A new smoothing-regularization approach for a maximum-likelihood estimation problem

We consider the problem min_i=1^m(aⁱ,x–b_ilogaⁱ, z) subject tox 0 which occurs as a maximum-likelihood estimation problem in several areas, and particularly in positron emission tomography. After noticing that this problem is equivalent to mind(b, Ax) subject tox 0, whered is the Kullback-Leibler information divergence andA, b are the matrix and vector with rows and entriesaⁱ,b_i, respectively, we suggest a regularized problem mind(b, Ax) + d(v, Sx), where is the regularization parameter,S is a smoothing matrix, andv is a fixed vector. We present a computationally attractive algorithm for the regularized problem, establish its convergence, and show that the regularized solutions, as goes to 0, converge to the solution of the original problem which minimizes a convex function related tod(v, Sx). We give convergence-rate results both for the regularized solutions and for their functional values.The research of A. N. Iusem was partially supported by CNPq Grant No. 301280/86-MA.