arXiv subjects

Paul Goulart

Publications and source records attributed to Paul Goulart.

At least 19 recordsLinked to original sources

arXiv2025

A warmstarting technique for general conic optimization in interior point methods

We propose a novel warmstarting method for primal-dual interior point methods based on a smoothing operator that generates a starting point on the central path from the previous optimum. Compared to traditional approaches that prioritize minimizing infeasibility residuals, our method focuses on maintaining proximity to the central path. Computation of a smoothing operator is efficient and can be parallelized for conic constraints. We also prove that the residual of the smoothed starting point remains comparable to the one before the smoothing step. The numerical tests show that the proposed warmstarting strategy can reduce iteration numbers and computational time effectively across test problems.

math.OC↗

arXiv2025

CuClarabel: GPU Acceleration for a Conic Optimization Solver

We present the GPU implementation of the general-purpose interior-point solver Clarabel for convex optimization problems with conic constraints. We introduce a mixed parallel computing strategy that processes linear constraints first, then handles other conic constraints in parallel. The GPU solver currently supports linear equality and inequality constraints, second-order cones, exponential cones, power cones and positive semidefinite cones of the same dimensionality. We demonstrate that integrating a mixed parallel computing strategy with GPU-based direct linear system solvers enhances the performance of GPU-based conic solvers, surpassing their CPU-based counterparts across a wide range of conic optimization problems. We also show that employing mixed-precision linear system solvers can potentially achieve additional acceleration without compromising solution accuracy.

math.OC↗

arXiv2025

Data-Driven Performance Guarantees for Parametric Optimization Problems

We propose a data-driven method to establish probabilistic performance guarantees for parametric optimization problems solved via iterative algorithms. Our approach addresses two key challenges: providing convergence guarantees to characterize the worst-case number of iterations required to achieve a predefined tolerance, and upper bounding a performance metric after a fixed number of iterations. These guarantees are particularly useful for online optimization problems with limited computational time, where existing performance guarantees are often unavailable or unduly conservative. We formulate the convergence analysis problem as a scenario optimization program based on a finite set of sampled parameter instances. Leveraging tools from scenario optimization theory enables us to derive probabilistic guarantees on the number of iterations needed to meet a given tolerance level. Using recent advancements in scenario optimization, we further introduce a relaxation approach to trade the number of iterations against the risk of violating convergence criteria thresholds. Additionally, we analyze the trade-off between solution accuracy and time efficiency for fixed-iteration optimization problems by casting them into scenario optimization programs. Numerical simulations demonstrate the efficacy of our approach in providing reliable probabilistic convergence guarantees and evaluating the trade-off between solution accuracy and computational cost.

math.OC↗

arXiv2025

Predictive Control Barrier Functions: Bridging model predictive control and control barrier functions

In this paper, we establish a connection between model predictive control (MPC) techniques and Control Barrier Functions (CBFs). Recognizing the similarity between CBFs and Control Lyapunov Functions (CLFs), we propose a MPC formulation that ensures invariance and safety without relying on explicit stability conditions. The value function of our proposed MPC is a CBF, which we refer to as the Predictive Control Barrier Function (PCBF), similar to traditional MPC formulations which encode stability by having value functions as CLFs. Our formulation is simpler than previous PCBF approaches and is based on weaker assumptions while proving a similar theorem that guarantees safety recovery. Notably, our MPC formulation does not require the value function to be strictly decreasing to ensure convergence to a safe invariant set. Numerical examples demonstrate the effectiveness of our approach in guaranteeing safety and constructing non-conservative CBFs.

math.OC↗

arXiv2025

Synthesis of safety certificates for discrete-time uncertain systems via convex optimization

We study the problem of co-designing control barrier functions and linear state feedback controllers for discrete-time linear systems affected by additive disturbances. For disturbances of bounded magnitude, we provide a semi-definite program whose feasibility implies the existence of a control law and a certificate ensuring safety in the infinite horizon with respect to the worst-case disturbance realization in the uncertainty set. For disturbances with unbounded support, we rely on martingale theory to derive a second semi-definite program whose feasibility provides probabilistic safety guarantees holding joint-in-time over a finite time horizon. We examine several extensions, including (i) encoding of different types of input constraints, (ii) robustification against distributional ambiguity around the true distribution, (iii) design of safety filters, and (iv) extension to general safety specifications such as obstacle avoidance.

math.OC↗

arXiv2023

Control of Cross-Directional Systems with Approximate Symmetries

Structural symmetries of linear dynamical systems can be exploited for decoupling the dynamics and reducing the computational complexity of the controller implementation. However, in practical applications, inexact structural symmetries undermine the ability to decouple the system, resulting in the loss of any potential complexity reduction. To address this, we propose substituting an approximation with exact structural symmetries for the original system model, thereby introducing an approximation error. We focus on internal model controllers for cross-directional systems encountered in large-scale and high-speed control problems of synchrotrons or the process industry and characterise the stability, performance, and robustness properties of the resulting closed loop. While existing approaches replace the original system model with one that minimises the Frobenius norm of the approximation error, we show that this can lead to instability or poor performance. Instead, we propose approximations that are obtained from semidefinite programming problems. We show that our proposed approximations can yield stable systems even when the Frobenius norm approximation does not. The paper concludes with numerical examples and a case study of a synchrotron light source with inexact structural symmetries. Exploiting structural symmetries in large-scale and high-speed systems enables faster sampling times and the use of more advanced control techniques, even when the symmetries are approximate.

eess.SY↗

arXiv2023

An Efficient IPM Implementation for A Class of Nonsymmetric Cones

We present an efficient implementation of interior point methods for a family of nonsymmetric cones, including generalized power cones, power mean cones and relative entropy cones, by exploiting underlying low-rank and sparse properties of Hessians of homogeneous self-concordant barrier functions. We prove that the augmented linear system in our interior point method is sparse and quasi-definite, enabling the use of sparse LDL factorization with a dual scaling strategy for nonsymmetric cones. Numerical results show that our proposed implementation for nonsymmetric cones performs much faster than the state-of-art solvers for spare problems and scales well for large problems.

math.OC↗

arXiv2023

The Past Does Matter: Correlation of Subsequent States in Trajectory Predictions of Gaussian Process Models

Computing the distribution of trajectories from a Gaussian Process model of a dynamical system is an important challenge in utilizing such models. Motivated by the computational cost of sampling-based approaches, we consider approximations of the model's output and trajectory distribution. We show that previous work on uncertainty propagation, focussed on discrete state-space models, incorrectly included an independence assumption between subsequent states of the predicted trajectories. Expanding these ideas to continuous ordinary differential equation models, we illustrate the implications of this assumption and propose a novel piecewise linear approximation of Gaussian Processes to mitigate them.

stat.ML↗

arXiv2023

A Unified Early Termination Technique for Primal-dual Algorithms in Mixed Integer Conic Programming

We propose an early termination technique for mixed integer conic programming for use within branch-and-bound based solvers. Our approach generalizes previous early termination results for ADMM-based solvers to a broader class of primal-dual algorithms, including both operator splitting methods and interior point methods. The complexity for checking early termination is $O(n)$ for each termination check assuming a bounded problem domain. We show that this domain restriction can be relaxed for problems whose data satisfies a simple rank condition, in which case each check requires an $O(n^2)$ solve using a linear system that must be factored only once at the root node. We further show how this approach can be used in hybrid model predictive control as long as system inputs are bounded. Numerical results show that our method leads to a moderate reduction in the total iterations required for branch-and-bound conic solvers with interior-point based subsolvers.

math.OC↗

arXiv2023

Burer-Monteiro ADMM for Large-scale SDPs

We propose a bilinear decomposition for the Burer-Monteiro method and combine it with the standard Alternating Direction Method of Multipliers algorithm for semidefinite programming. Bilinear decomposition reduces the degree of the augmented Lagrangian from four to two, which makes each of the subproblems a quadratic programming and hence computationally efficient. Our approach is able to solve a class of large-scale SDPs with diagonal constraints. We prove that our ADMM algorithm converges globally to a first-order stationary point, and show by exploiting the negative curvature that the algorithm converges to a point within $O(1-1/r)$ of the optimal objective value. Additionally, the proximal variant of the algorithm can solve block-diagonally constrained SDPs with global convergence to a first-order stationary point. Numerical results show that both our ADMM algorithm and the proximal variant outperform the state-of-art Riemannian manifold algorithms and can reach the global optimum empirically.

math.OC↗

arXiv2022

Safeguarded Anderson acceleration for parametric nonexpansive operators

This paper describes the design of a safeguarding scheme for Anderson acceleration to improve its practical performance and stability when used for first-order optimisation methods. We show how the combination of a non-expansiveness condition, conditioning constraints, and memory restarts integrate well with solver algorithms that can be represented as fixed point operators with dynamically varying parameters. The performance of the scheme is demonstrated on seven different QP and SDP problem types, including more than 500 problems. The safeguarded Anderson acceleration scheme proposed in this paper is implemented in the open-source ADMM-based conic solver COSMO.

math.OC↗

arXiv2021

Learning ODE Models with Qualitative Structure Using Gaussian Processes

Recent advances in learning techniques have enabled the modelling of dynamical systems for scientific and engineering applications directly from data. However, in many contexts explicit data collection is expensive and learning algorithms must be data-efficient to be feasible. This suggests using additional qualitative information about the system, which is often available from prior experiments or domain knowledge. We propose an approach to learning a vector field of differential equations using sparse Gaussian Processes that allows us to combine data and additional structural information, like Lie Group symmetries and fixed points. We show that this combination improves extrapolation performance and long-term behaviour significantly, while also reducing the computational cost.

cs.LG↗

arXiv2020

COSMO: A conic operator splitting method for convex conic problems

This paper describes the Conic Operator Splitting Method (COSMO) solver, an operator splitting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. A number of benchmarks against other state-of-the-art solvers for a variety of problems show the effectiveness of our approach. Our Julia implementation is open-source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.

math.OC↗

arXiv2020

Approximate Dynamic Programming for Delivery Time Slot Pricing: a Sensitivity Analysis

We consider the revenue management problem of finding profit-maximising prices for delivery time slots in the context of attended home delivery. This multi-stage optimal control problem admits a dynamic programming formulation that is intractable for realistic problem sizes due to the so-called "curse of dimensionality". Therefore, we study three approximate dynamic programming algorithms both from a control-theoretical perspective and in a parametric numerical case study. Our numerical analysis is based on real-world data, from which we generate multiple scenarios to stress-test the robustness of the pricing policies to errors in model parameter estimates. Our theoretical analysis and numerical benchmark tests show that one of these algorithms, namely gradient-bounded dynamic programming, dominates the others with respect to computation time and profit-generation capabilities of the delivery slot pricing policies that it generates. Finally, we show that uncertainty in the estimates of the model parameters further increases the profit-generation dominance of this approach.

math.OC↗

arXiv2020

Gradient-Bounded Dynamic Programming for Submodular and Concave Extensible Value Functions with Probabilistic Performance Guarantees

We consider stochastic dynamic programming problems with high-dimensional, discrete state-spaces and finite, discrete-time horizons that prohibit direct computation of the value function from a given Bellman equation for all states and time steps due to the "curse of dimensionality". For the case where the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function in the realm of dual dynamic programming. We show that the proposed algorithm terminates after a finite number of iterations. Furthermore, we derive probabilistic guarantees on the value accumulated under the associated policy for a single realisation of the dynamic program and for the expectation of this value. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.

math.OC↗

arXiv2020

Gradient-Bounded Dynamic Programming with Submodular and Concave Extensible Value Functions

We consider dynamic programming problems with finite, discrete-time horizons and prohibitively high-dimensional, discrete state-spaces for direct computation of the value function from the Bellman equation. For the case that the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function similar to dual dynamic programming. We then show that the proposed algorithm terminates after a finite number of iterations. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.

math.OC↗

arXiv2020

A clique graph based merging strategy for decomposable SDPs

Chordal decomposition techniques are used to reduce large structured positive semidefinite matrix constraints in semidefinite programs (SDPs). The resulting equivalent problem contains multiple smaller constraints on the nonzero blocks (or cliques) of the original problem matrices. This usually leads to a significant reduction in the overall solve time. A further reduction is possible by remerging cliques with significant overlap. The degree of overlap for which this is effective is dependent on the particular solution algorithm and hardware to be employed. We propose a novel clique merging approach that utilizes the clique graph to identify suitable merge candidates. We show its performance by comparing it with two existing methods on selected problems from a benchmark library. Our approach is implemented in the latest version of the conic ADMM-solver COSMO.

math.OC↗

arXiv2020

Output feedback stochastic MPC with packet losses

The paper considers constrained linear systems with stochastic additive disturbances and noisy measurements transmitted over a lossy communication channel. We propose a model predictive control (MPC) law that minimizes a discounted cost subject to a discounted expectation constraint. Sensor data is assumed to be lost with known probability, and data losses are accounted for by expressing the predicted control policy as an affine function of future observations, which results in a convex optimal control problem. An online constraint-tightening technique ensures recursive feasibility of the online optimization and satisfaction of the expectation constraint without bounds on the distributions of the noise and disturbance inputs. The cost evaluated along trajectories of the closed loop system is shown to be bounded by the optimal predicted cost. A numerical example is given to illustrate these results.

math.OC↗