Minimization of quadratic functionals on cones in Hilbert spaces

AbstractThis paper concerns a problem of minimization of a quadratic functional on a cone in a Hilbert space. First, a simple auxiliary minimization in R2, is solved. The obtained results allow the construction of many different algorithms that solve the primary minimization problem. However, only two algorithms are considered in the paper. The first one concerns a general case, i.e., the case when the only restrictions imposed on the cone are its closeness and convexity. This algorithm uses a projection on the cone technique. The second one can be applied if it is assumed, additionally, that the considered Hilbert space is separable and has Riesz basis. The cone is, in this case, defined as a collection of points whose expansions with respect to this basis have nonnegative coefficients.Weak convergence of both algorithms is proved. The rates of their convergence are estimated by the convergence of certain series of reals. Possible applications to an optimal control problem are suggested in the introduction and in the conclusion