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We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of this class is that the inner function may fail to be locally Lipschitz continuous. It covers a range of important, yet challenging, applications, including inverse optimal value optimization and problems under value-at-risk constraints. We propose an asymptotic decomposition of the composite function that guarantees epi-convergence to the original function, leading to necessary optimality conditions for the corresponding minimization problem. The proposed decomposition also enables us to design a numerical algorithm such that any accumulation point of the generated sequence, if it exists, satisfies the newly introduced optimality conditions. These results expand on the study of so-called amenable functions introduced by Poliquin and Rockafellar in 1992, which are compositions of convex functions with smooth maps, and the prox-linear methods for their minimization. To demonstrate that our algorithmic framework is practically implementable, we further present verifiable termination criteria and preliminary numerical results. Funding: Financial support from the National Science Foundation Division of Computing and Communication Foundations [Grant CCF-2416172] and Division of Mathematical Sciences [Grant DMS-2416250] and the National Cancer Institute, National Institutes of Health [Grant 1R01CA287413-01] is gratefully acknowledged. 

In this paper, we focus on the computation of the nonparametric maximum likelihood es- timator (NPMLE) in multivariate mixture models. Our approach discretizes this infinite dimensional convex optimization problem by setting fixed support points for the NPMLE and optimizing over the mixing proportions. We propose an efficient and scalable semis- mooth Newton based augmented Lagrangian method (ALM). Our algorithm outperforms the state-of-the-art methods (Kim et al., 2020; Koenker and Gu, 2017), capable of handling n ≈ 106 data points with m ≈ 104 support points. A key advantage of our approach is its strategic utilization of the solution’s sparsity, leading to structured sparsity in Hessian computations. As a result, our algorithm demonstrates better scaling in terms of m when compared to the mixsqp method (Kim et al., 2020). The computed NPMLE can be directly applied to denoising the observations in the framework of empirical Bayes. We propose new denoising estimands in this context along with their consistent estimates. Extensive nu- merical experiments are conducted to illustrate the efficiency of our ALM. In particular, we employ our method to analyze two astronomy data sets: (i) Gaia-TGAS Catalog (Anderson et al., 2018) containing approximately 1.4 × 106 data points in two dimensions, and (ii) a data set from the APOGEE survey (Majewski et al., 2017) with approximately 2.7 × 104 data points. 

In this paper, we have studied a decomposition method for solving a class of non-convex two-stage stochastic programs, where both the objective and constraints of the second-stageproblem are nonlinearly parameterized by the first-stage variables. Due to the failure of the Clarkeregularity of the resulting nonconvex recourse function, classical decomposition approaches such asBenders decomposition and (augmented) Lagrangian-based algorithms cannot be directly generalizedto solve such models. By exploring an implicitly convex-concave structure of the recourse function,we introduce a novel decomposition framework based on the so-called partial Moreau envelope. Thealgorithm successively generates strongly convex quadratic approximations of the recourse functionbased on the solutions of the second-stage convex subproblems and adds them to the first-stage mas-ter problem. Convergence has been established for both a fixed number of scenarios and a sequentialinternal sampling strategy. Numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm.