The augmented Lagrangian method (ALM) is one of the most useful methods for constrained optimization. Its convergence has been well established under convexity assumptions or smoothness assumptions, both assumptions. ALM may experience oscillations and divergence when underlying problem simultaneously nonconvex nonsmooth. In this paper, we consider linearly with a (in particular, weakly convex)...

In this paper, we present a novel variational wavelet inpainting based on the total variation (TV) regularization and l1-norm fitting term. The goal of model is to recover incomplete coefficients in presence impulsive noise. By incorporating Moreau envelope, proposed for can better handle non-differentiability A modified primal dual fixed-point algorithm developed proximity operator solve model...

We study statistical estimators computed using iterative optimization methods that are not run until completion. Classical results on maximum likelihood (MLEs) assert a one-step estimator (OSE), in which single Newton-Raphson iteration is performed from starting point with certain properties, asymptotically equivalent to the MLE. further develop these early-stopping by deriving properties of de...

In this paper we develop proximal methods for statistical learning. Proximal point algorithms are useful in statistics and machine learning for obtaining optimization solutions for composite functions. Our approach exploits closedform solutions of proximal operators and envelope representations based on the Moreau, Forward-Backward, Douglas-Rachford and Half-Quadratic envelopes. Envelope repres...

Compensated convex transforms have been introduced for extended real valued functions defined over Rn. In their application to image processing, interpolation and shape interrogation, where one deals with a bounded domain, the implicit assumption was made that function coincides its transform at boundary of data domain. this paper, we introduce local compensated in open subsets ? Rn by making s...

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (2016) when applied to log-concave probability distributions that are restricted to a convex body K. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with K. Explicit convergence bounds i...

We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate O(k−1/4). Our result yields the first complexity guarantees for the stochastic proximal point algorithm on weakly convex problems...

A new algorithm, named the Parametric Legendre Transform (PLT) algorithm, to compute the Legendre-Fenchel conjugate of a convex function of one variable is presented. It returns a parameterization of the graph of the conjugate except for some affine parts corresponding to nondifferentiable points of the function. The approach is extended to the computation of the Moreau envelope, resulting in a...

Bauschke, Lucet, and Trienis [SIAM Rev., 50 (2008), pp. 115–132] developed the concept of the proximal average of two convex functions. In this work we show the relationship between the proximal average and the Moreau envelope and exploit this relationship to develop stability theory for a generalized proximal average function. This approach allows us to extend the concept of the proximal avera...