Aleksandr Y. Aravkin We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term \(f(x) = \frac{1}{2} \|F(x)\|_2^2\) and a nonsmooth term \(h\). Both \(f\) and \(h\) may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of \(h\) using a first-order method such as … Read more
We develop a trust-region method for minimizing the sum of a smooth term f and a nonsmooth term h, both of which can be nonconvex. Each iteration of our method minimizes apossibly nonconvex model of f+h in a trust region. The model coincides with f+h in value and subdifferential at the center. We establish global … Read more
Block tridiagonal systems appear in classic Kalman smoothing problems, as well in generalized Kalman smoothing, where problems may have nonsmooth terms, singular covariance, constraints, nonlinear models, and unknown parameters. In this paper, first we interpret all the classic smoothing algorithms as different approaches to solve positive definite block tridiagonal linear systems. Then, we obtain new … Read more
The solution of classic discrete-time, finite-horizon linear quadratic regulator (LQR) problem is well known in literature. By casting the solution to be a static state-feedback, we propose a new method that trades off low LQR objective value with closed-loop stability. CitationTo appear on the special issue on the 21st IFAC World Congress 2020, IFAC PapersOnLine.ArticleDownload … Read more
Level-set optimization formulations with data-driven constraints minimize a regularization functional subject to matching observations to a given error level. These formulations are widely used, particularly for matrix completion and sparsity promotion in data interpolation and denoising. The misfit level is typically measured in the l2 norm, or other smooth metrics. In this paper, we present … Read more
Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, vari- able selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of … Read more
An optimization approach is proposed to construct sparse portfolios with mean-reverting price behaviors. Our objectives are threefold: (i) design a multi-asset long-short portfolio that best fits an Ornstein-Uhlenbeck process in terms of maximum likelihood, (ii) select portfolios with desirable characteristics of high mean reversion and low variance though penalization, and (iii) select a parsimonious portfolio … Read more
Common numerical methods for constrained convex optimization are predicated on efficiently computing nearest points to the feasible region. The presence of a design matrix in the constraints yields feasible regions with more complex geometries. When the functional components are gauges, there is an equivalent optimization problem—the gauge dual– where the matrix appears only in the … Read more
Machine learning theory typically assumes that training data is unbiased and not adversarially generated. When real training data deviates from these assumptions, trained models make erroneous predictions, sometimes with disastrous effects. Robust losses, such as the huber norm are designed to mitigate the effects of such contaminated data, but they are limited to the regression … Read more
Convex optimization problems arising in applications often have favorable objective functions and complicated constraints, thereby precluding first-order methods from being immediately applicable. We describe an approach that exchanges the roles of the objective and constraint functions, and instead approximately solves a sequence of parametric level-set problems. A zero-finding procedure, based on inexact function evaluations and … Read more