It was recently shown [7, 9] that "properly built" linear and polyhedral estimates nearly attain minimax accuracy bounds in the problem of recovery of unknown signal from noisy observations of linear images of the signal when the signal set is an ellitope. However, design of nearly optimal estimates relies upon solving semidefinite optimization problems with matrix variables, what puts the synthesis of such estimates beyond the rich of the standard Interior Point algorithms of semidefinite optimization even for moderate size recovery problems. Our goal is to develop First Order Optimization algorithms for the computationally efficient design of linear and polyhedral estimates. In this paper we (a) explain how to eliminate matrix variables, thus reducing dramatically the design dimension when passing from Interior Point to First Order optimization algorithms and (2) develop and analyse a dedicated algorithm of the latter type -- Composite Truncated Level method.

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