arXiv subjects

Ryan Cory-Wright

Publications and source records attributed to Ryan Cory-Wright.

At least 19 recordsLinked to original sources

arXiv2026

A Shrinkage Path Heuristic for Wasserstein Distributionally Robust Optimization

Wasserstein distributionally robust optimization (DRO) is a versatile and widely adopted framework for decision-making under uncertainty, yet its standard deterministic reformulations generally contain non-convex inner subproblems that are challenging to solve. To address this issue, we propose a shrinkage path heuristic that reduces the solution of a DRO problem to a one-dimensional search over the line segment connecting the (typically benign) sample average approximation (SAA) and the (more demanding but practically solvable) classical robust optimization solution. We derive a priori suboptimality bounds in stylized settings and, for the general case, a posteriori bounds obtained by applying a similar heuristic to a dual formulation. Numerical experiments on a multi-item newsvendor and an appointment scheduling problem show that the shrinkage path heuristic attains 85-110% (resp. 45-70%) of the out-of-sample performance improvements of Wasserstein DRO over SAA, at a fraction of the computational cost.

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arXiv2026

msPCA: An R Package for Sparse PCA with Multiple Components

We present msPCA: an open-source R package for sparse principal component analysis with multiple components. It implements an alternating maximization algorithm to generate a set of sparse loading vectors that collectively explain a large fraction of the variance in a dataset, while remaining non-redundant. The algorithm supports two definitions of non-redundancy: either orthogonality of the loading vectors or zero pairwise correlation between principal components (PCs). In the reported benchmarks, msPCA solves sparse PCA problems with thousands of features, achieving competitive runtimes while producing sparse components with controlled feasibility violations and a high fraction of variance explained.

stat.ML↗

arXiv2026

Compact Lifted Relaxations for Low-Rank Optimization

We develop tractable convex relaxations for rank-constrained quadratic optimization problems over $n \times m$ matrices, a setting for which tractable relaxations are typically only available when the objective or constraints admit spectral structure. We derive lifted semidefinite relaxations that do not require such spectral terms. Although a direct lifting introduces a large semidefinite constraint in dimension $n^2 + nm + 1$, we prove that many blocks of the moment matrix are redundant and derive an equivalent compact relaxation that only involves two semidefinite constraints of dimension $nm + 1$ and $n+m$, respectively. We also derive a new class of valid inequalities for low-rank problems, which we call projection cuts, that exploit the fact that rank constraints are inherited by linear images of a low-rank matrix, to strengthen our low-rank relaxations substantially. For matrix completion and reduced-rank regression problems, among others, we exploit additional structure to obtain even more compact formulations involving semidefinite matrices of dimension at most the sum of the two dimensions of the low-rank decision matrix (i.e., of size at most $n+m$). Overall, we obtain scalable semidefinite bounds for a broad class of low-rank quadratic problems.

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arXiv2026

Pricing Discrete and Nonlinear Markets With Semidefinite Relaxations

Nonconvexities in markets with discrete decisions and nonlinear constraints make efficient pricing challenging, often necessitating subsidies. A prime example is the unit commitment (UC) problem in electricity markets, where costly subsidies are commonly required. We propose a new pricing scheme for nonconvex markets with both discreteness and nonlinearity, by convexifying nonconvex structures through a semidefinite programming (SDP) relaxation and deriving prices from the relaxation's dual variables. When the choice set is bounded, we establish strong duality for the SDP, which allows us to extend the envelope theorem to the value function of the relaxation. This extension yields a marginal price signal for demand, which we use as our pricing mechanism. We demonstrate that under certain conditions-for instance, when the relaxation's right hand sides are linear in demand-the resulting lost opportunity cost is bounded by the relaxation's optimality gap. This result highlights the importance of achieving tight relaxations. The proposed framework applies to nonconvex electricity market problems, including for both direct current and alternating current UC. Our numerical experiments indicate that the SDP relaxations are often tight, reinforcing the effectiveness of the proposed pricing scheme. Across a suite of IEEE benchmark instances, the lost opportunity cost under our pricing scheme is, on average, 46% lower than that of the commonly used fixed-binary pricing scheme.

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arXiv2025

Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether

Advances in AI have shown great potential in contributing to the acceleration of scientific discovery. Symbolic regression can fit interpretable models to data, but these models are not necessarily derivable from established theory. Recent systems (e.g., AI-Descartes, AI-Hilbert) enforce derivability from prior knowledge. However, when existing theories are incomplete or incorrect, these machine-generated hypotheses may fall outside the theoretical scope. Automatically finding corrections to axiom systems to close this gap remains a central challenge in scientific discovery. We propose a solution: an open-source algebraic geometry-based system that, given an incomplete axiom system expressible as polynomials and a hypothesis that the axioms cannot derive, generates a minimal set of candidate axioms that, when added to the theory, provably derive the (possibly noisy) hypothesis. We illustrate the efficacy of our approach by showing that it can reconstruct key axioms required to derive the carrier-resolved photo-Hall effect, Einstein's relativistic laws, and several other laws.

cs.AI↗

arXiv2025

The Need for Verification in AI-Driven Scientific Discovery

Artificial intelligence (AI) is transforming the practice of science. Machine learning and large language models (LLMs) can generate hypotheses at a scale and speed far exceeding traditional methods, offering the potential to accelerate discovery across diverse fields. However, the abundance of hypotheses introduces a critical challenge: without scalable and reliable mechanisms for verification, scientific progress risks being hindered rather than being advanced. In this article, we trace the historical development of scientific discovery, examine how AI is reshaping established practices for scientific discovery, and review the principal approaches, ranging from data-driven methods and knowledge-aware neural architectures to symbolic reasoning frameworks and LLM agents. While these systems can uncover patterns and propose candidate laws, their scientific value ultimately depends on rigorous and transparent verification, which we argue must be the cornerstone of AI-assisted discovery.

cs.AI↗

arXiv2025

Stability Regularized Cross-Validation

We revisit the problem of ensuring strong test set performance via cross-validation, and propose a nested k-fold cross-validation scheme that selects hyperparameters by minimizing a weighted sum of the usual cross-validation metric and an empirical model-stability measure. The weight on the stability term is itself chosen via a nested cross-validation procedure. This reduces the risk of strong validation set performance and poor test set performance due to instability. We benchmark our procedure on a suite of $13$ real-world datasets, and find that, compared to $k$-fold cross-validation over the same hyperparameters, it improves the out-of-sample MSE for sparse ridge regression and CART by $4\%$ and $2\%$ respectively on average, but has no impact on XGBoost. It also reduces the user's out-of-sample disappointment, sometimes significantly. For instance, for sparse ridge regression, the nested k-fold cross-validation error is on average $0.9\%$ lower than the test set error, while the $k$-fold cross-validation error is $21.8\%$ lower than the test error. Thus, for unstable models such as sparse regression and CART, our approach improves test set performance and reduces out-of-sample disappointment.

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arXiv2025

Improved Approximation Algorithms for Orthogonally Constrained Problems Using Semidefinite Optimization

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite relaxation and propose a randomized rounding algorithm to generate feasible solutions from the relaxation. Second, we derive constant-factor approximation guarantees for our algorithm. When optimizing for $m$ orthonormal vectors in dimension $n$, we leverage strong duality and semidefinite complementary slackness to show that our algorithm achieves a $1/3$-approximation ratio. For any $m$ of the form $2^q$ for some integer $q$, we also construct an instance where the performance of our algorithm is exactly $(m+2)/(3m)$, which can be made arbitrarily close to $1/3$ by taking $m \rightarrow + \infty$, hence showing that our analysis is tight.

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arXiv2023

Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach

We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental problem in Operations Research and Machine Learning which arises in various applications, including data compression, latent semantic indexing, collaborative filtering, and medical imaging. We introduce a novel formulation for SLR that directly models its underlying discreteness. For this formulation, we develop an alternating minimization heuristic that computes high-quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We also develop a custom branch-and-bound algorithm that leverages our heuristic and convex relaxations to solve small instances of SLR to certifiable (near) optimality. Given an input $n$-by-$n$ matrix, our heuristic scales to solve instances where $n=10000$ in minutes, our relaxation scales to instances where $n=200$ in hours, and our branch-and-bound algorithm scales to instances where $n=25$ in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of rank, sparsity, and mean-square error while maintaining a comparable runtime.

stat.ML↗

arXiv2023

Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. In recent years, data-driven scientific discovery has emerged as a viable competitor in settings with large amounts of experimental data. Unfortunately, data-driven methods often fail to discover valid laws when data is noisy or scarce. Accordingly, recent works combine regression and reasoning to eliminate formulae inconsistent with background theory. However, the problem of searching over the space of formulae consistent with background theory to find one that best fits the data is not well-solved. We propose a solution to this problem when all axioms and scientific laws are expressible via polynomial equalities and inequalities and argue that our approach is widely applicable. We model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivstellensatz certificates. The optimization techniques leveraged in this paper allow our approach to run in polynomial time with fully correct background theory under an assumption that the complexity of our derivation is bounded), or non-deterministic polynomial (NP) time with partially correct background theory. We demonstrate that some famous scientific laws, including Kepler's Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from axioms and experimental data.

cs.AI↗

arXiv2023

Efficient Cross-Validation for Sparse Linear Regression

Given a high-dimensional covariate matrix and a response vector, ridge-regularized sparse linear regression selects a subset of features that explains the relationship between covariates and the response in an interpretable manner. To choose hyperparameters that control the sparsity level and amount of regularization, practitioners commonly use k-fold cross-validation. However, cross-validation substantially increases the computational cost of sparse regression as it requires solving many mixed-integer optimization problems (MIOs) for each hyperparameter combination. To address this computational burden, we derive computationally tractable relaxations of the k-fold cross-validation loss, facilitating hyperparameter selection while solving $50$--$80\%$ fewer MIOs in practice. Our computational results demonstrate, across eleven real-world UCI datasets, that exact MIO-based cross-validation can be competitive with mature software packages such as glmnet and L0Learn.

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arXiv2023

Decarbonizing OCP

We present our collaboration with the OCP Group, one of the world's largest producers of phosphate and phosphate-based products, to reduce OCP's carbon emissions significantly. We study the problem of decarbonizing OCP's electricity supply by installing a mixture of solar panels and batteries to minimize its time-discounted investment cost plus the cost of satisfying its remaining demand via the national grid. OCP is currently designing its renewable investment strategy, using insights gleaned from our optimization model, and has pledged to invest 130 billion MAD (approx. 13 billion USD) in a green initiative by 2027, a subset of which involves decarbonization. We immunize our model against deviations between forecast and realized solar generation output via a combination of robust and distributionally robust optimization. To account for variability in daily solar generation, we propose a data-driven robust optimization approach that prevents excessive conservatism by averaging across uncertainty sets. To protect against variability in seasonal weather patterns induced by climate change, we invoke distributionally robust optimization techniques. Under a 10 billion MAD investment by OCP, the proposed methodology reduces the carbon emissions which arise from OCP's energy needs by more than 70% while generating a net present value (NPV) of 5 billion MAD over twenty years. Moreover, a 20 billion MAD investment induces a 95% reduction in carbon emissions and generates an NPV of around 2 billion MAD. To fulfill the Paris climate agreement, rapidly decarbonizing the global economy in a financially sustainable fashion is imperative. Accordingly, this work develops a robust optimization methodology that enables OCP to decarbonize at a profit by purchasing solar panels and batteries. Moreover, the methodology could be applied to decarbonize other industrial consumers.

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arXiv2023

Disjunctive Branch-and-Bound for Certifiably Optimal Low-Rank Matrix Completion

Low-rank matrix completion consists of computing a matrix of minimal complexity that recovers a given set of observations as accurately as possible. Unfortunately, existing methods for matrix completion are heuristics that, while highly scalable and often identifying high-quality solutions, do not provide an instance-wise certificate of optimality. We reexamine matrix completion with an optimality-oriented eye. We reformulate low-rank matrix completion problems as convex problems over the non-convex set of projection matrices and implement a disjunctive branch-and-bound scheme that solves them to certifiable optimality. Further, we derive a novel and often near-exact class of convex relaxations by decomposing a low-rank matrix as a sum of rank-one matrices and incentivizing that two-by-two minors in each rank-one matrix have determinant zero. In numerical experiments, our new convex relaxations decrease the optimality gap by two orders of magnitude compared to existing attempts, and our disjunctive branch-and-bound scheme solves $n \times m$ rank-$k$ matrix completion problems to certifiable optimality or near optimality in hours for $\max \{m, n\} \leq 2500$ and $k \leq 5$. Moreover, this reduction in the training error translates into an average $2\%$--$50\%$ reduction in the test set error compared with alternating minimization-based methods.

cs.LG↗

arXiv2023

A Stochastic Benders Decomposition Scheme for Large-Scale Stochastic Network Design

Network design problems involve constructing edges in a transportation or supply chain network to minimize construction and daily operational costs. We study a stochastic version where operational costs are uncertain due to fluctuating demand and estimated as a sample average from historical data. This problem is computationally challenging, and instances with as few as 100 nodes often cannot be solved to optimality using current decomposition techniques. We propose a stochastic variant of Benders decomposition that mitigates the high computational cost of generating each cut by sampling a subset of the data at each iteration and nonetheless generates deterministically valid cuts, rather than the probabilistically valid cuts frequently proposed in the stochastic optimization literature, via a dual averaging technique. We implement both single-cut and multi-cut variants of this Benders decomposition, as well as a variant that uses clustering of the historical scenarios. To our knowledge, this is the first single-tree implementation of Benders decomposition that facilitates sampling. On instances with 100-200 nodes and relatively complete recourse, our algorithm achieves 5-7% optimality gaps, compared with 16-27% for deterministic Benders schemes, and scales to instances with 700 nodes and 50 commodities within hours. Beyond network design, our strategy could be adapted to generic two-stage stochastic mixed-integer optimization problems where second-stage costs are estimated via a sample average.

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arXiv2022

Sparse PCA With Multiple Components

Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods-such as iteratively computing one sparse PC and deflating the covariance matrix-that do not guarantee the orthogonality, let alone the optimality, of the resulting solution when we seek multiple mutually orthogonal PCs. We challenge this status by reformulating the orthogonality conditions as rank constraints and optimizing over the sparsity and rank constraints simultaneously. We design tight semidefinite relaxations to supply high-quality upper bounds, which we strengthen via additional second-order cone inequalities when each PC's individual sparsity is specified. Further, we derive a combinatorial upper bound on the maximum amount of variance explained as a function of the support. We exploit these relaxations and bounds to propose exact methods and rounding mechanisms that, together, obtain solutions with a bound gap on the order of 0%-15% for real-world datasets with p = 100s or 1000s of features and r \in {2, 3} components. Numerically, our algorithms match (and sometimes surpass) the best performing methods in terms of fraction of variance explained and systematically return PCs that are sparse and orthogonal. In contrast, we find that existing methods like deflation return solutions that violate the orthogonality constraints, even when the data is generated according to sparse orthogonal PCs. Altogether, our approach solves sparse PCA problems with multiple components to certifiable (near) optimality in a practically tractable fashion.

math.OC↗

arXiv2021

Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality

Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than $p=100s$ of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a cutting-plane method which solves the problem to certifiable optimality at the scale of selecting k=5 covariates from p=300 variables, and provides small bound gaps at a larger scale. We also propose a convex relaxation and greedy rounding scheme that provides bound gaps of $1-2\%$ in practice within minutes for $p=100$s or hours for $p=1,000$s and is therefore a viable alternative to the exact method at scale. Using real-world financial and medical datasets, we illustrate our approach's ability to derive interpretable principal components tractably at scale.

math.OC↗

arXiv2021

A new perspective on low-rank optimization

A key question in many low-rank problems throughout optimization, machine learning, and statistics is to characterize the convex hulls of simple low-rank sets and judiciously apply these convex hulls to obtain strong yet computationally tractable convex relaxations. We invoke the matrix perspective function - the matrix analog of the perspective function - and characterize explicitly the convex hull of epigraphs of simple matrix convex functions under low-rank constraints. Further, we combine the matrix perspective function with orthogonal projection matrices-the matrix analog of binary variables which capture the row-space of a matrix-to develop a matrix perspective reformulation technique that reliably obtains strong relaxations for a variety of low-rank problems, including reduced rank regression, non-negative matrix factorization, and factor analysis. Moreover, we establish that these relaxations can be modeled via semidefinite constraints and thus optimized over tractably. The proposed approach parallels and generalizes the perspective reformulation technique in mixed-integer optimization and leads to new relaxations for a broad class of problems.

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arXiv2021

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization problems over the non-convex set of orthogonal projection matrices. Furthermore, we design outer-approximation algorithms to solve low-rank problems to certifiable optimality, compute lower bounds via their semidefinite relaxations, and provide near-optimal solutions through rounding and local search techniques. We implement these numerical ingredients and, for the first time, solve low-rank optimization problems to certifiable optimality. Using currently available spatial branch-and-bound codes, not tailored to projection matrices, we can scale our exact (resp. near-exact) algorithms to matrices with up to 30 (resp. 600) rows/columns. Our algorithms also supply certifiably near-optimal solutions for larger problem sizes and outperform existing heuristics, by deriving an alternative to the popular nuclear norm relaxation which generalizes the perspective relaxation from vectors to matrices. All in all, our framework, which we name Mixed-Projection Conic Optimization, solves low-rank problems to certifiable optimality in a tractable and unified fashion.

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