We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on (0,+\infty), and includes as a special case the entropy-like proximal algorithm [Eggermont, <i>Linear Algebra Appl.</i>, 130 (1990), pp. 2542], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the

This article studies estimation of partially linear hazard regression models for multivariate survival data. A profile pseudopartial likelihood estimation method is proposed under the marginal hazard model framework. The estimation on the parameters for the linear part is accomplished by maximization of a pseudopartial likelihood profiled over the nonparametric part. This enables us to obtain-consistent estimators of the parametric component. Asymptotic normality is obtained for the estimates of both the linear and nonlinear parts. The new technical challenge is that the nonparametric component is indirectly estimated through its integrated derivative function from a local polynomial fit. An algorithm of fast implementation of our proposed method is presented. Consistent standard error estimates using sandwich-type ideas are also developed, which facilitates inferences for the model. It is shown that the

. Motivated by block partitioned problems arising from group sparsity representation and generalized noncooperative potential games, this paper presents a basic decomposition method for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an optimization problem is given by the sum of two nonseparable functions minus a sum of separable, pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1 , i.e., differentiable with a Lipschitz gradient, while the other summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By taking advantage of the (negative) pointwise maximum structure in the objective, the developed algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth problems. This aim is accomplished by combining the alternating direction method of multipliers (ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall problem into subproblems each involving the individual blocks. To arrive at a stationary solution of the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex function in each pointwise maximum. In order to lessen the potential computational burden in each iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is established.

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