An m-moment minimum error method is constructed for quadratic optimization in Hilbert space, with proved convergence, optimality among Krylov methods, and numerical tests on Helmholtz, heat, and thermoacoustics inverse problems.
Polyak Minorant Method for Convex Optimization
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abstract
In 1963 Boris Polyak suggested a particular step size for gradient descent methods, now known as the Polyak step size, that he later adapted to subgradient methods. The Polyak step size requires knowledge of the optimal value of the minimization problem, which is a strong assumption but one that holds for several important problems. In this paper we extend Polyak's method to handle constraints and, as a generalization of subgradients, general minorants, which are convex functions that tightly lower bound the objective and constraint functions. We refer to this algorithm as the Polyak Minorant Method (PMM). It is closely related to cutting-plane and bundle methods.
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On the construction of a gradient method of quadratic optimization, optimal from the point of view of minimizing the distance to the exact solution
An m-moment minimum error method is constructed for quadratic optimization in Hilbert space, with proved convergence, optimality among Krylov methods, and numerical tests on Helmholtz, heat, and thermoacoustics inverse problems.