We consider the problem of minimizing the sum of a smooth function h with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function P and a surjective linear map M. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions h and P are semialgebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the 1/2 regularization. Finally, when M is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex h. We illustrate our theoretical finding with a variety of applications such as signal denoising and sparse optimisation problems.

We propose a computational efficient yet simple numerical algorithm to solve the surface eikonal equation on general implicit surfaces. The method is developed based on the embedding idea and the fast sweeping methods. We first approximate the solution to the surface eikonal equation by the Euclidean weighted distance function defined in a tubular neighbourhood of the implicit surface, and then apply the efficient fast sweeping method to numerically compute the corresponding viscosity solution. Unlike some other embedding methods which require the radius of the computational tube satisfies h = O ( x ) for some h = O ( x ) , our approach allows h = O ( x ) . This implies that the total number of grid points in the computational tube is optimal and is given by h = O ( x ) for a co-dimensional one surface in h = O ( x ) . The method can be easily extended to general static HamiltonJacobi equation defined on implicit

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In this paper, we present an efficient \GQR\ algorithm for solving the linear response eigenvalue problem $\scrH\bx ={\lambda}\bx $, where $\scrH$ is $\bPi^-$-symmetric with respect to $\Gamma_0 = \diag(I_n,-I_n)$. Based on newly introduced $\Gamma$-orthogonal transformations, the \GQR\ algorithm preserves the $\bPi^-$-symmetric structure of $\scrH$ throughout the whole process, which guarantees the computed eigenvalues to appear pairwise $(\lambda,-\lambda)$ as they should. With the help of a newly established implicit $\Gamma$-orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the \GQR\ algorithm. Numerical experiments are given to show the effectiveness of the algorithm.