In this paper, we study, for the first time, nonconvex minimax separable quadratic optimization problems with multiple separable quadratic constraints and their second-order cone programming (SOCP) relaxations. Under suitable conditions, we establish exact SOCP relaxation for minimax nonconvex separable quadratic programs. We show that various important classes of specially structured minimax quadratic optimization problems admit exact SOCP relaxations under easily verifiable conditions. These classes include some minimax extended trust-region problems, minimax uniform quadratic optimization problems, max dispersion problems, and some robust quadratic optimization problems under bounded data uncertainty. The present work shows that nonconvex minimax separable quadratic problems with quadratic constraints, which contain a hidden closed and convex epigraphical set, exhibit exact SOCP relaxations.

We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both corrupted and noise free, are equally penalized in data fitting, leading to insurmountable difficulty in balancing regularization and data fitting. In this paper, we propose to combine TV regularization with nonconvex \emph{smoothly clipped absolute deviation} (SCAD) penalty for data fitting (TVSCAD for short). Our motivation is simply that data fitting should be enforced only when an observed data is not severely corrupted, while for those data more likely to be severely corrupted, less or even null penalization should be enforced. A \emph{difference of convex functions} algorithm is adopted to solve the nonconvex TVSCAD model, resulting in solving a sequence of TVL1-equivalent problems, each of which can then be solved efficiently by the alternating direction method of multipliers. Theoretically, we establish global convergence to a critical point of the nonconvex objective function. The R-linear and at-least-sublinear convergence rate results are derived for the cases of anisotropic and isotropic TV, respectively. Numerically, experimental results are given to show that the TVSCAD approach improves those of the TVL1 significantly, especially for cases with high level impulsive noise, and is comparable with the recently proposed iteratively corrected TVL1 method [\textit{Inverse Problems}, 32 (2016), pp.~085004].

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The alternating direction method of multipliers (ADMM) is a popular and efficient first-order method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of the inertial proximal point method and the proximal ADMM, for linearly constrained separable convex optimization. This class of methods are of inertial nature because at each iteration the proximal ADMM is applied to a point extrapolated at the current iterate in the direction of last movement. The recently proposed inertial primal-dual algorithm [A. Chambolle and T. Pock, On the ergodic convergence rates of a first-order primaldual algorithm, preprint, 2014, Algorithm 3] and the inertial linearized ADMM [C. Chen, S. Ma, and J. Yang, arXiv:1407.8238, eq. (3.23)] are covered as special cases. The proposed algorithmic framework is very general in the sense that the weighting matrices in the proximal terms are allowed to be only positive semidefinite, but not necessarily positive definite as required by existing methods of the same kind. By setting the two proximal terms to zero, we obtain an inertial variant of the classical ADMM, which is to the best of our knowledge new. We carry out a unified analysis for the entire class of methods under very mild assumptions. In particular, convergence, as well as asymptotic $o(1/\sqrt{k})$ and nonasymptotic $O(1/\sqrt{k})$ rates of convergence, are established for the best primal function value and feasibility residues, where k denotes the iteration counter. The global iterate convergence of the generated sequence is established under an additional assumption. We also present extensive experimental results on total variation–based image reconstruction problems to illustrate the profits gained by introducing the inertial extrapolation steps.