This article develops duality principles and related convex dual formulations suitable for the local optimization of non-convex primal formulations for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a Ginzburg-Landau type equation.

In this paper we describe a nonparametric maximum likelihood (NPML) method for estimating multivariate mixing distributions. Given $N$ independent observations, convexity theory shows that the NPML estimator is discrete with at most $N$ support points. The original infinite NPML problem then becomes the finite dimensional problem of finding the location and probability of the support points. The probability of the support points is found by a Primal-Dual Interior-Point method; the location of the support points is found by an Adaptive Grid method. Our method is able to handle high-dimensional and complex multivariate mixture models.An important application is discussed for the problem of population pharmacokinetics and a non-trivial example is treated.Our algorithm has been successfully applied in hundreds of published pharmacometric studies. In addition to population pharmacokinetics, this research also applies to empirical Bayes estimation and many other areas of applied mathematics.

The fast and reliable processing of medical images is of paramount importance to adequately generate data to feed machine learning algorithms that can prevent and diagnose health issues. Here we benchmark different compressed sensing techniques applied to magnetic resonance imaging as a means to reduce the acquisition time spent in the collection of data and signals that form the image. We show that by using these techniques, it is possible to reduce the number of signals needed and, therefore, substantially decrease the time to acquire the measurements. To this end, we have considered and compared different algorithms: the Iterative Re-Weighted Least Squares, the Iterative Soft Thresholding Algorithm, the Iterative Hard Thresholding Algorithm, the Primal Dual Algorithm and the Log-Barrier Algorithm. We have implemented such algorithms in different analysis programs that have been used to perform the reconstruction of the images and found that the Iterative Soft Thresholding Algorithm gives the optimal results. We found that the images obtained with this algorithm have less quality than the original ones, but the quality is good enough to distinguish each body structure and detect any health problems.