In this paper we propose two algorithms for the quadratic constrained matrix problem—the problem of computing the best possible estimate of an unknown matrix with known bounds on individual entries, known row and column totals, and known weighted totals of subsets of individual entries. The problem has been widely studied since it frequently appears as a “core” problem in a variety of application areas. To date, the diagonal case of this problem has received the majority of attention from researchers; here we focus primarily on the solution of problems with any positive definite quadratic form as minimand. We also provide results of computational experiments with our two algorithms and with our implementation of an algorithm of Bachem and Korte.