Conditions of optimality for constrained optimization proble

Masserli, E. J.

Polak, E.

English

NASA Technical Reports Server

Second Order Conditions of Optimality for Constrained 
Optimization Problems in Finite Dimensional Spaces 
by. 
E. Polak and E. J. Messerli 
ABSTRACT 
First-order necessary conditions of optimality f o r  many problems in 
optimal control, nonlinear programming, and the calculus of variations can be 
obtained by transcribing these problems into a simple canonical form. 
dimensional space this form reads: 
(1) Basic Problem: 
functions and S2 a given subset of En. 
all x E ~2 satisfying r(x) = 0, f(x^) - < f(x). 
is of the form: 
(2) If 
vector + E Emtl with +O < 0 such that 
In finite 
Let f : En --f E', r : En -Em be continuously differentiable 
Find 2 E C2 such that r(G) = 0, and, for 
_ _  Roughly, the most general necessary condition for the Basic Problem (1) 
is an optimal solution to the Basic Problem (1), then there is a nonzero. 
- m .  
i= 1 
( $o Vf(g) t Z 4' Vri(G), 6x) - < 0 for  all 6x in a convex cone which "approxi: 
mates" the set S2 at the optimal solution 2. 
such as when the gradients Vf(g), Vr1(2), . . . , orm(;) a re  linearly dependent, lead- 
ing to a need f o r  auxiliary necessary conditions for these cases. 
In this paper the special case of linear dependence caused by the gradient 
Vf(G) being zero is investigated. A new condition, involving second order partial 
derivatives of the equality constraint function r( e ) ,  and first  order partial deri-  
vatives of the equality constraint function r( . ), This new condition 
holds for  all perturbation vectors in the convex cone associated with the generel 
(first order) necessary condition (2)  - a set which is larger than the one usually 
This general necessary condition can be satisfied trivially in some cases, 
. 
is obtained. 
considered in obtaining second order conditions. 
The research reported herein was supported by the National Aeronautics and 
Space Administration under Grant NsG-354, Supplement 4. 
The authors a re  with the Electronics Research Laboratory, Department of Elec- 
trical Engineering and Computer Sciences, University of California, Berkeley. 
https://ntrs.nasa.gov/search.jsp?R=19680017930 2020-03-12T10:43:45+00:00Z
Second Order Conditions of Optimality for Constrained 
Optimization Problems in Finite Dimensional Spaces 
SUMMARY 
Introduction: In the last few years it has been shown [l, 21 that problems of the 
calculus of variations, nonlinear programming and optimal control can be treated 
in a unified manner as  far  a s  necessary conditions of optimality a re  concerned. 
This was done by establishing that all these problems can be transcribed into a 
simple canonical form, for which necessary conditions were developed. Special- 
ized necessary conditions of optimality for any particular problem then followed 
from the structure of the problem. 
as follows: 
1. 
functions, and let 52 be a subset of En. 
and (ii) for every x in 52 with r(x) = 0, f(x) > f(G). 
min(f(x) Ir(x) = 0, q(x) < 0 )  , where f:En-+Ef, r:En+ Em and q:En+ Ek are  continu- 
ously differentiable, is recognized to be the Basic Problem (1) with R = {x lq(x) < 0)  . 
Examples of the transcription of discrete optimal control problems to Basic Prob- 
lem form may be found in [l] . 
Before giving the necessary condition for the Basic Problem (1) we require 
an "approximation" of the set 52 at a given point. 
2. 
the constraint set R at  x if for any collection { 6x1, * .  . , bXk] of linearly indepen- 
dent vectors in C(2, R) there exists an E > 0 (possibly depending on G, 6x1, : . . , 6xk)J 
and a continuous map P(  ) from the convex hull (co) of (0, 6x1, . . ., 6xk) into 
S2 - such that P (  6x) = E 6x t O ( E  6x) where I I O(E bx) I I/ 11. E 6x1 I 4 0 as E --t 0, uniformly 
for 6xe CO(O, 6x1, . , . , 6xk} . 
following one. 
3. Fundamental Theorem: If 2 is a solution to the Basic Problem (l), and
C(x, 52 
.$-=A$-. ,, . . . $ J ~ )  in Em'1 with Go C 0 such that for every 6x in the closure, 
For finite dimensional problems the canonical form mentioned above reads 
n Basic Problem: Let f : En+E',  r : E +Em be continuously differentiable 
Find a vector E 52 such that: (i) r(2) = 0 
Thus, for example, the usual Nonlinear Programming Problem, 
1 . -  
Definition: A convex cone C(E,Q) will  be called a conical approximation of * 
The most general necessary condition fo r  the Basic Problem (1) is the 
is a conical approximation of 0 a t  2, then there exists a nonzero vector 
- 
* d  
C(2,0), of C(G,R): 
4. 
The research reported herein was supported by the National Aeronautics and 
Space Administration under Grant NsG- 354, Supplement 4. 
The authors a re  with the Electronics Research Laboratory, Department of Elec- 
trical Engineering and Computer Sciences, University of California, Berkeley. 
Note that the Fundamental Theorem ( 3 )  may become degenerate in two 
The first occurs when +O must be chosen to be zero, and hence no infor- 
This 
ways. 
mation about the cost function f( 0 )  enters into the necessary condition (4). 
most often occurs when there is only one x E $2 satisfying r(x) = 0, and may be 
avoided by introducing a regularity condition, such as the Kuhn- Tucker constraint 
qualification, on r( 0 ) and a. The Fundamental Theorem also becomes degener- 
ate when the vectors Vf(z), Vrl(f) ,  . . ., arm(;) a r e  linearly dependent since then 
m 
one can always choose a + # 0 which satisfies +OVf(;) + Z 
i= 1 
hence (4), without reference to the optimality of 2. 
desirable to have a second-order necessary condition. 
cases when a second-order condition is also meaningful. 
C(z, S2) there a re  “critical” vectors y which satisfy (Vf(G), y) = 0 and 
(Vrl(x), y) = 0 
these vectors relations of the form: 
5. 
or 
6. 
1Ci Vri(G) = 0, and 
When a degeneracy in the first-order condition occurs, it is obviously 
However, there a re  other 
Thus, suppose that in 
. A  
i = 1, . . , m. Then, under suitable assumptions, one obtains fo r  
(3 Y 2 0 T a2f y - 5  ax 
(x) - r; u 2 (XI )  Y 2 0 y (-+x)- 22 x - 
ax i= 1 ax i= 1 ax 
2 m i a Z r i . +  k i a q  2 i A ~ a f  A 
2 
(see, fo r  example [ 3 ] ,  [4] ). 
In this paper we consider a special case of degeneracy in the first-order 
condition (4), namely the case when Vf(2) = 0, which causes the vectors Vf(;), 
Vrl(G), Vrz(G), However, we shall not re- 
strict ourselves to critical directions only as  in [ 3 ] ,  [4], and, instead, we shall 
obtain a condition similar to (4), but with 6x - (f)  bx playing the role of 
. . , arm(;) to be linearly dependent. 
Ta2f 
ax2 
(Vf(+, 6x). 
11. A Second-Order Condition: Let us assume that is a solution to the Basic 
Problem (1) such that Vf(2) = 0, and suppose that f (  0 )  is twice continuously dif- 
ferentiable. Then to the Fundamental Theorem ( 3 )  we can add the following new 
second-order condition: 
7. is a solution to the Basic Problem (1) such that Vf(G) = 0, 
and C(2, S2) is a conical approximation of S2 at the point G, then the ray R, 
8. R =  { Y E E  I y =  ~ ( - l , O , O , O . e . , O )  @ L O }  
has no points in the interior of the set  L defined by: 
Theorem: If 
A m t l  
7 
An equivalent statement of the Fundamental Theorem ( 3 )  is that the ray R 
given by (8) has no points in the interior of the set Lo defined by: 
Since L 
essence of the statement of Theorem (3 )  in the original form given. 
there is a curved surface which separates R and L - rather than a plane. 
fact a parboloid of the form: 
is also a convex cone, Lo and R must be separated, which is the 
0 
Since L defined in (9) is not in general convex it is natural to inquire if 
In 
11. with Lo < 0 N 0 0  i i 2  g,(y)= x Y + x ( Y  1 - i= 1 
is the logical candidate, which, on substituting Y E  L gives the following quad- 
ratic in dx, 
We a re  thus led to the following consequence of Theorem (7). 
13. Theorem: If 2 is a solution to the Basic Problem (1) such that V f ( f )  = 0, 
and C(G,!i2) is a conical approximation of 0 at 2, then there is a nonzero vector 
X E  Emtl with A'< - 0 such that for every 6x in C(;,!i2), 
0 15. Moreover, if the ray R (8) is not a boundary ray of the set L ( 9 ) ,  X may 
be taken as  -1. 
16. 
hO = 0. 
fication (15) allows a nontrivial statement f o r  many problems. 
Remark: The relation (14) can always be satisfied trivially i f  we allow 
While there a re  cases in which Xo must be chosen to be zero, the quali- 
Thus, let u s  again consider the Nonlinear Programming Problem, 
17. 
n m  , where f : E-E' is assumed twice continuously differentiable, and r : E + E  
q : En+ Ek a re  continuously differentiable, 
i Define I(x) = { i f  (1, . , ., k} I ql(x) = 0)  and I C(x) = {yl (Vq (x), y > (  0 i c  I(x)).  
We now' obtainAthe following condition from Theorem (14). 
Theorem: If x is a solution to  the Nonlinear Programming Problem (17) 18. 
such that Vf(2) = 0, IC(2 )  i s  not empty, and the Kuhn-Tucker Constraint Qualifi- 
cation [ 5 ]  i s  satisfied, then there is a vector X in Em such that 
. 111. Conclusions: We have shown in this paper that, when first-order necessary 
conditions of optimality fail because the gradient of the cost function at  the optimal 
point is  zero, it is possible to replace these first-order conditions with a 
condition. This new condition, involving second-order partial derivatives of the 
cost function and first-order partial derivatives of the equality constraint func- 
tion, holds for all perturbation vectors in a set which is larger than the one 
usually considered in obtaining second-order conditions. 
-3-  
References : 
1. M. Canon, C. Cullum, and E. Polak, "Constrained Minimization Problems 
in Finite Dimensional Spaces, ' I  J. SLAM Control, Vol. 4, No. 3, 1966, 
pp. 528-547 
2. N. Da Cunha and E. Polak, "Constrained Minimization under Vector-Valued 
Criteria in Linear Topological Spaces, 
Academic Press, New York, 1967. 
in Mathematical Theory of Control, 
3. M. R. Hestenes, Calculus of Variations and Optimal Control Theory, 
Wiley, New York, 1966. 
J. SIAMApp. Math., Vol. 15, No. 3, pp. 641-652, 1967. 
5. H. W. Kuhn and A. W. Tucker, "Nonlinear Programming, I'  Proc. of the 
Second Berkeley Symposium on Mathematical Statistics and Probability, 
University of California Press, Berkeley, California, 19 51, pp. 481-492. 
4. G. P. McCormick, "Second Order Conditions for Constrained Minima, 
- 4- 


Second order conditions of optimality for constrained optimization problems in finite dimensional spaces

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server