Global Minimum Point of a Convex Function. J. M. Soriano. Depatiamento de An&is Mater&tic0. Facultad de Matem'ticas. Universidad de Se&la. Apart. 1160.
Global Minimum Point of a Convex Function J. M. Soriano Depatiamento de An&is Facultad de Matem’ticas Universidad de Se&la
Mater&tic0
Apart. 1160 41080 Sevilla, Spain
ABSTRACT In this paper we prove the existence of an unique global minimum point of a convex function under some smoothness conditions. Our proof permits us to calculate numerically such a minimum point utilizing a constructive homotopy method.
1.
INTRODUCTION
With the invention of high-speed computers, large-scale problems from such diverse fields as economics, agriculture, military planning, and flows in networks became at least potentially solvable, a lot of them being extremum problems. The great importance of extremum problems in applied mathematics leads us to the general study of the extremum of functions from R” to R. It is not easy to know the extremum points, for differentiable functions because it is not always possible to solve the equation Vf = 0 to calculate critical points. Convex functions have a particularly simple extremal structure [2], and there exist algorithms to calculate extremum points, supposing its existence. However, it is not easy to prove the existence of extremum even in the case of convex differentiable functions [2, 31. Therefore, it is very important to give sufficient conditions to guarantee this existence.
2.
PROOF
OF A UNIQUE
GLOBAL
MINIMUM
POINT
Given a strictly convex function f from R” to R, we prove the existence of a unique global minimum point for f if the following condition is verified: APPLIED MATHEMATlCS
AND COMPUTATION
55:213-218
(1993)
213
0 Elsevier Science Publishing Co., Inc., 1993 655 Avenue of the Americas, New York, NY 10010
0096-3003/93/$6.00
J. M. SORIANO
214
Sf( X)/6X,
lim
>
X-m
0
foranyvalueofi
This proof is founded on the continuation to determine
E l,...,n.
(1)
xi
that point numerically
method
and the method can serve
as we have shown in [S, 61; see also [I],
and [7].
THEOREM 1. Let f: R” + R, f E C3(R”) be a strictly convex function (1). Th en there exists an unique minimum point for f.
verifying
PROOF.
We
have Vf: R” *
R”, with Of E C2(R”>. We construct
the
function
H:R”[O,
l] --f R”,
= (1 -t)x+tVf(X).
liI(X,t)
1) Let us first prove that zero is a regular value for H. Since f is strictly convex, zero is a regular value for Of, and hence
rank
S2f(X)
( i. . . 6X&$
Moreover,
the former
bilinear transformation. Let us now consider
l 0. Therefore, t(s)
dt/ds is
=
strictly
Minimum 4)
of Convex
217
Functions
Let us now show that Y * can only be continued
IlXll. This is a consequence
of the condition
by bounded
(1) of the theorem
values of
because
the
equation tVf(X)+(l-t)X=O leads us to ‘f(
“)/sxi
=-
1-t
-t
‘i
’
i = 1,.
. . ,n,
(5)
and t E (0, 11,(1 - t/ - t) =G 0 ( or - ~0 when t --f 0). But (5) is absurd for a sufficiently great 1)X 1) since it implies lim
sf(x>/xi
< xi
IIXII-fm
o
’
against the hypothesis. 5) The right extreme point of the maximal continuation of Y * belongs to the hyperplane t = 1. That follows from det H’,(Y) (i = 1,. . . , n) being continuously differentiable, and the system (4) autonomous; so it is possible to continue
Y * to the boundary of R"[O,l] with s E [0, + a) [4]. Clearly, that right extreme point T+ of the maximal prolongation of Y * cannot coincide with the initial value t = 0 by 3), and the trajectory of this prolongation is only defined for bounded values of X by 4). Therefore, (A,l)
T+= 6)
E R”(1).
Finally, let’s note that A is a minimum lim
H(X,t)
local point for f(X)
since
= 0
(X,t)-r(A,l) implies that Vf( A) = 0 due to the continuity of H. Theorem B [2, p. 1241 implies that this minimum
is global and
A
is the
unique minimum point for f.
8
We don’t develop here the numerical aspect, but is has been developed similar conditions i; other papers of ours [k, 6j.
in
REFERENCES 1
C. B. Garcia and W. I. Zangwill, Pathways Equilibria, London,
Prentice-Hall 1981.
Series
to Solution,
in Computational
Fixed
Points
and
Mathematics. Prentice-Hall,
J. M. SORIANO
218 2
A. W. Roberts
3
R. T. Rockafellar, University
4
and D. E. Varberg,
Press,
I. G. Petrovski,
Convex Princeton, Ordinary
Convex functions. Princeton
Analysis,
New Jersey, Differential
Pure Appl.
Mathematical
Math. 57 (1973).
Series
28 Princeton
1970. Dover
Equations,
Publications,
New York,
1966. 5
J. M. Soriano,
Sobre
Reu. Real Acad. 6
J.
M.
7
E. Zeidler,
Collect.
Soriano,
Cienc.
las existencia Exact.
A special
Math. 41(1):45-58
New York,
Nonlinear 1985.
y el cilculo
Fis. Nutur.
type
de ceros
Madrid
of triangulation
de funciones
LXXXII(3-4):523-531
in numerical
nonlinear
regulares, (1988). analysis,
(1990).
Functional
Analysis
and its Appkcations,
Springer-Verlag,
