An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization

This paper focuses on the minimization of a sum of a twice continuously differentiable function $f$ and a nonsmooth convex function. We propose an inexact regularized proximal Newton method by an approximation of the Hessian $\nabla^2\!f(x)$ involving the $\varrho$th power of the KKT residual. For $\varrho=0$, we demonstrate the global convergence of the iterate sequence for the KL objective function and its $R$-linear convergence rate for the KL objective function of exponent $1/2$. For $\varrho\in(0,1)$, we establish the global convergence of the iterate sequence and its superlinear convergence rate of order $q(1\!+\!\varrho)$ under an assumption that cluster points satisfy a local H\"{o}lderian local error bound of order $q\in(\max(\varrho,\frac{1}{1+\varrho}),1]$ on the strong stationary point set; and when cluster points satisfy a local error bound of order $q>1+\varrho$ on the common stationary point set, we also obtain the global convergence of the iterate sequence, and its superlinear convergence rate of order $\frac{(q-\varrho)^2}{q}$ if $q>\frac{2\varrho+1+\sqrt{4\varrho+1}}{2}$. A dual semismooth Newton augmented Lagrangian method is developed for seeking an inexact minimizer of subproblem. Numerical comparisons with two state-of-the-art methods on $\ell_1$-regularized Student's $t$-regression, group penalized Student's $t$-regression, and nonconvex image restoration confirm the efficiency of the proposed method

On the Origin of Surface Ozone Episode in Shanghai over Yangtze River Delta during a Prolonged Heat Wave

A heat wave with temperatures over 35°C and sunny stagnant meteorological conditions occurred in Shanghai from 27 July to 5 August 2015, leading to a sustained episode of high ozone lasting 12 days. We have conducted a detailed source apportionment of surface ozone, by precursor source category and region, using a photochemical transport model. In this episode, a southwesterly wind prevailed over the Yangtze River Delta, and therefore precursors from the local Shanghai region and the region immediately to the south of Shanghai are the two major contributors (in total 90%) to ozone in Shanghai. The source apportionment reveals that local industrial sources and energy/biogenic sources in neighbouring regions are the principal causes for the high levels of ozone. By examining the contributions from individual physical and chemical processes, we show that ozone concentrations start to rise rapidly in the morning because chemical production dominates as the solar radiation increases, and while there is little removal by deposition when ozone remains low. In general, chemical production, horizontal advection and vertical diffusion contribute to increase ozone concentration during daytime, and deposition and vertical advection reduce ozone concentrations