arXiv subjects

Stephen Boyd

Publications and source records attributed to Stephen Boyd.

At least 55 records · Page 3Linked to original sources

arXiv2024

Portfolio Optimization with Cumulative Prospect Theory Utility via Convex Optimization

We consider the problem of choosing a portfolio that maximizes the cumulative prospect theory (CPT) utility on an empirical distribution of asset returns. We show that while CPT utility is not a concave function of the portfolio weights, it can be expressed as a difference of two functions. The first term is the composition of a convex function with concave arguments and the second term a composition of a convex function with convex arguments. This structure allows us to derive a global lower bound, or minorant, on the CPT utility, which we can use in a minorization-maximization (MM) algorithm for maximizing CPT utility. We further show that the problem is amenable to a simple convex-concave (CC) procedure which iteratively maximizes a local approximation. Both of these methods can handle small and medium size problems, and complex (but convex) portfolio constraints. We also describe a simpler method that scales to larger problems, but handles only simple portfolio constraints.

math.OC↗

arXiv2024

Robust Bond Portfolio Construction via Convex-Concave Saddle Point Optimization

The minimum (worst case) value of a long-only portfolio of bonds, over a convex set of yield curves and spreads, can be estimated by its sensitivities to the points on the yield curve. We show that sensitivity based estimates are conservative, \ie, underestimate the worst case value, and that the exact worst case value can be found by solving a tractable convex optimization problem. We then show how to construct a long-only bond portfolio that includes the worst case value in its objective or as a constraint, using convex-concave saddle point optimization.

math.OC↗

arXiv2024

Disciplined Saddle Programming

We consider convex-concave saddle point problems, and more generally convex optimization problems we refer to as $\textit{saddle problems}$, which include the partial supremum or infimum of convex-concave saddle functions. Saddle problems arise in a wide range of applications, including game theory, machine learning, and finance. It is well known that a saddle problem can be reduced to a single convex optimization problem by dualizing either the convex (min) or concave (max) objectives, reducing a min-max problem into a min-min (or max-max) problem. Carrying out this conversion by hand can be tedious and error prone. In this paper we introduce $\textit{disciplined saddle programming}$ (DSP), a domain specific language (DSL) for specifying saddle problems, for which the dualizing trick can be automated. The language and methods are based on recent work by Juditsky and Nemirovski arXiv:2102.01002 [math.OC], who developed the idea of conic-representable saddle point programs, and showed how to carry out the required dualization automatically using conic duality. Juditsky and Nemirovski's conic representation of saddle problems extends Nesterov and Nemirovski's earlier development of conic representable convex problems; DSP can be thought of as extending disciplined convex programming (DCP) to saddle problems. Just as DCP makes it easy for users to formulate and solve complex convex problems, DSP allows users to easily formulate and solve saddle problems. Our method is implemented in an open-source package, also called DSP.

math.OC↗

arXiv2024

Markowitz Portfolio Construction at Seventy

More than seventy years ago Harry Markowitz formulated portfolio construction as an optimization problem that trades off expected return and risk, defined as the standard deviation of the portfolio returns. Since then the method has been extended to include many practical constraints and objective terms, such as transaction cost or leverage limits. Despite several criticisms of Markowitz's method, for example its sensitivity to poor forecasts of the return statistics, it has become the dominant quantitative method for portfolio construction in practice. In this article we describe an extension of Markowitz's method that addresses many practical effects and gracefully handles the uncertainty inherent in return statistics forecasting. Like Markowitz's original formulation, the extension is also a convex optimization problem, which can be solved with high reliability and speed.

q-fin.PM↗

arXiv2024

Fast Path Planning Through Large Collections of Safe Boxes

We present a fast algorithm for the design of smooth paths (or trajectories) that are constrained to lie in a collection of axis-aligned boxes. We consider the case where the number of these safe boxes is large, and basic preprocessing of them (such as finding their intersections) can be done offline. At runtime we quickly generate a smooth path between given initial and terminal positions. Our algorithm designs trajectories that are guaranteed to be safe at all times, and detects infeasibility whenever such a trajectory does not exist. Our algorithm is based on two subproblems that we can solve very efficiently: finding a shortest path in a weighted graph, and solving (multiple) convex optimal-control problems. We demonstrate the proposed path planner on large-scale numerical examples, and we provide an efficient open-source software implementation, fastpathplanning.

cs.RO↗

arXiv2023

A Simple Method for Predicting Covariance Matrices of Financial Returns

We consider the well-studied problem of predicting the time-varying covariance matrix of a vector of financial returns. Popular methods range from simple predictors like rolling window or exponentially weighted moving average (EWMA) to more sophisticated predictors such as generalized autoregressive conditional heteroscedastic (GARCH) type methods. Building on a specific covariance estimator suggested by Engle in 2002, we propose a relatively simple extension that requires little or no tuning or fitting, is interpretable, and produces results at least as good as MGARCH, a popular extension of GARCH that handles multiple assets. To evaluate predictors we introduce a novel approach, evaluating the regret of the log-likelihood over a time period such as a quarter. This metric allows us to see not only how well a covariance predictor does over all, but also how quickly it reacts to changes in market conditions. Our simple predictor outperforms MGARCH in terms of regret. We also test covariance predictors on downstream applications such as portfolio optimization methods that depend on the covariance matrix. For these applications our simple covariance predictor and MGARCH perform similarly.

econ.EM↗

arXiv2023

Value-Gradient Iteration with Quadratic Approximate Value Functions

We propose a method for designing policies for convex stochastic control problems characterized by random linear dynamics and convex stage cost. We consider policies that employ quadratic approximate value functions as a substitute for the true value function. Evaluating the associated control policy involves solving a convex problem, typically a quadratic program, which can be carried out reliably in real-time. Such policies often perform well even when the approximate value function is not a particularly good approximation of the true value function. We propose value-gradient iteration, which fits the gradient of value function, with regularization that can include constraints reflecting known bounds on the true value function. Our value-gradient iteration method can yield a good approximate value function with few samples, and little hyperparameter tuning. We find that the method can find a good policy with computational effort comparable to that required to just evaluate a control policy via simulation.

math.OC↗

arXiv2023

Tractable Evaluation of Stein's Unbiased Risk Estimate with Convex Regularizers

Stein's unbiased risk estimate (SURE) gives an unbiased estimate of the $\ell_2$ risk of any estimator of the mean of a Gaussian random vector. We focus here on the case when the estimator minimizes a quadratic loss term plus a convex regularizer. For these estimators SURE can be evaluated analytically for a few special cases, and generically using recently developed general purpose methods for differentiating through convex optimization problems; these generic methods however do not scale to large problems. In this paper we describe methods for evaluating SURE that handle a wide class of estimators, and also scale to large problem sizes.

math.ST↗

arXiv2023

Implementation of an Oracle-Structured Bundle Method for Distributed Optimization

We consider the problem of minimizing a function that is a sum of convex agent functions plus a convex common public function that couples them. The agent functions can only be accessed via a subgradient oracle; the public function is assumed to be structured and expressible in a domain specific language (DSL) for convex optimization. We focus on the case when the evaluation of the agent oracles can require significant effort, which justifies the use of solution methods that carry out significant computation in each iteration. To solve this problem we integrate multiple known techniques (or adaptations of known techniques) for bundle-type algorithms, obtaining a method which has a number of practical advantages over other methods that are compatible with our access methods, such as proximal subgradient methods. First, it is reliable, and works well across a number of applications. Second, it has very few parameters that need to be tuned, and works well with sensible default values. Third, it typically produces a reasonable approximate solution in just a few tens of iterations. This paper is accompanied by an open-source implementation of the proposed solver, available at \url{https://github.com/cvxgrp/OSBDO}.

math.OC↗

arXiv2023

Bounds on Efficiency Metrics in Photonics

In this paper, we present a method for computing bounds for a variety of efficiency metrics in photonics, such as the focusing efficiency or the mode purity. We focus on the special case where the objective function can be written as the ratio of two quadratic functions of the field and show that there exists a simple semidefinite programming relaxation for this problem. We provide a numerical example of bounding the maximal mode conversion purity for a device of given size. This paper is accompanied by an open source Julia package for basic simulations and bounds.

math.OC↗

arXiv2023

Computing Tighter Bounds on the $n$-Queens Constant via Newton's Method

In recent work Simkin shows that bounds on an exponent occurring in the famous $n$-queens problem can be evaluated by solving convex optimization problems, allowing him to find bounds far tighter than previously known. In this note we use Simkin's formulation, a sharper bound developed by Knuth, and a Newton method that scales to large problem instances, to find even sharper bounds.

math.OC↗

arXiv2023

Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models

Laplacian regularized stratified models (LRSM) are models that utilize the explicit or implicit network structure of the sub-problems as defined by the categorical features called strata (e.g., age, region, time, forecast horizon, etc.), and draw upon data from neighboring strata to enhance the parameter learning of each sub-problem. They have been widely applied in machine learning and signal processing problems, including but not limited to time series forecasting, representation learning, graph clustering, max-margin classification, and general few-shot learning. Nevertheless, existing works on LRSM have either assumed a known graph or are restricted to specific applications. In this paper, we start by showing the importance and sensitivity of graph weights in LRSM, and provably show that the sensitivity can be arbitrarily large when the parameter scales and sample sizes are heavily imbalanced across nodes. We then propose a generic approach to jointly learn the graph while fitting the model parameters by solving a single optimization problem. We interpret the proposed formulation from both a graph connectivity viewpoint and an end-to-end Bayesian perspective, and propose an efficient algorithm to solve the problem. Convergence guarantees of the proposed optimization algorithm is also provided despite the lack of global strongly smoothness of the Laplacian regularization term typically required in the existing literature, which may be of independent interest. Finally, we illustrate the efficiency of our approach compared to existing methods by various real-world numerical examples.

stat.ML↗

arXiv2022

A Light-Weight Multi-Objective Asynchronous Hyper-Parameter Optimizer

We describe a light-weight yet performant system for hyper-parameter optimization that approximately minimizes an overall scalar cost function that is obtained by combining multiple performance objectives using a target-priority-limit scalarizer. It also supports a trade-off mode, where the goal is to find an appropriate trade-off among objectives by interacting with the user. We focus on the common scenario where there are on the order of tens of hyper-parameters, each with various attributes such as a range of continuous values, or a finite list of values, and whether it should be treated on a linear or logarithmic scale. The system supports multiple asynchronous simulations and is robust to simulation stragglers and failures.

cs.LG↗

arXiv2022

Portfolio Construction with Gaussian Mixture Returns and Exponential Utility via Convex Optimization

We consider the problem of choosing an optimal portfolio, assuming the asset returns have a Gaussian mixture (GM) distribution, with the objective of maximizing expected exponential utility. In this paper we show that this problem is convex, and readily solved exactly using domain-specific languages for convex optimization, without the need for sampling or scenarios. We then show how the closely related problem of minimizing entropic value at risk can also be formulated as a convex optimization problem.

math.OC↗

arXiv2022

Strategic Asset Allocation with Illiquid Alternatives

We address the problem of strategic asset allocation (SAA) with portfolios that include illiquid alternative asset classes. The main challenge in portfolio construction with illiquid asset classes is that we do not have direct control over our positions, as we do in liquid asset classes. Instead we can only make commitments; the position builds up over time as capital calls come in, and reduces over time as distributions occur, neither of which the investor has direct control over. The effect on positions of our commitments is subject to a delay, typically of a few years, and is also unknown or stochastic. A further challenge is the requirement that we can meet the capital calls, with very high probability, with our liquid assets. We formulate the illiquid dynamics as a random linear system, and propose a convex optimization based model predictive control (MPC) policy for allocating liquid assets and making new illiquid commitments in each period. Despite the challenges of time delay and uncertainty, we show that this policy attains performance surprisingly close to a fictional setting where we pretend the illiquid asset classes are completely liquid, and we can arbitrarily and immediately adjust our positions. In this paper we focus on the growth problem, with no external liabilities or income, but the method is readily extended to handle this case.

math.OC↗

arXiv2022

Portfolio Construction as Linearly Constrained Separable Optimization

Mean-variance portfolio optimization problems often involve separable nonconvex terms, including penalties on capital gains, integer share constraints, and minimum position and trade sizes. We propose a heuristic algorithm for such problems based on the alternating direction method of multipliers (ADMM). This method allows for solve times in tens to hundreds of milliseconds with around 1000 securities and 100 risk factors. We also obtain a bound on the achievable performance. Our heuristic and bound are both derived from similar results for other optimization problems with a separable objective and affine equality constraints. We discuss a concrete implementation in the case where the separable terms in the objective are piecewise quadratic, and we empirically demonstrate its effectiveness for tax-aware portfolio construction.

math.OC↗

arXiv2022

Operator Splitting for Adaptive Radiation Therapy with Nonlinear Health Dynamics

We present an optimization-based approach to radiation treatment planning over time. Our approach formulates treatment planning as an optimal control problem with nonlinear patient health dynamics derived from the standard linear-quadratic cell survival model. As the formulation is nonconvex, we propose a method for obtaining an approximate solution by solving a sequence of convex optimization problems. This method is fast, efficient, and robust to model error, adapting readily to changes in the patient's health between treatment sessions. Moreover, we show that it can be combined with the operator splitting method ADMM to produce an algorithm that is highly scalable and can handle large clinical cases. We introduce an open-source Python implementation of our algorithm, AdaRad, and demonstrate its performance on several examples.

physics.med-ph↗

arXiv2022

Confidence bands for a log-concave density

We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is $k$-affine.

math.ST↗