f(x) -f(x*))( >) > (Vf(x*),u(x,x*)) + -K(z, V2f(x*)z)
2
+ p(x, x*)0(x, x*)
The function f is said to be (strictly) (, , p, 0, m)-sonvex
on X if it is (strictly) (0, , p, 0, m)-sonvex at each x* E X.
Definition 1.5 The function f is said to be (strictly)
(0, q, p, 0, m)-pseudosonvex at x* if there exist functions
O: R -R,:XxX - R",p:Xx X - R, and 0:Xx
X - [", and a positive integer m, such that for each x E
X(x f x*) and z E R,Vf (x*),(x (x*))(+>)(Z,Vf(x*)z)
-> (f (X) -f (x*)) ( > )> 0p(x, x*)0(x, x*) I'"
equivalently,
5 (x) -f(x*))(<;)<o - (Vf(x*),u(x,x*)) +-2(z, V2f(x*)z)
< - p(x, x*)IO(x, x*)1 m.
The function f is said to be (strictly) (0, , p, 0, im)-pseu-
dosonvex on X if it is (strictly) (0, , p, 0, m)-pseudosonvex
at each x* E X.
Definition 1.6 The function f is said to be (prestrictly)
(, , p, 0, m)-quasisonvex at x* if there exist functions
S: -R,:Xx X ->R",p:X x X - R, and 0:Xx
X - D", and a positive integer m, such that for each x E X
and z E EW,
4(f(x) -f(x*))(<) 0 -(Vf(x*),u(x,x*))
1
+ (z, V2f(x*)z) -p(x, x*)0(x, x*) I'",
2
equivalently
KVf(x*),(xx*))+KzV2f(x*)z) > - p(x,x*) 0(x,x*) In
2
4b(f(x) _f(x*))(>) >0.Springer
Math Sci (2016) 10:185-199
187
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Zalmai, G. J. & Verma, Ram U. Generalized parameter-free duality models in discrete minmax fractional programming based on second-order optimality conditions, article, November 8, 2016; London, UK. (https://digital.library.unt.edu/ark:/67531/metadc967166/m1/3/: accessed May 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.