  Latent variable models have been playing a central role in psychometrics and
related fields. In many modern applications, the inference based on latent
variable models involves one or several of the following features: (1) the
presence of many latent variables, (2) the observed and latent variables being
continuous, discrete, or a combination of both, (3) constraints on parameters,
and (4) penalties on parameters to impose model parsimony. The estimation often
involves maximizing an objective function based on a marginal
likelihood/pseudo-likelihood, possibly with constraints and/or penalties on
parameters. Solving this optimization problem is highly non-trivial, due to the
complexities brought by the features mentioned above. Although several
efficient algorithms have been proposed, there lacks a unified computational
framework that takes all these features into account. In this paper, we fill
the gap. Specifically, we provide a unified formulation for the optimization
problem and then propose a quasi-Newton stochastic proximal algorithm.
Theoretical properties of the proposed algorithms are established. The
computational efficiency and robustness are shown by simulation studies under
various settings for latent variable model estimation.
