About

Stephen P. Boyd is a professor at the Department of Electrical Engineering at Stanford University. His research focuses on optimization, control, and systems theory, with a particular emphasis on convex optimization and its applications. Boyd has contributed extensively to the development of algorithms and theoretical foundations in these areas, and his work is widely recognized for its impact on both academic research and practical problem-solving in engineering and data science. He has supervised numerous PhD students and postdoctoral researchers, many of whom have gone on to prominent roles in academia and industry. Boyd is also known for his involvement in educational initiatives, including the development of courses and software related to convex optimization. His contributions have helped shape modern approaches to large-scale optimization problems, and he continues to be an influential figure in the field.

Research topics

• Computer Science
• Mathematical optimization
• Artificial Intelligence
• Mathematics
• Machine Learning
• Algorithm
• Geometry
• Applied mathematics
• Mathematical analysis

Selected publications

• CuClarabel: GPU Acceleration for a Conic Optimization Solver

  ACM Transactions on Mathematical Software · 2026-05-21

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• An Operator Splitting Method for Large-Scale CVaR-Constrained Quadratic Programs

  Optimization and Engineering · 2026-04-25

  preprintOpen accessSenior author

  Abstract We introduce a fast and scalable method for solving quadratic programs with conditional value-at-risk (CVaR) constraints. While these problems can be formulated as standard quadratic programs, the number of variables and constraints grows linearly with the number of scenarios, making general-purpose solvers impractical for large-scale problems. Our method combines operator splitting with a specialized $$O(m\log m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>log</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> algorithm for projecting onto CVaR constraints, where m is the number of scenarios. The method alternates between solving a linear system and performing parallel projections, onto CVaR constraints using our specialized algorithm and onto box constraints by simple clipping. Numerical examples from several application domains demonstrate that our method outperforms general-purpose solvers by several orders of magnitude on problems with up to millions of scenarios. Our method is implemented in an open-source package called CVQP.

  PublisherDOI

• Enhancing a Risk Model by Adding Transient Statistical Factors

  ArXiv.org · 2026-05-13

  articleOpen access

  Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

  PublisherOA PDF

• Enhancing a Risk Model by Adding Transient Statistical Factors

  arXiv (Cornell University) · 2026-05-13

  preprintOpen access

  Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

  PublisherDOI

• Automatic Generation of Explicit Quadratic Programming Solvers

  ArXiv.org · 2025-06-13

  preprintOpen accessSenior author

  We consider a family of convex quadratic programs in which the coefficients of the linear objective term and the righthand side of the constraints are affine functions of a parameter. It is well known that the solution of such a parametrized quadratic program is a piecewise affine function of the parameter. The number of (polyhedral) regions in the solution map can grow exponentially in problem size, but when the number of regions is moderate, a so-called explicit solver is practical. Such a solver computes the coefficients of the affine functions and the linear inequalities defining the polyhedral regions offline; to solve a problem instance online it simply evaluates this explicit solution map. Potential advantages of an explicit solver over a more general purpose iterative solver can include transparency, interpretability, reliability, and speed. In this paper we describe how code generation can be used to automatically generate an explicit solver from a high level description of a parametrized quadratic program. Our method has been implemented in the open-source software CVXPYgen, which is part of CVXPY, a domain specific language for general convex optimization.

  PublisherOA PDFDOI

• Correction: Tax-Aware Portfolio Construction Via Convex Optimization

  Journal of Optimization Theory and Applications · 2025-01-30

  articleOpen access
  PublisherOA PDFDOI

• Aging-Aware Battery Control via Convex Optimization

  ArXiv.org · 2025-05-13

  preprintOpen access

  We consider the task of controlling a battery while balancing two competing objectives that evolve over different time scales. The short-term objective, such as arbitrage or load smoothing, improves with more battery cycling, while the long-term objective is to maximize battery lifetime, which discourages cycling. Using a semi-empirical aging model, we formulate this problem as a convex optimization problem. We use model predictive control (MPC) with a convex approximation of aging dynamics to optimally manage the trade-off between performance and degradation. Through simulations, we quantify this trade-off in both economic and smoothing applications.

  PublisherOA PDFDOI

• Solving Large Multicommodity Network Flow Problems on GPUs

  ArXiv.org · 2025-01-29

  preprintOpen accessSenior author

  We consider the all-pairs multicommodity network flow problem on a network with capacitated edges. The usual treatment keeps track of a separate flow for each source-destination pair on each edge; we rely on a more efficient formulation in which flows with the same destination are aggregated, reducing the number of variables by a factor equal to the size of the network. Problems with hundreds of nodes, with a total number of variables on the order of a million, can be solved using standard generic interior-point methods on CPUs; we focus on GPU-compatible algorithms that can solve such problems much faster, and in addition scale to much larger problems, with up to a billion variables. Our method relies on the primal-dual hybrid gradient algorithm, and exploits several specific features of the problem for efficient GPU computation. Numerical experiments show that our primal-dual multicommodity network flow method accelerates state of the art generic commercial solvers by $100\times$ to $1000\times$, and scales to problems that are much larger. We provide an open source implementation of our method.

  PublisherOA PDFDOI

• A Note on Optimal Product Pricing

  Operations Research Forum · 2025-01-01

  articleOpen accessSenior author
  PublisherOA PDFDOI

• Adaptive Strategies for Pension Fund Management

  ArXiv.org · 2025-08-18

  preprintOpen accessSenior author

  This paper proposes a simulation-based framework for assessing and improving the performance of a pension fund management scheme. This framework is modular and allows the definition of customized performance metrics that are used to assess and iteratively improve asset and liability management policies. We illustrate our framework with a simple implementation that showcases the power of including adaptable features. We show that it is possible to dissipate longevity and volatility risks by permitting adaptability in asset allocation and payout levels. The numerical results show that by including a small amount of flexibility, there can be a substantial reduction in the cost to run the pension plan as well as a substantial decrease in the probability of defaulting.

  PublisherOA PDFDOI

Recent grants

• Semidefinite Programming for Weight Design in Fast Converging Distributed Algorithms

  NSF · $175k · 2004–2007

• Sensors: GOALI: Networked Estimation and Decision Computing for Structural Health Monitoring

  NSF · $280k · 2005–2009

Frequent coauthors

• Shane Barratt

  Stanford University

  57 shared

• Donald Eckman

  University of California, San Diego

  49 shared

• Jorge I. Poveda

  49 shared

• Richard Bellman

  49 shared

• Steven Diamond

  47 shared

• Lieven Vandenberghe

  University of California, Los Angeles

  42 shared

• Roger Reynolds

  41 shared

• Bernard Septimus

  41 shared

Labs

• TeachingPI