arXiv subjects

Jake Roth

Publications and source records attributed to Jake Roth.

3 recordsLinked to original sources

arXiv2026

Perturbation Duality for Robust and Distributionally Robust Optimization: Short and General Proofs

Duality is a foundational tool in robust and distributionally robust optimization (RO/DRO), underpinning both analytical insights and tractable reformulations. Whereas RO/DRO duality is commonly established through minimax arguments or conic duality, we use perturbation duality to obtain new, more general results with short proofs. We show that this perspective provides a natural and unifying framework for deriving RO/DRO dual formulations, proving the associated duality results, and diagnosing the regularity assumptions on which they depend. First, guided by perturbation duality, we establish new duality theorems for a recent DRO framework that unifies several canonical models, including $\phi$-divergence and Wasserstein models, through optimal transport subject to conditional moment constraints. Our results resolve an open conjecture on this DRO duality by clarifying the role of compactness: compactness itself is not necessary, but can be replaced by perturbation-based regularity conditions. Second, we revisit \emph{robust duality}, commonly described as \emph{primal-worst equals dual-best.} Using bifunctions, we unify dual-best formulations appearing in the literature and derive concise perturbation-based proofs that streamline recent results. Overall, the paper positions perturbation duality as a versatile and underutilized tool for RO and DRO, offering both conceptual unification and technical generality across a broad class of models.

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arXiv2024

Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction

Superquantiles have recently gained significant interest as a risk-aware metric for addressing fairness and distribution shifts in statistical learning and decision making problems. This paper introduces a fast, scalable and robust second-order computational framework to solve large-scale optimization problems with superquantile-based constraints. Unlike empirical risk minimization, superquantile-based optimization requires ranking random functions evaluated across all scenarios to compute the tail conditional expectation. While this tail-based feature might seem computationally unfriendly, it provides an advantageous setting for a semismooth-Newton-based augmented Lagrangian method. The superquantile operator effectively reduces the dimensions of the Newton systems since the tail expectation involves considerably fewer scenarios. Notably, the extra cost of obtaining relevant second-order information and performing matrix inversions is often comparable to, and sometimes even less than, the effort required for gradient computation. Our developed solver is particularly effective when the number of scenarios substantially exceeds the number of decision variables. In synthetic problems with linear and convex diagonal quadratic objectives, numerical experiments demonstrate that our method outperforms existing approaches by a large margin: It achieves speeds more than 750 times faster for linear and quadratic objectives than the alternating direction method of multipliers as implemented by OSQP for computing low-accuracy solutions. Additionally, it is up to 25 times faster for linear objectives and 70 times faster for quadratic objectives than the commercial solver Gurobi, and 20 times faster for linear objectives and 30 times faster for quadratic objectives than the Portfolio Safeguard optimization suite for high-accuracy solution computations.

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arXiv2023

On $O(n)$ Algorithms for Projection onto the Top-$k$-sum Sublevel Set

The \emph{top-$k$-sum} operator computes the sum of the largest $k$ components of a given vector. The Euclidean projection onto the top-$k$-sum sublevel set serves as a crucial subroutine in iterative methods to solve composite superquantile optimization problems. In this paper, we introduce a solver that implements two finite-termination algorithms to compute this projection. Both algorithms have $O(n)$ complexity of floating point operations when applied to a sorted $n$-dimensional input vector, where the absorbed constant is \emph{independent of $k$}. This stands in contrast to an existing grid-search-inspired method that has $O(k(n-k))$ complexity, a partition-based method with $O(n+D\log D)$ complexity, where $D\leq n$ is the number of distinct elements in the input vector, and a semismooth Newon method with a finite termination property but unspecified floating point complexity. The improvement of our methods over the first method is significant when $k$ is linearly dependent on $n$, which is frequently encountered in practical superquantile optimization applications. In instances where the input vector is unsorted, an additional cost is incurred to (partially) sort the vector, whereas a full sort of the input vector seems unavoidable for the other two methods. To reduce this cost, we further derive a rigorous procedure that leverages approximate sorting to compute the projection, which is particularly useful when solving a sequence of similar projection problems. Numerical results show that our methods solve problems of scale $n=10^7$ and $k=10^4$ within $0.05$ seconds, whereas the most competitive alternative, the semismooth Newton-based method, takes about $1$ second. The existing grid-search method and Gurobi's QP solver can take from minutes to hours.

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