Monotonic function

Abstract: Iterative shrinkage/thresholding (1ST) algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems. This class of problems results from combining a linear observation model with a nonquadratic regularizer (e.g., total variation or wavelet-based regularization). It happens that the convergence rate of these 1ST algorithms depends heavily on the linear observation operator, becoming very slow when this operator is ill-conditioned or ill-posed. In this paper, we introduce two-step 1ST (TwIST) algorithms, exhibiting much faster convergence rate than 1ST for ill-conditioned problems. For a vast class of nonquadratic convex regularizers (lscrP norms, some Besov norms, and total variation), we show that TwIST converges to a minimizer of the objective function, for a given range of values of its parameters. For noninvertible observation operators, we introduce a monotonic version of TwIST (MTwIST); although the convergence proof does not apply to this scenario, we give experimental evidence that MTwIST exhibits similar speed gains over IST. The effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples.

Abstract: The authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem. In addition, they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm, the monopolist, the Bertrand oligopolist, consumer and growth theory, game theory, and general equilibrium analysis. Copyright 1994 by The Econometric Society.