0 for i = 1 ,..., N.
This assumption is further discussed in Section 4. We also change the distance used in Aj, in agreement with the procedure of Area et al. [3] and with the greater role that the rows of C play, since we force the sum of each row to a prescribed constant. More precisely, let Jt = {j: (i,j) e J) and note that, because of (A), Jt is nonempty for i = 1 ,..., N. Then, if C and D are in Ay, let
d'x(C, D) = \Y, (maxllogCCyD^lYT. Note that dx = d'x for a = oo only. Finally, let =
{CeAy:CiJ=CJi}.
(6)
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PROBLEM 1 Find the symmetric matrix C e Aj such that dJjC, C°) = min da(C, C°) over all symmetric matrices C e Aj.
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We can then state the problem of finding the consistent contact matrix closest to the assigned one, as follows. PROBLEM
matrices
2 Find the matrix CeAnB CeAnB.
such that d'a(C, C°) = min d'x(C, C°) over all
Finding a solution to Problem 2 appears to be rather difficult. When n = 2, the solution (unique for a > 1) can be found, but its explicit expression depends on the value of several inequalities. For instance, setting p = p°l2 < 1 and q = p21 < 1, if T\V > T2q, p ^ i and T2(\ -q)^
Txp, then C 1 2 = {TYpT2q)112
for all values of a > 1.
PROBLEM 2a Given a matrix C e B, find the matrix De A such that dx(C, D) = min da(C, D) over all matrices D e A.
2b Given a matrix D e A such that Da > 0 for i = 1 ,..., N, find the matrix CeB such that d'x(D, C) = mind'a(D, C) over all matrices C e B.
PROBLEM
Problem 2a is just Problem 1; therefore we again have Knox's solution. As for Problem 2b, we have the following result. PROPOSITION
1 / / 1 ^ a < oo, there exists a unique solution of Problem 2b:
^ (7) Proof. Note first that all elements in Aj, and therefore in A and B, have strictly positive diagonal elements, because of assumption (A). Clearly the value of C,y influences only the ith term in the summation in (6). Therefore, for any fixed i, the values of CiJt with j e Jt, will be such as to minimize max|log(Cy/Dy)|
(8)
under the constraint that Ci} are nonnegative and sum up to Tt. To simplify the notation, let Jt = {i,jl ,...,],}, where 0 ^ r < n — 1, and rename xk = ClJk, yk = DlJk (k = 1 ,..., r), y0 = Dih and T = T,. The minimization of (8) can be restated as the problem of finding positive x, ,..., xr, such that X*-i x*
2, there does not seem to be any explicit formula. It is certainly possible to use numerical minimization techniques; however, they only guarantee convergence to a local minimum, not necessarily a global one. The procedure of Area et al. [3] provides an hybrid solution to this problem. In fact, each step can be read as follows: given a matrix C e B find the matrix DHe A that minimizes dx(C, £>„); then, find the matrix C + 1 eB that minimizes d'x(D", C + 1). One may hope that this procedure will converge to a matrix C = D that will be in A n B (this will indeed be proved in the following section) and that this limit C will conserve some minimizing property (which does not seem to be generally true). In order to show that indeed algorithm (4) has the above property, we study the solution to the following subproblems.
254
ANDREA PUGLIESE
minimize F(xi,..., xr) = max] max |log(xj/.y,)|, log 0-1
(9)
r
The matrix C of the thesis then corresponds to
Note that we have log
= F(Xl ,., xr)
for all j = 1,..., r.
Now take x ^ x, choose _/ with Xj ¥= Xj, and suppose, for instance, X*=o ^t ^ T. If x^ < Xj, we have
,..., xr)
= F(x t ,..., x r ).
If Xj > Xj and x t > xk for all /c = 1 ,..., r, we have T — £ l = j x t < T — ^ ^
t
xk, and
so log
= log log
>log
= F(x!,..., x r ).
In any case, we have therefore F(xx ,..., x r ) > F(x t ,..., x r ), that is, the thesis. When a = 00, C,j will minimize (8) only for j such that log
=
max
/
T.
When i does not have this property, Ctt can be chosen somewhat arbitrarily. Note that if in Problem 2a we substitute d'x for da, or in Problem 2b dx for d'a, the solution is not necessarily unique, and certainly has no simple expression. The choice of this hybrid step is therefore mainly dictated by simplicity. 3. Convergence of the approximating algorithm We now consider the convergence of algorithm (4). It can be rewritten as follows: given a probability matrix p° satisfying assumption (A), we define the symmetric matrix C° by "0 _ y —
(10)
Then, given a symmetric matrix C, let (ii)
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ii-oy
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and, using (7) and (10), we obtain
a;1 = (F(c-)) y .= c&o/d? dp1*2.
(12)
Let S = {C symmetric nonnegative matrices} and n = {CeS
such that dk > 0 for all / = 1,..., N } ,
THEOREM
2
Matrix C converges (as n -* oo) to a matrix C such that
£ Cy = 7J V i = 1 ,..., JV. For its proof, we need several lemmas. Most of the results can be extended to the case where the geometric mean (xy)112 is replaced by a generalized mean m(x, y). We start with some simple but useful properties of F. LEMMA 3 Let CeS such that Cu > Ofor i = 1,..., N; let C = F(C). Without loss of generality, assume dl = maX|_j N dt and dN = min ( = 1 N dt. Then 12
'.
12
The equality {djd^ only if dt = dN.
(13) u2
= 3, holds only ifdt = dx; and the equality (d,/dN)
max