UNCLASSIFIED. .n ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA UNCLASSIFIED

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1 UNCLASSIFIED.n ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA UNCLASSIFIED

2 NOTICE: When government or other drawings, specifications or other data are used for any purpose other than in connection with a definitely related government procurement operation, the U. S. Government thereby incurs no responsibility, nor any obligation whatsoever; and the fact that the Government may have foiraulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use or sell any patented invention that may in any way be related thereto.

3 ./» v RESEARCH REPORT 5 22 MAY 1961 I.E.R t CC^ SUPPORTS OF A CONVEX FUNCTION <3C oo by E. Eisenberg >- CZ CD N OPERATIONS RESEARCH CENTER INSTITUTE OF ENGINEERING RESEARCH UNIVERSITY OF C A L I F 0 R N I A - B E R K E L E Y

4 SUPPORTS OF A CONVEX FUNCTION by 1 E. Eisenberg Operations Research Center University of California, Berkeley 22 May Research Report 5 This research was supported in part by the Office of Naval Research under contract Nonr-222(83) with the University of California. Reproduction in whole or in part, is t ermitted for any purpose of the United States Government.

5 SUPPORTS OF A CONVEX FUNCTION Let C be a real, symmetric, mxm, positive-semi-definite matrix. Let R = -Ux,, -Mx,,..^x.., )x. isa real number, i = 1,..., m ^ y, and let [I mi ; K c: R be a polyhedral Dlyhedral convex cone, i. e., there exists a real mxn matrix A such that K=- x xer : x xer m and xa ^0 y. Consider the function ^/: K -" R T 1/2 defined by ^(x) = (xcx ) for all x e K. U, of all supports of 0, where We wish to characterize the set, (1) U = R m r\ u xek =>ux T^ (xcx T ) 1/2 Let R = K fy -lir I ir > 0 } and consider the set (2) V = v(3xer m, TTCR^ T T and v = ira + xc, xcx < 1, xa <. 0 We shall demonstrate: THEOREM: U := V. We first show: LEMMA 1 x, yer m =>(xcy T ) < (xcx T ) (ycy T ). Proof: If x, yer consider the polynomial p(m = X xcx + ZX.xCy + ycy T (x+ X.y)C(x+ Ky). Since C is positive-semi-definite, p(x.) > 0 for all real numbers X, and thus the discriminant of p is non-positive, i.e.,

6 4(xCy T ) - 4(xCx T ) (ycy T ) < 0. q.e.d. As an immediate application of Lemma 1 we show: LEMMA 2 Vc U Proof: Let vev, then there exist x c R, w e R suchthat v = TTA +XC, T m T 1 T T* T 1 xcx < 1. Now if yer, ya < 0, then vy = yair + xcy and vy < xcy T, because ya < 0, ir T > 0 and yair < 0. Thus, vy < (xcx T P (ycy 1 )^, T T Ji T by Lemma 1, and vy < (ycy ), because xcx < 1. Thus, v eu. q. e. d. From the fact that C is positive-semi-definite, it follows that: LEMMA 3 The set V is convex. T T Proof: If x. er, TT. er, x. A< 0, u, = ir, A + x, C, x,cx, < 1, X-, er, for k k + k kk k kk k + T k = 1, Z and V + X = 1, then - X. u, +\,u. = '.X ^ + X 172)A + + (X.x. + X. x,)c, (A..x, + X x.)a <.0 f \ jx, + \ x 2 fr m, X ir, + X. IT er", T and (X x, + X x z )C{>..x, + V x.) - 1 < < (X^^ X 2 x z )C(X 1 x 1 +X^x <,) J - X^jCx^- X z x z Cx 2 ^ x i x z x.cx, - Zx.Cx, + x.cx, - - X.X^ (x, - x- > )C(x 1 - x ) < 0, because C is positive-semi- definite, q.e.d. -Z-

7 LEMMA 4 The set V is closed. Proof: Let - w, I be a sequence with w.er, k = 1, 2 We define the (pseudo) norm of w., denoted " w.!, to be the smallest non-negative integer p such that there exists a k and for all k > k, x, has at most p nonzero componerts. Now, suppose u is in the closure of V, i.e., there exist sequences - u, >, i ^u [ an^ l^-u i suc^ that (3) ^k 611^ ' x k errn ' "k^ ^k^ + X k C rp j K x k A < 0 and y^x^ < 1. / 1,^,. and -^ u, > converges to u Suppose the sequence 4 x, > is bounded, then we may assume, having taken an appropriate subsequence, that for some xer, -< x, > ~* x and thus, by (3), xa < 0 and xcx < 1. Now, ya <, 0 ==>u, y - x, Cy = IT, A y = T T T yair < 0, all k =^uy - xcy < 0. Thus the system. y er ya < 0 (u- xc)y T > Ü has no solution and by the usual feasibility theorenn for linear inequalities (see e.q. (4) or (5)) the system: 3-

8 IT er n ira = u - xc has a solution, and thus uev. We have just demonstrated that Since «{-.} is bounded, then uev. - x. ^ + -jx, Al ^ m + n, it is always possible to choose - Xi,. r and i xr, > satisfying (3) and such that Wl + IM is minimal. We shall show next that if "I x are v r ' "i ^t, f ^ chosen, then -i x,! xs ii indeed bounded, thus completing the proof. Suppose then that - x, [ is not bounded, i.e., ^ = T 1/2 ^^Xv ^ ~' ' x > and we may assume that I x, I > 0 for all k. Let z k = 1^1 k = 1, 2... then -j z, is bounded and we may assume that there is a zer such that the z, converge to z and z = 1. From (3) it follows that z.a < 0 and T z.cz, < 1 T : r- for all k. Thus, za < 0 and zcz < 0. But then, ' X k' T m from Lemma 1, zcy = 0 for all yer, and zc = 0. Summarizing: (4),m zer,za<0, zc=0. Note that if z has a nonzero component, then infinitely many x,' s must have the same component nonzero, this follows from the fact that z is the limit of. As a consequence, if J X. > is any sequence of real "kl numbers, then -{x, + X., z INK i If za ^ 0, and a J, j = 1,..., n.

9 th denotes the j column of A, let X., = max k j za j za J < 0 Then we may replace, in (3), x. by x i+^u z because X., za J + x a J < 0 for all j, and (x, + X., z)a < 0, also ac = 0 and thus (x k +X.,z)C = x, C, T T (x, + X, z)c(x, + X, z) = x, Cx. < 1. However each (x, + X, z)a has at least one more zero component than x. A, contradicting the minimality of I i ^ir I + i x lr^'i I * Thus, za = 0 and we may replace, in (3), x, by x, + X, z for an arbitrary sequence < X, I. But z ^ 0 and we can define X so that x, + X, z has at least one more zero component than x, has, thus j i x, + X z > < N ^i. r However, (x, + X z)a = x. A, and -i (x, + X, z)a y \ = \ < ^A. f I» contradicting the minimality assumption. Lastly, we show: q. e. d. LEMMA 5 u o v Proof: Suppose u ^ V. By Lemmat 3 and 4 V is a closed convex set, hence there is a hyperplane which separates u strongly from V (see [4]), Thus there exist xer and aer suchthat T T ux > a > vx all v e V. -5-

10 Now, if ircr + then v = TTA is in V (taking x = 0 in the definition of V), Thus xair = TTA X < a for all ir e R, and xa < 0, xe K. Also v = 0 is in V, so that a > 0. If ueu then 0 < a < ux T < (xcx 1^) 1 ' 2, T thus xcx > 0 and v = xc,. e : ; T.i/2 (xcx ) v ' c on s equ e ntly. _ T.l/2.. xcx T. ^ T.l/2 (xcx ) > a > y Y/2. ~ t xcx ' (xcx ) a contradiction. Thus u ^ U. q.e.d. Note: A direct application of Lemmas 2 and 5 yields the theorem stated at the beginning. -6-

11 REFERENCES 1. Barankin, E. W. and R. Dorfman, On Quadratic Programming, Berkeley: University of California Press, Dantzig, G. B., "Quadratic Programming: A Variant of the Wolfe- Markowitz Algorithms," Berkeley: Operations Research Center, University of California, Research Report Z, 14 April Eisenberg, E., "Duality in Homogeneous Programming," to appear in Proceedings Amer. Math. Society. 4. Fenchel, W., Convex Sets, Cones, and Functions, Princeton: Princeton University Lecture Notes, Gale, D., H. W. Kuhn and A.W. Tucker, "Linear Programming and the Theory of Games," Activity Analysis of Production and Allocation, New York: Cowles Commission Monograph 13, pp Kuhn, H.W. and A.W. Tucker, "Non-Linear Programming," Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley: University of California Press, pp Markowitz, H., Portfolio Selection -- An Efficient Diversification of Investments, New York: John Wiley and Sons, Wolfe, P., "A Duality Theorem for Non-Linear Programming, " Santa Monica: The RAND Corporation, Report P-2028, 30 June I