E L which minimizes q'o on the set of all z C L that satisfy the constraints Ap(z) < 0 for i = -,u ... (b) lim sos 0 (y, 6) - (oj(zo) -hi(y)h = 0 for each i = 1, ..., m o > D. 6.
GENERAL NECESSARY CONDITIONS FOR OPTIMIZATION PROBLEMS* BY HUBERT HALKIN AND LUCIEN W. NEUSTADT DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA (SAN DIEGO), AND DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITY OF SOUTHERN CALIFORNIA, LOS ANGELES
Communicated by Solomon Lefschetz, August 3, 1966
Introduction.-In the derivation of necessary conditions for various optimization problems one should distinguish two basic elements: the first element is common to all problems and deals with optimization in itself, whereas the second element is an exercise in ordinary differential equations, difference equations, or partial differential equations, according to the particular nature of the problem under consideration. The present paper is a contribution to the first of these elements: we give a very general maximum principle for a mathematical programing problem over an arbitrary set. Although nonlinear optimal control problems have been our original motivation, we shall show, by an example, that the results of this paper include the standard Kuhn-Tucker necessary conditions and generalize them from finite-dimensional spaces to arbitrary linear spaces. 1\oreover, this paper includes and extends all the important necessary conditions in optimal control (with or without restricted phase coordinates). The reader will find in Dubovitskiy and Milyutin,2 Gamkrelidze,4' Halkin,6 7 and Neustadt9' 10 a suitable background for the present paper. 1. Problem.-Given a set L and real-valued functions go, i = -Iu, ..., 0, ... m defined on L (where m and ,u are given nonnegative integers), find an element zo E L which minimizes q'o on the set of all z C L that satisfy the constraints Ap(z) < 0 for i = -,u, . . ., -1 and s(pZ) = 0 for i = 1, . . ., m. 2. Assumption.-We shall make the following assumption: there exist convex sets H, (i = -,u, .. ., m) in a real linear space 8, real valued functions hi (i = -, m) defined on the corresponding Hi, and a set Al c 8 with the following properties: m m (i) O C n Hi and M c nHi; i=-A
i=-A
(ii) M is convex; (iii) the functionals hi are convex for i < 0 and linear for i > 0, and h1(O) = 0 for each i; and (iv) for every subset A of M which consists of m + 1 elements in general position [the elements yl, , Ym+' are in general position if the vectors (yj - Ym+i), for j = 1, .. ., m, are linearly independent], there exist a set D c (0,1) and a mapping 0 from [A] X D ([A ] is the convex hull of A) into L such that (a) O CD, (b) lim sos 0 (y, 6) - (oj(zo) -hi( y)h =< 0 for each i = 1,- ..., m o >D 6 . 0 for each i =
uniformly over [A], and (c) for every 5 C D and every i = 1, .. ., m the mappings (pi 0 (-,S) are continuous on [A]. In (iv) (c) continuity on [A ] is to be understood with respect to the ordinary Euclidean, finite-dimensional topology on [A]. -
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3. Maximum Principle.-Let zo be a solution of the preceding problem satisfy..., a~such that ing the above hypotheses. Then there exist real numbers am,,, m (a) j ahi(y) < O for all y E M; i =-
(-y) at 0 if i < 0; and (5) aipi(zo) = Oif i < 0. In addition, if for some subset J of -ju, ..., 0} the point 0 is an internal point (ref. 3, p. 410) of Hi for all i C J, and if 0 F M, then there exist linear functionals li(i F J) on 8 such that (e) E-J ail (y) + i=-E ath(y) < 0 for all y F M, and iE i q J (nq) lI(y) < hi(y) for all y F Hi and every i C J.
Moreover, if 8 is a topological linear space and, for some i FJ, 0 is an interior point of Hi and hf is continuous at 0, then 1i is continuous on 3. 4. Proof of the Maximum Principle.-Let us first suppose that .01(Zo) = 0 for each i < 0 and that m > 0. Let h be the linear mapping from M into Rm (Euclidean m-space) defined by the relation h(y) = (hi(y), . .., hm(y)). Let M = {y:y F M, hi(y) < 0 for i = -, ..., 0} and let K = h(M). It follows at once that ? and K are convex sets in 8 and Rm, respectively. Let us prove that 0 is not an interior point of K. Indeed, suppose the contrary, so that there is a closed m simplex S c K such that 0 is an interior point of S. We shall construct a map r from S into L and a map (I from L into Rm such that (i) D e r is a continuous map from S into Rm; (ii) x - D (x) F S for all x C S; (iii) x F S and D ° t(x) = 0 imply that (Pi
^ (z) < 0 if i < 0
(Po o (x) < vo(zo)
° (X) = 0 if i > 0. It follows from Brouwer's fixed-point theorem and (i), (ii) that there exists an xi E S such that xi - 4' (x1) = xi, i.e., 4' a t(x,) = 0. Let z1 = t(x1). The point Z1 F L. Then, from (iii), we have (p,(zj) < 0 if i < 0, spo(zi) < spo(zo) and ((z1) = 0 for i > 0, which contradicts the optimality of zo. Let h(y), j = 1, ..., m + 1, be the vertices of S, where yj G Aforj = 1, m + 1. It is easily seen that yi, ..., Ym+' are in general position. For the set A = {yi ... .X Ym+i}X let D C (0,1) and 0: [A ] X D -. L be given as per part (iv) of the m+1 m+1 assumption. If x = E fj1h(yj) F S, where E I = 1 and ft > 0 for each j, (pi
j=1
j=1
m+1
let
t(x)
=
0
( j=1 E f31yj, 6*), where 5* is some element of D which will be defined
later. If E L, let 4' (z) = -*
(Vi(z), ..., (pm(Z)). The number 5* is determined
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as follows: Let E, > 0 be such that {x: x E Rm, lxI < c-d c S, and let -2E2 = max, < j< m+1,-.f.< i < o h1(yj). By definition of M, E2> 0, and it follows from the convexity of the hi that hi(y) < -62 for all y C [A] and i < 0. Now let "* C D be such that, for every y E [A ],
cPi O (y, 6*) -fp(zo)
-
hi(b*y)
)f i° °(26*)hi(y)|
0, the statements (a), (fi),
Setting aj = fli for i = - ..., 0 and aj = (y), (6) of the maximum principle follow at once from (2) and (3). If m = 0, it is easy to show, on the basis of our hypotheses, that { (h-A(y), .... ..., < 0 for each i} = . Consequently, (a), ho(y)): y C M} nf {a II0): (i), (-y), and (6) follow as before. If (pi(zo) < 0 for some i < 0, we shall suppose, without loss of generality, that sOj(zo) =O for i = -M', ...,-1, andsoi(zo) < Ofor i = -,A ....,-a'-1. We can repeat the preceding arguments with ,u replaced by .t'. The only change that need be made is to choose 6* C D sufficiently small (when achieving the contradiction) so that (io D(x) < O for x C S and for i = -it, .. ., -g' -1 as well as for i = - 1 (it is easily seen that this can be done on the basis of our assumption). We then arrive at relations (a), (fi), and (y), but with ,u replaced by is'. Setting aj = 0 for i < -.', we obtain the desired relations (a), (d3), ("y), and (6). Let us show that if 0 C M and J is as indicated in the theorem statement, then there exist linear functionals li(i C J) on 8 such that (e) and (X) are satisfied. Let J =il . . ., ik}. We define the following two subsets of 8 X R': V, = (yX): y E Hi,, X > hil(y)}
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m
V =
{(Y"):
is C llI, )\
0 for all y E Hi, and 11* # 0. But this is impossible since 0 is an internal point of H1,. Thus, since (0,1) E V1, cl > 0. If we set li,(y) = -crll1*(y), then it is straightforward to show that m
aicll,(Y)
+
E-,
lil(y)
aih(y)
< 0 for all y E M
hi,(y) for all y C Hi,.
h12(Y)} W2 =
{(Yi):
y C M, X
