An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

Some Dynamic and Steady-State Properties of Threshold Auto-Regressions with Applications to Stationarity and Local Explosivity

The purpose of this paper is to investigate the dynamics and steady-state properties of threshold autoregressive models with exogenous states that follow Markovian processes. Markovian processes are widely used in applied economics although their statistical properties have not been explored in detail. We use characteristic functions to carry out the analysis, and this allows us to describe limiting distributions for processes not considered in the literature previously. We also calculate analytical expressions for some moments. Furthermore, we see that we can have locally explosive processes that are explosive in one regime whilst being strongly stationary overall. This is explored through simulation analysis, where we also show how the distribution changes when the explosive state becomes more frequent although the overall process remains stationary. In doing so, we are able to relate our analysis to asset prices which exhibit similar distributional properties.

Non-Parametric Signal Interpolation

This paper considers the problem of interpolation (smoothing) of a partially observable Markov random sequence. For the dynamic observation models, an equation for the interpolation of the posterior probability density is derived. The main goal of this paper is to consider the smoothing problem for the case of unknown distributions of an unobservable component of a random Markov sequence. Successful results were obtained for the strongly stationary Markov processes with mixing and for the conditional density belonging to the exponential family of densities. The resulting method is based on the empirical Bayes approach and kernel nonparametric estimation. The equation for the optimal smoothing estimator is derived in the form independent of unknown distributions of an unobservable process. Such form of the equation allows to use the nonparametric estimates for some conditional functionals in the equation given a set of dependent observations. To compare the nonparametric estimators with optimal mean square smoothing estimators in Kalman scheme, simulation results are given.

A LOCALLY SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

We propose a locally smoothing method for some mathematical programs with complementarity constraints, which only incurs a local perturbation on these constraints. For the approximate problem obtained from the smoothing method, we show that the Mangasarian–Fromovitz constraints qualification holds under certain conditions. We also analyse the convergence behaviour of the smoothing method, and present some sufficient conditions such that an accumulation point of a sequence of stationary points for the approximate problems is a C-stationary point, an M-stationary point or a strongly stationary point. Numerical experiments are employed to test the performance of the algorithm developed. The results obtained demonstrate that our algorithm is much more promising than the similar ones in the literature.

Notes on Convergence Properties for a Smoothing-Regularization Approach to Mathematical Programs with Vanishing Constraints

We give some improved convergence results about the smoothing-regularization approach to mathematical programs with vanishing constraints (MPVC for short), which is proposed in Achtziger et al. (2013). We show that the Mangasarian-Fromovitz constraints qualification for the smoothing-regularization problem still holds under the VC-MFCQ (see Definition 5) which is weaker than the VC-LICQ (see Definition 7) and the condition of asymptotic nondegeneracy. We also analyze the convergence behavior of the smoothing-regularization method and prove that any accumulation point of a sequence of stationary points for the smoothing-regularization problem is still strongly-stationary under the VC-MFCQ and the condition of asymptotic nondegeneracy.

Representation of Strongly Stationary Stochastic Processes

A generalization of the orthogonality conditions for a stochastic process to represent strongly stationary processes up to a fixed order is presented. The particular case of non-normal delta correlated processes, and the probabilistic characterization of linear systems subjected to strongly stationary stochastic processes are also discussed.