An inexact linearized proximal algorithm for a class of DC composite optimization problems and applications

This paper is concerned with a class of DC composite optimization problems which, as an extension of the convex composite optimization problem and the DC program with nonsmooth components, often arises from robust factorization models of low-rank matrix recovery. For this class of nonconvex and nonsmooth problems, we propose an inexact linearized proximal algorithm (iLPA) which in each step computes an inexact minimizer of a strongly convex majorization constructed by the partial linearization of their objective functions. The generated iterate sequence is shown to be convergent under the Kurdyka-{\L}ojasiewicz (KL) property of a potential function, and the convergence admits a local R-linear rate if the potential function has the KL property of exponent $1/2$ at the limit point. For the latter assumption, we provide a verifiable condition by leveraging the composite structure, and clarify its relation with the regularity used for the convex composite optimization. Finally, the proposed iLPA is applied to a robust factorization model for matrix completions with outliers, DC programs with nonsmooth components, and $\ell_1$-norm exact penalty of DC constrained programs, and numerical comparison with the existing algorithms confirms the superiority of our iLPA in computing time and quality of solutions.

Advertising strategy for profit-maximization: a novel practice on Tmall's online ads manager platforms

Ads manager platform gains popularity among numerous e-commercial vendors/advertisers. It helps advertisers to facilitate the process of displaying their ads to target customers. One of the main challenges faced by advertisers, especially small and medium-sized enterprises, is to configure their advertising strategy properly. An ineffective advertising strategy will bring too many ``just looking'' clicks and, eventually, generate high advertising expenditure unproportionally to the growth of sales. In this paper, we present a novel profit-maximization model for online advertising optimization. The optimization problem is constructed to find optimal set of features to maximize the probability that target customers buy advertising products. We further reformulate the optimization problem to a knapsack problem with changeable parameters, and introduce a self-adjusted algorithm for finding the solution to the problem. Numerical experiment based on statistical data from Tmall show that our proposed method can optimize the advertising strategy given expenditure budget effectively.