-invex-type functions on differentiable manifolds.
M.Ferrara1
Abstract. In this paper the Author introduces new properties about invex
functions in the frame of mathematical programming on differentiable
manifolds.
Mathematics subject Classification(2000): Primary 90C30, Secondary 53B20.
Journal of Economic Literature Classification: C00, C61.
Key Words: Generalized invexity, Riemannian manifolds, Kuhn-Tucker conditions.
1. Introduction
The aim of this paper is to make new remarks about results of invex
programming obtained on Riemannian manifolds. In 1994 C. Udriste [see [11]]
provides an exhaustive exposition of convex programming on Riemannian
manifolds. Here, the Author proved the classic Kuhn-Tucker Theorem for
the problem
min f x 
xA
where f: MR is a geodesically convex function, (M,g) denotes a
Riemannian manifold and
A  x  M : hi (x)  0; i  1,...., s
where hi: MR is a geodesically convex function for all i = 1,.....,s
Such a generalization is based on the fact that some properties of convex
programs on Euclidean spaces can be used in the case of a Riemannian
manifold.
In 1994, R. Pini has introduced a definition of the invex function for
functions defined on general differentiable manifolds. In fact in [7] Pini
proved that the Kuhn-Tucker conditions are necessary and sufficient in
order that a point is a solution for the problem
min f(x )
xS


where f: RnR is -invex and S  x  R n : h i x   0; i  1,....s , with
1
Associate Professor, Faculty of Economics, University of Messina, Via dei Verdi, 75,
98122 Messina (ITALY). E-mail adress: [email protected]
1
hi: RnR -invex for all i = 1,...., s. The main result of the present paper is
the proof of the validity of some theorems introduced in [1], with reference
to differentiable manifolds.
2. Notation and definitions.
This section, contains an outline of some important notions, concerning
Riemannian geometry and which will be used in the sequel.
Definition 2.1
A differentiable manifold of dimension n is a set M and a family of injective
mappings x α : U α  R n  M of open sets U α of R n into M such that:
(1)
x (U


)  M;
(2) for any pair ,  with x α (U α )  x β (U β )  W  . The set x α1 (W ) and
x β1 (W) are open sets in Rn and the mappings xβ1  x α : x -1α (W)  xβ (W) are
differentiable:
(3) the familyU, x is maximal relative to the conditions (1) e (2), each
other family V, ysatisfying (1) and (2) is included in U, x.
The pair U, x with p  x (U) is called a parametrization (or system of
coordinates) of M at p; x  (U) is then called a coordinate neighbourhood
at p.
We know that each differentiable manifold can be made into a topological
space using the family of finite intersection of coordinate neighbourhoods as
a base of the topology. Usually, we can suppose that the manifold is an
Hausdorff space with countable base.
Definition 2.2
Let M be a differentiable manifold. A differentiable function : (-, )  M
is called a curve in M. Suppose that (0) = p and let  be the set of functions
f: M  R differentiable at p. The tangent vector to the curve  at t = 0 is a
function  (0):   R given by:
α' (0) f 
df  α 
f
dt t  0
A tangent vector to M at p is the tangent vector at t = 0 of some curve
: (-, )M with (0) = p. The set of all tangent vectors to M at p will be
indicate by TpM and it will be called tangent space to M at p.
2
Let f: M  R a differentiable function. Its differential dfp at p is a linear
mapping beetwen TpM and Tf(p) R  R.
Therefore dfp is an element of the dual space of TpM.
After these notions, now, we consider only Riemannian manifold.
Definition 2.4
A Riemannian metric g on a differentiable manifolds M is a correspondence
which associates to each point p of M an inner product gp on the tangent
space TpM such that, if x: U  Rn M is a system of coordinates around p,
 


(q),
(q)  g ij (x(q))  is a differentiable function for each
then g p 
 x

x j
 i

q  U.
Definition 2.5
Let M be a differentiable manifold and g is a Riemannian metric on M. The
pair (M, g) is called Riemannian manifold.
As it happens in Rn, we can introduce some differential operators on a
Riemannian manifolds. Let f be a function of class C1 (M), then grad f (or df)
is defined by:
gp (grad fp, v) = dfp (v)  v  Tp M
It is clear that grad fp  Tp M.
Definition 2.6
A Riemannian manifold M is geodesically complete when for all p  M, any
geodesic  starting from p is defined for all values of the parameter t  R.
In the case of complete Riemannian manifolds, one of the most important
results is the Hopf-Rinow Theorem. This theorem states that the following
assertions are equivalent:
1.
M is geodesically complete;
2.
M is complete metric space endowed with the metric d defined in this
way:
p,qM d(p,q)= inf l()   :a, bM piecewise differentiable
curve; a = p, b = q
3.
The closed and bounded sets of M are compact.
In addition the Hopf-Rinow Theorem states that M is a geodesically
complete manifold, so that for each p, q  M there exist a geodesic arc 
joining p to q, moreover  is such that d(p, q) = l(), i. e. . minimizes the
length of the curves passing through p and q. This important property of
Hopf-Rinow Theorem explaines the reason why, when considering a
geodesically convex function defined on a Riemannian manifolds (M,g), M
must be a complete manifold.
3
3. Generalized invex functions on differentiable manifold.
Considering the concept of geodesically convex function we assume that:
a function f: M  R is geodesically convex when its restrictions to each
geodesic arc of M are convex.
Definition 3.1
The function f : M  R is geodesically convex iff for each geodesic arc : a, b 
M and for each 0, 1 we have:
f(  ((1   ) a  b )  (1   ) f ( (a))   f ( (b))
We can have the usual notion of convexity when we consider the manifold
(M, g) as the space Rn endowed with the Euclidean space.
At this point, the concept of invexity, born in the Euclidean case, can be also
extended to differentiable manifolds. As it has been shown by R.Pini
see7, if we consider a complete Riemannian manifold (M,g) (definition
2.6) there exists a map  (see definition below) which has the same role
played in classic definitions of invexity in Euclidean spaces. Hence the class
of -invex functions are the geodesically convex functions on a Riemannian
manifold.
Definition 3.2
Let (M, g) be a complete Riemannian manifolds of dimension n and A  M.
Let
f : M  R be a differentiable function and  a map such that:
(3.1)
: M x M  TM
We know that a function  see 1 is said to be -invex at y  X  Rn iff:
(x) – (y)  ((x, y),  (y))
xX
Now, we consider a complete Riemannian manifolds (M,g).We have the
following:
Definition 3.3
The function f : M  R is said to be -invex on M iff :
f(p) – f(q)  ( (p, q)) dfq for every p, q  M
where  is a map such that (p, q)  T q M  p, q  M,
f is said to be -pseudoinvex at q  A if p  A, ( (p, q)) dfq  0  f(p) – f(q)  0
and f is said -quasiinvex at qA if pA, f(p) – f(q)  0  ((p, q)) dfq  0.
4
Starting from above definition, we can introduce the following:
Definition 3.4
f is -inf invex at qA0, with respect to A  M, iff:
inf f p  f q   inf ηp, q  dfq
pA0
pA0
f is -supinvex at qA0, with respect to A  M iff:
sup f p  f q   sup ηp, q  dfq
p A0
p A0
f is -inf-pseudoinvex at q, with respect to A0 iff:
inf ηp, q dfq  0  inf f p  f q   0
pA0
p A0
f is -sup-quasiinvex at qA0 iff:
sup f p  f q   0  sup ηp, q  dfq  0 .
p A0
p A0
We have the following general definitions:
Definition 3.5
f is -inf-invex (-sup invex, -inf pseudoinvex, -supquasiinvex) on
AM, with respect A0 iff it is -inf invex (-sup invex, -inf pseudoinvex,
-supquasiinvex) at any point q  A  M.
The previously properties introduced can be used in a classic framework of
scalar optimization problem:
inf f p where A0  p  A hjp   0,
pA0
j  1,...., m 
Definition 3.6
The problem (P) is -inf-sup invex at q  A, iff f:MR is -inf-invex at q,
with respect to A0, and hj with j = 1,…m are -sup invex at q with respect to
A0. (P) is -inf-sup invex on A iff it is -inf-sup invex at every point q  A.
Let consider us (D) a classical Wolfe dual in A  M of (P)
m


( D) sup  p , V  f  p    v j h j  p ,
j 1


5
m


where V   p,v   Mx Rm grad f  p    v j grad h j  p   0
j 1


4. Sufficency of inf-sup invexity in Kuhn-Tucker conditions.
Let f : MR and hj :MR be two functions differentiable on M of class C1.
A Kuhn-Tucker point (or K-T point) is a pair (p0, v)  M x Rm satisfying
the following conditions:
m
grad f  p0    v j h j  p0   0
(4.1)
j 1
m
v h  p   0
(4.2)
j 1
j
j
0
for q  A0.
Let J0 (q) the set of active constraints at q such that:
J 0 q   j h j q0   0
Theorem 4.1
Let (p0, v) be any Kuhn-Tucker point of the problem (P). Iff there exists
some , such that f is -inf pseudoinvex at p0 with respect to A0 and for
every j  J0 (p0), hj is -sup.quasiinvex at q  A, with respect A0, then p0 is
an optimum solution of (P).
Proof. Starting from the conditions (4.1) and (4.2) one has:
inf η p, p0  grad f  p0   inf
p A0
p A0
  v η p, p dh  p 
jJ 0  p 0 
j
0
j
0
Obviously,
inf
p A0
  v η p, p grad h  p     v
jJ 0  p 0 
j
0
j

0
jJ 0 p 0
j
sup η p, p0 grad h j  p0 
p A

We know that sup h j  p   h j  p0   0 whenever j J0(p0), it follows finally
p A0
from:
  (v ) sup η p, p grad h  p   0
jJ 0  p 0 
j
0
p A0
6
j
0
and hence, one has
inf  f  p   f  p0   0 
p A0
Corollary 1.
Let (p0, v) be a K-T point of the problem (P). Iff there exists some , such
that f is -infinvex at p0 with respect A0, and for every j  J0 (p), hj is supinvex at q  A with respect A0, then p0 is an optimum solution of (P).
Corollary 2.
Assume that there exists some , such that (P) is -inf-supinvex on A. Then
for every K-T point (p0, v), p0 is an optimum solution.
Remark
We also assume that:
j  J 0*  p0   sup η p, p0 grad h j  p0   0
(•)
pA
Consequentely, one obtains:
Corollary 3.
Let (p0, v) be any point of the problem (P). Iff there exists some , such that
the constraint function satisfy (•) and the objective function satisfies:
inf
p A0
 f  p   f  p0   0
  p, p0  grad f  p0   0  pinf
A
(••)
0
then p0 is an optimum solution.
5. Necessity of inf-sup invexity.
We can call Kuhn-Tucker invex on A  M the problem (P) iff there exists a
map :MxMTM such that:
 f  p   f q   η p, q  d f q

p, q, A0  η p, q grad h j q   0 whenever h j q   0
 for j  1,2,.........., m

7
Theorem 5.1
Every K-T point of problem (P) is a global minimizer iff (P) is K-T invex.
We can exstend these results using the concepts introduced considering
a Riemannian manifold (M,g) of dimension n.
One has the following:
Theorem 5.2
A K-T point (p0, v) is a global minimizer of the problem (P) iff there exists a
map :MxMTM, such that (•) and (••) hold.
Proof. The sufficency follows from corollary 3 of theorem 4.1. For the
necessity one observes that K-T invexity implies the conditions of the
theorem.
It is also obvious the following:
Theorem 5.3
Every K-T point of problem (P) is a global minimizer iff there exists a
map :MxMTM, such that f is -infpseudo invex (or -inf-invex) on
A0, and for every q  A0 , hj is -sup quasiinvex (-sup-invex) at q,
whenever j J0(q).
Acknowledgements.
The author is grateful to Proff. F.Giannessi and M.I.Stoka for their attention,
kindness and fruitful discussions.
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