Lieven Vandenberghe

Anders Hansson, Zhang Liu, Lieven Vandenberghe

We present a subspace system identification method based on weighted nuclear norm approximation. The weight matrices used in the nuclear norm minimization are the same weights as used in standard subspace identification methods. We show that the inclusion of the weights improves the performance in terms of fit on validation data. As a second benefit, the weights reduce the size of the optimization problems that need to be solved. Experimental results from randomly generated examples as well as from the Daisy benchmark collection are reported. The key to an efficient implementation is the use of the alternating direction method of multipliers to solve the optimization problem.

Martin S. Andersen, Joachim Dahl, Lieven Vandenberghe

Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern, and its dual cone, the cone of sparse matrices with the same pattern that have a positive semidefinite completion. Efficient large-scale algorithms for evaluating these barriers and their derivatives are important in interior-point methods for nonsymmetric conic formulations of sparse semidefinite programs. The algorithms are based on the multifrontal method for sparse Cholesky factorization.

Walaa M. Moursi, Lieven Vandenberghe

The Douglas-Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper closed convex functions; more generally two maximally monotone operators. Recent results concerned with linear rates of convergence of the method require additional properties of the underlying monotone operators, such as strong monotonicity and cocoercivity. In this paper, we study the case when one operator is Lipschitz continuous but not necessarily a subdifferential operator and the other operator is strongly monotone. This situation arises in optimization methods which involve primal-dual approaches. We provide new linear convergence results in this setting.

Anni Zhao, Ayca Ermis, Jeffrey Robert Vitt et al.

Objective: Accurate noninvasive estimation of intracranial pressure (ICP) remains a major challenge in critical care. We developed a bespoke machine learning algorithm that integrates system identification and ranking-constrained optimization to estimate mean ICP from noninvasive signals. Methods: A machine learning framework was proposed to obtain accurate mean ICP values using arbitrary noninvasive signals. The subspace system identification algorithm is employed to identify cerebral hemodynamics models for ICP simulation using arterial blood pressure (ABP), cerebral blood velocity (CBv), and R-wave to R-wave interval (R-R interval) signals in a comprehensive database. A mapping function to describe the relationship between the features of noninvasive signals and the estimation errors is learned using innovative ranking constraints through convex optimization. Patients across multiple clinical settings were randomly split into testing and training datasets for performance evaluation of the mapping function. Results: The results indicate that about 31.88% of testing entries achieved estimation errors within 2 mmHg and 34.07% of testing entries between 2 mmHg to 6 mmHg from the nonlinear mapping with constraints. Conclusion: Our results demonstrate the feasibility of the proposed noninvasive ICP estimation approach. Significance: Further validation and technical refinement are required before clinical deployment, but this work lays the foundation for safe and broadly accessible ICP monitoring in patients with acute brain injury and related conditions.

Stefan Vlaski, Lieven Vandenberghe, Ali H. Sayed

The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required to be differentiable or even continuous. The individual, regularized, cost functions are distributed across a strongly-connected network of agents and the Pareto optimal solution is sought by appealing to a multi-agent diffusion strategy. To this end, the regularizers are smoothed by means of infimal convolution and it is shown that the Pareto solution of the approximate, smooth problem can be made arbitrarily close to the solution of the original, non-smooth problem. Performance bounds are established under conditions that are weaker than assumed before in the literature, and hence applicable to a broader class of adaptation and learning problems.