Despite progress in language model (LM) capabilities, evaluations have thus far focused on models' performance on tasks that humans have previously solved, including in programming (SWE-Bench) and mathematics (FrontierMath). We therefore propose testing models' ability to design and implement algorithms in an open-ended benchmark: We task LMs with writing code that efficiently solves computationally challenging problems in computer science, physics, and mathematics. Our AlgoTune benchmark consists of 120 tasks collected from domain experts and a framework for validating and timing LM-synthesized solution code, which is compared to reference implementations from popular open-source packages.In addition, we develop a baseline LM agent, AlgoTuner, and evaluate its performance across a suite of frontier models.AlgoTuner achieves an average 1.58x speedup against reference solvers, including methods from packages such as SciPy, scikit-learn and CVXPY.However, we find that current models fail to discover algorithmic innovations, instead preferring surface-level optimizations. We hope that AlgoTune catalyzes the development of LM agents exhibiting creative problem solving beyond state-of-the-art human performance.

As first-order optimization methods become the method of choice for solving large-scale optimization problems, optimization solvers based on first-order algorithms are being built. Such general-purpose solvers must robustly detect infeasible or misspecified problem instances, but the computational complexity of first-order methods for doing so has yet to be formally studied. In this work, we characterize the optimal accelerated rate of infeasibility detection. We show that the standard fixed-point iteration achieves a $\mathcal{O}(1/k^2)$ and $\mathcal{O}(1/k)$ rates, respectively, on the normalized iterates and the fixed-point residual converging to the infimal displacement vector, while the accelerated fixed-point iteration achieves $\mathcal{O}(1/k^2)$ and $\tilde{\mathcal{O}}(1/k^2)$ rates. We then provide a matching complexity lower bound to establish that $\Theta(1/k^2)$ is indeed the optimal accelerated rate.

Recently, the anchor acceleration, an acceleration mechanism distinct from Nesterov's, has been discovered for minimax optimization and fixed-point problems, but its mechanism is not understood well, much less so than Nesterov acceleration. In this work, we analyze continuous-time models of anchor acceleration. We provide tight, unified analyses for characterizing the convergence rate as a function of the anchor coefficient $\beta(t)$, thereby providing insight into the anchor acceleration mechanism and its accelerated $\mathcal{O}(1/k^2)$-convergence rate. Finally, we present an adaptive method inspired by the continuous-time analyses and establish its effectiveness through theoretical analyses and experiments.

Despite the broad use of fixed-point iterations throughout applied mathematics, the optimal convergence rate of general fixed-point problems with nonexpansive nonlinear operators has not been established. This work presents an acceleration mechanism for fixed-point iterations with nonexpansive operators, contractive operators, and nonexpansive operators satisfying a Hölder-type growth condition. We then provide matching complexity lower bounds to establish the exact optimality of the acceleration mechanisms in the nonexpansive and contractive setups. Finally, we provide experiments with CT imaging, optimal transport, and decentralized optimization to demonstrate the practical effectiveness of the acceleration mechanism.