We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We pr...

Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when the optimization variables are nonlinearly coupled in some nonconvex constraints. In this work, we propose an algorithm named penalty dual decomposition (PDD...

In this paper, we propose a new decomposition approach named the proximal primal dual algorithm (Prox-PDA) for smooth nonconvex linearly constrained optimization problems. The proposed approach is primal-dual based, where the primal step minimizes certain approximation of the augmented Lagrangian of the problem, and the dual step performs an approximate dual ascent. The approximation used in th...

In this work we consider the regularization of vectorial data such as color images. Based on the observation that edge alignment across image channels is a desirable prior for multichannel image restoration, we propose a novel scheme of minimizing the rank of the image Jacobian and extend this idea to second derivatives in the framework of total generalized variation. We compare the proposed co...

The problem of minimizing sum-of-nonconvex functions (i.e., convex functions that are average of non-convex ones) is becoming increasingly important in machine learning, and is the core machinery for PCA, SVD, regularized Newton’s method, accelerated non-convex optimization, and more. We show how to provably obtain an accelerated stochastic algorithm for minimizing sumof-nonconvex functions, by...

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tack...

In the paper, we study the mini-batch stochastic ADMMs (alternating direction method of multipliers) for the nonconvex nonsmooth optimization. We prove that, given an appropriate mini-batch size, the mini-batch stochastic ADMM without variance reduction (VR) technique is convergent and reaches the convergence rate of O(1/T ) to obtain a stationary point of the nonconvex optimization, where T de...