arXiv subjects

Lieven Vandenberghe

Publications and source records attributed to Lieven Vandenberghe.

14 recordsLinked to original sources

arXiv2026

New complexity bounds for primal--dual interior-point algorithms in conic optimization

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 $O(\vartheta^{1/2}\ln(1/ε))$ 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.

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arXiv2026

Noninvasive Intracranial Pressure Estimation Using Subspace System Identification and Bespoke Machine Learning Algorithms: A Learning-to-Rank Approach

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. 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. 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. Our results demonstrate the feasibility of the proposed noninvasive ICP estimation approach. 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.

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arXiv2022

Linear optimization over homogeneous matrix cones

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.

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arXiv2022

Bregman three-operator splitting methods

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.

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arXiv2020

Capacities and Optimal Input Distributions for Particle-Intensity Channels

This work introduces the particle-intensity channel (PIC) as a model for molecular communication systems and characterizes 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 an 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.

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arXiv2020

T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method

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 is confirmed by an equivalence theorem. The methodology is illustrated with several examples.

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arXiv2019

Regularized Diffusion Adaptation via Conjugate Smoothing

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.

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arXiv2018

Douglas-Rachford splitting for a Lipschitz continuous and a strongly monotone operator

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.

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arXiv2017

Centralized Network Utility Maximization over Aggregate Flows

We study a network utility maximization (NUM) decomposition in which the set of flow rates is grouped by source-destination pairs. We develop theorems for both single-path and multipath cases, which relate an arbitrary NUM problem involving all flow rates to a simpler problem involving only the aggregate rates for each source-destination pair. The optimal aggregate flows are then apportioned among the constituent flows of each pair. This apportionment is simple for the case of $α$-fair utility functions. We also show how the decomposition can be implemented with the alternating direction method of multipliers (ADMM) algorithm.

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arXiv2016

Semidefinite representations of gauge functions for structured low-rank matrix decomposition

This paper presents generalizations of semidefinite programming formulations of 1-norm optimization problems over infinite dictionaries of vectors of complex exponentials, which were recently proposed for superresolution, gridless compressed sensing, and other applications in signal processing. Results related to the generalized Kalman-Yakubovich-Popov lemma in linear system theory provide simple, constructive proofs of the semidefinite representations of the penalty functions used in these applications. The connection leads to several extensions to gauge functions and atomic norms for sets of vectors parameterized via the nullspace of matrix pencils. The techniques are illustrated with examples of low-rank matrix approximation problems arising in spectral estimation and array processing.

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arXiv2013

Reduced-Complexity Semidefinite Relaxations of Optimal Power Flow Problems

We propose a new method for generating semidefinite relaxations of optimal power flow problems. The method is based on chordal conversion techniques: by dropping some equality constraints in the conversion, we obtain semidefinite relaxations that are computationally cheaper, but potentially weaker, than the standard semidefinite relaxation. Our numerical results show that the new relaxations often produce the same results as the standard semidefinite relaxation, but at a lower computational cost.

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arXiv2013

Decomposition in conic optimization with partially separable structure

Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general non-polyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators.

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arXiv2012

Subspace System Identification via Weighted Nuclear Norm Optimization

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.

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arXiv2012

Logarithmic barriers for sparse matrix cones

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.

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