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A general method for the optimization of convex or concave cost functions over the intersection of convex constraint sets is described for applications in signal reconstruction and restorarion. A unique continuous function is obtained, under certain conditions, by employing Fenchel's duality theorem to give a finite dimensional dual problem. This approach allows the solution of very complex constrained problems by separating the constraints into those related to the solution and those related to additive noise statistics. The method is applied to computed tomography with noisy data of which the noise covariance and bounds on the solution are known approximately. A fast implementation of the required optimization procedure is given and the resulting solution is shown to be significantly better than a suboptimal feasible solution.