We study a class of methods for solving convex programs, which are based on nonquadratic Augmented Lagrangians for which the penalty parameters are functions of the multipliers. This gives rise to lagrangians which are nonlinear in the multipliers. Each augmented lagrangian is speciied by a choice of a penalty function ' and a penalty-updating function. The requirements on ' are mild, and allow...

A number of variable selection methods have been proposed involving nonconvex penalty functions. These methods, which include the smoothly clipped absolute deviation (SCAD) penalty and the minimax concave penalty (MCP), have been demonstrated to have attractive theoretical properties, but model fitting is not a straightforward task, and the resulting solutions may be unstable. Here, we demonstr...

A number of variable selection methods have been proposed involving nonconvex penalty functions. These methods, which include the smoothly clipped absolute deviation (SCAD) penalty and the minimax concave penalty (MCP), have been demonstrated to have attractive theoretical properties, but model fitting is not a straightforward task, and the resulting solutions may be unstable. Here, we demonstr...

The exactness of the penalization for the exact l1 penalty function method used for solving nonsmooth constrained optimization problems with both inequality and equality constraints are presented. Thus, the equivalence between the sets of optimal solutions in the nonsmooth constrained optimization problem and its associated penalized optimization problem with the exact l1 penalty function is es...

We introduce an algebraic operator framework to study discounted penalty functions in renewal risk models. For inter-arrival and claim size distributions with rational Laplace transform, the usual integral equation is transformed into a boundary value problem, which is solved by symbolic techniques. The factorization of the differential operator can be lifted to the level of boundary value prob...

Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty func...

Hansen, Kooperberg, and Sardy (1998) introduced a family of continuous, piecewise linear functions defined over adaptively selected triangulations of the plane as a general approach to statistical modeling of bivariate densities, regression and hazard functions. These triograms enjoy a natural affine equivariance that offers distinct advantages over competing tensor product methods that are mor...

In this paper, we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone are speciŽ c members of nonlinear regular...

In high-dimensional model selection problems, penalized least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a data-driven weighted linear combination of convex loss functions, together with weighted L(1)-penalty. It is completely data-adaptive and does not require prior knowled...