SUCCESSIVE APPROXIMATE ALGORITHM FOR BEST APPROXIMATION FROM A POLYHEDRON

Suppose K is the intersection of a finite number of closed half-spaces {K-i} in a Hilbert space X, and x is an element of X\K. Dykstra's cyc lic projections algorithm is a known method to determine an approximat e solution of the best approximation of x from K, which is denoted by P-K(x). Dykstra's algorithm reduces the problem to an iterative scheme which involves computing the best approximation from the individual K -i. It is known that the sequence {x(j)} generated by Dykstra's method converges to the best approximation P-K(x). But since it is difficult to find the definite value of an upper bound of the error \\x(j)/-P-K (x)\\, the applicability of the algorithm is restrictive. This paper i ntroduces a new method, called the successive approximate algorithm, b y which one can generate a finite sequence x(0), x(1), ..., x(k) with x(k) = P-K(x). In addition, the error \\x(j) - P-K(x)\\ is monotone de creasing and has a definite upper bound easily to be determined. So th e new algorithm is very applicable in practice. (C) 1998 Academic Pres s.