Objective: Accurate noninvasive estimation of intracranial pressure (ICP) remains a major challenge in critical care. We developed a machine learning algorithm that integrates system identification and rankingconstrained optimization to estimate mean ICP from noninvasive signals. Methods: A machine learning framework was proposed to obtain estimated 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 and 6 mmHg from the nonlinear mapping with constraints. Conclusion: Our results demonstrate the feasibility of the proposed noninvasive ICP estimation approach. Significance: This work presents a proof-of-concept framework for noninvasive ICP estimation in patients with acute brain injury and related conditions. Although the findings demonstrate the feasibility of the proposed framework, important limitations remain, particularly regarding estimation accuracy, cohort diversity, and validation across clinical settings. Additional algorithm development, expanded multicenter datasets, and rigorous prospective validation are therefore required to improve performance and establish clinical utility.

We provide improved complexity results for symmetric primal–dual interior-point algorithms in conic optimization. The results follow from new uniform bounds on a key complexity measure for primal–dual metrics at pairs of primal and dual points. The complexity measure is defined as the largest eigenvalue of the product of the Hessians of the primal and dual barrier functions, normalized by the proximity of the points to the central path. For algorithms based on self-scaled barriers for symmetric cones, we determine the exact value of the complexity measure. In the significantly more general case of self-concordant barriers with negative curvature, we provide the asymptotically tight upper bound of 4/3. This result implies O(ϑ^1/2ln(1/ε)) iteration complexity for a variety of symmetric (and some nonsymmetric) primal–dual interior-point algorithms. Finally, in the case of general self-concordant barriers, we give improved bounds for some variants of the complexity measure.

We present an algorithm for computing the minimum-rank positive semidefinite completion of a sparse matrix with a chordal sparsity pattern.This problem is tractable, in contrast to the minimumrank positive semidefinite completion problem for general sparsity patterns.We also present a similar algorithm for the Euclidean distance matrix completion with minimum embedding dimension.The two algorithms use efficient recursions over a clique tree associated with the chordal sparsity pattern.As an application, we use the minimum-rank completion method as a rounding technique to convert the solution of a sparse semidefinite optimization problem with non-unique solutions to an optimal solution of lower rank.In experiments with semidefinite relaxations of optimal power flow problems, the minimum-rank completion often results in solutions of lower rank than the solutions computed by interior-point solvers.

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. We describe transitive subsets of the automorphism groups of the cones and their duals, and important properties of the composition of log-det barrier functions with the automorphisms in this set. Next, we consider extensions to linear slices of the positive semidefinite cone, i.e., intersection of the positive semidefinite cone with a linear subspace, and review conditions that make the cone homogeneous. In the third part of the paper we give a high-level overview of the classical algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation. We conclude by discussing the role of homogeneous cone structure in primal-dual symmetric interior-point methods.

The software and data in this repository are a snapshot of the software and data that were used in the research reported in the paper Disjoint Bilinear Optimization: A Two-stage Robust Optimization Perspective by Jianzhe Zhen, Ahmadreza Marandi, Danique de Moor, Dick den Hertog, and Lieven Vandenberghe.

The paper presents primal-dual proximal splitting methods for convex optimization, in which generalized Bregman distances are used to define the primal and dual proximal update steps. The methods extend the primal and dual Condat-Vu algorithms and the primal-dual three-operator (PD3O) algorithm. The Bregman extensions of the Condat-Vu algorithms are derived from the Bregman proximal point method applied to a monotone inclusion problem. Based on this interpretation, a unified framework for the convergence analysis of the two methods is presented. We also introduce a line search procedure for stepsize selection in the Bregman dual Condat-Vu algorithm applied to equality-constrained problems. Finally, we propose a Bregman extension of PD3O and analyze its convergence.

The purpose of this article is to develop and study a decentralized 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 multiagent 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 nonsmooth 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.

In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this end, a solution scheme based on a blending of three popular robust optimization techniques is proposed. For disjoint bilinear optimization problems with a polyhedral feasible region and a general convex feasible region, we show that, under mild regularity conditions, the convex relaxations of the original bilinear formulation and its two-stage robust reformulation obtained from a reformulation-linearization-based technique and linear decision rules, respectively, are equivalent. For generic bilinear optimization problems, the convex relaxations from the reformulation-linearization-based technique are generally tighter than the one from linear decision rules. Numerical experiments on bimatrix games, synthetic disjoint bilinear problem instances, and convex maximization problems demonstrate the efficiency and effectiveness of the proposed solution scheme. Summary of Contribution: Computing solutions for disjoint bilinear optimization problems are of much interest in real-life applications, yet they are, in general, computationally intractable. This paper proposes a computationally tractable approximation as well as a convergent algorithm to the optimal values of such problems. Extensive computational experiments on (i) (constrained) bimatrix games, (ii) synthetic disjoint bilinear problems, and (iii) convex maximization problems are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

OBJECTIVE:Many physiological signals are degraded by significant corruptions that limit their usefulness. One example is cerebral blood flow velocity (CBFV) signals, measured by transcranial Doppler, which are susceptible to large errors from patient motion. In this paper, we propose a method to remove artifacts and impute sections of missing data in these signals. APPROACH:The method exploits the low-order dynamical relationship between CBFV, arterial blood pressure and, where available, intracranial pressure. It enhances the measured signals by fitting them to a low-order dynamical model, using convex regularization terms that improve robustness to large deviations and missing data. The method is based on a convex optimization formulation and utilizes recent work in trace norm approximation and subspace system identification. MAIN RESULTS:Simulations demonstrate that the method successfully removes real CBFV artifacts and can impute missing data with reasonable accuracy. Performance was improved when intracranial pressure data was available. CONCLUSION:The methods presented can be used by researchers to remove artifacts and estimate missing sections in CBFV signals. The general approach may be applied to other biomedical signal processing settings. SIGNIFICANCE:This low-order dynamical approach has ongoing applications in noninvasive intracranial pressure estimation.

The primal-dual hybrid gradient (PDHG) algorithm proposed by Esser, Zhang, and Chan, and by Pock, Cremers, Bischof, and Chambolle is known to include as a special case the Douglas–Rachford splitting algorithm for minimizing the sum of two convex functions. We show that, conversely, the PDHG algorithm can be viewed as a special case of the Douglas–Rachford splitting algorithm.

This work introduces the particle-intensity channel (PIC) as a new model for molecular communication systems that includes imperfections at both transmitter and receiver and provides a new characterization of the capacity limits as well as properties of the optimal (capacity-achieving) input distributions for such channels. In the PIC, the transmitter encodes information, in symbols of a given duration, based on the probability of particle release, and the receiver detects and decodes the message based on the number of particles detected during the symbol interval. In this channel, the transmitter may be unable to control precisely the probability of particle release, and the receiver may not detect all the particles that arrive. We model this channel using a generalization of the binomial channel and show that the capacity-achieving input distribution for this channel always has mass points at probabilities of particle release of zero and one. To find the capacity-achieving input distributions, we develop a novel and efficient algorithm we call dynamic assignment Blahut-Arimoto (DAB). For diffusive particle transport, we also derive the conditions under which the input with two mass points is capacity-achieving.

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples.

The Douglas–Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper, closed, and convex functions and, 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 based on primal–dual approaches. We provide new linear convergence results in this setting.

This paper develops an efficient method to compute the binomial channel capacity and applies it to the molecular channel. The binomial channel (with parameter n) takes the success probability for a Bernoulli trial as input and produces the number of successes in trials as output. The input alphabet is the unit interval and the output alphabet is the set of integers from zero to n. Despite the fact that the input alphabet is uncountably infinite the capacity-achieving input distributions turn out to have a small finite support that evolves gracefully as n increases. The ellipsoid algorithm was previously used to compute the binomial channel capacity, but convergence is rather slow even with a well-chosen initial condition. The Dynamic Assignment Blahut-Arimoto (DAB) algorithm starts with the capacity-achieving mass point locations for the n-1 case and exploits Csiszàr's Min-Max Capacity Theorem to check convergence and adjust mass point locations to achieve a much faster convergence rate, unlocking the potential for the capacity and corresponding input distribution to be computed for larger values of n.

Signal processing applications of semidefinite optimization are often rooted in sum-of-squares representations of nonnegative trigonometric polynomials. Interior-point solvers for semidefinite optimization can handle constraints of this form with a per-iteration-complexity that is cubic in the degree of the trigonometric polynomial. The purpose of this paper is to discuss first-order methods with a lower complexity per iteration. The methods are based on generalized proximal operators defined in terms of the Itakura-Saito distance. This is the Bregman distance defined by the negative entropy function. The choice for the Itakura-Saito distance is motivated by the fact that the associated generalized projection on the set of normalized nonnegative trigonometric polynomials can be computed at a cost that is roughly quadratic in the degree of the polynomial. The generalized projection is the key operation in generalized proximal first-order methods that use Bregman distances instead of the squared Euclidean distance. The paper includes numerical results with Auslender and Teboulle's accelerated proximal gradient method for Bregman distances.

We consider T-optimal experiment design problems for discriminating multivariate polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. The original optimality criterion is reformulated as a convex optimization problem with a moment cone constraint. In the case of univariate regression models, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. For general multivariate cases, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When the relaxation hierarchy converges, the optimal discrimination design can be recovered from the optimal moment matrix, and its optimality confirmed by an equivalence theorem. The methodology is illustrated with several examples.

This groundbreaking textbook combines straightforward explanations with a wealth of practical examples to offer an innovative approach to teaching linear algebra. Requiring no prior knowledge of the subject, it covers the aspects of linear algebra - vectors, matrices, and least squares - that are needed for engineering applications, discussing examples across data science, machine learning and artificial intelligence, signal and image processing, tomography, navigation, control, and finance. The numerous practical exercises throughout allow students to test their understanding and translate their knowledge into solving real-world problems, with lecture slides, additional computational exercises in Julia and MATLAB®, and data sets accompanying the book online. Suitable for both one-semester and one-quarter courses, as well as self-study, this self-contained text provides beginning students with the foundation they need to progress to more advanced study.

We analyze the proximal Newton method for minimizing a sum of a self-concordant function and a convex function with an inexpensive proximal operator. We present new results on the global and local convergence of the method when inexact search directions are used. The method is illustrated with an application to L1-regularized covariance selection, in which prior constraints on the sparsity pattern of the inverse covariance matrix are imposed. In the numerical experiments the proximal Newton steps are computed by an accelerated proximal gradient method, and multifrontal algorithms for positive definite matrices with chordal sparsity patterns are used to evaluate gradients and matrix-vector products with the Hessian of the smooth component of the objective.