arXiv subjects

Sebastian Pokutta

Publications and source records attributed to Sebastian Pokutta.

At least 91 records · Page 5Linked to original sources

arXiv2025

A Frank-Wolfe-based primal heuristic for quadratic mixed-integer optimization

We propose a primal heuristic for quadratic mixed-integer problems. Our method extends the Boscia framework -- originally a mixed-integer convex solver leveraging a Frank-Wolfe-based branch-and-bound approach -- to address nonconvex quadratic objective and constraints. We reformulate nonlinear constraints, introduce preprocessing steps, and a suite of heuristics including rounding strategies, gradient-guided selection, and large neighborhood search techniques that exploit integer-feasible vertices generated during the Frank-Wolfe iterations. Computational results demonstrate the effectiveness of our method in solving challenging MIQCQPs, achieving improvements on QPLIB instances within minutes and winning first place in the Land-Doig MIP Computational Competition 2025.

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arXiv2025

Neural Sum-of-Squares: Certifying the Nonnegativity of Polynomials with Transformers

Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property, i.e., it can be written as a sum of squares of other polynomials. In practice, however, certifying the SOS criterion remains computationally expensive and often involves solving a Semidefinite Program (SDP), whose dimensionality grows quadratically in the size of the monomial basis of the SOS expression; hence, various methods to reduce the size of the monomial basis have been proposed. In this work, we introduce the first learning-augmented algorithm to certify the SOS criterion. To this end, we train a Transformer model that predicts an almost-minimal monomial basis for a given polynomial, thereby drastically reducing the size of the corresponding SDP. Our overall methodology comprises three key components: efficient training dataset generation of over 100 million SOS polynomials, design and training of the corresponding Transformer architecture, and a systematic fallback mechanism to ensure correct termination, which we analyze theoretically. We validate our approach on over 200 benchmark datasets, achieving speedups of over $100\times$ compared to state-of-the-art solvers and enabling the solution of instances where competing approaches fail. Our findings provide novel insights towards transforming the practical scalability of SOS programming.

cs.LG↗

arXiv2025

Don't Be Greedy, Just Relax! Pruning LLMs via Frank-Wolfe

Pruning is a common technique to reduce the compute and storage requirements of Neural Networks. While conventional approaches typically retrain the model to recover pruning-induced performance degradation, state-of-the-art Large Language Model (LLM) pruning methods operate layer-wise, minimizing the per-layer pruning error on a small calibration dataset to avoid full retraining, which is considered computationally prohibitive for LLMs. However, finding the optimal pruning mask is a hard combinatorial problem and solving it to optimality is intractable. Existing methods hence rely on greedy heuristics that ignore the weight interactions in the pruning objective. In this work, we instead consider the convex relaxation of these combinatorial constraints and solve the resulting problem using the Frank-Wolfe (FW) algorithm. Our method drastically reduces the per-layer pruning error, outperforms strong baselines on state-of-the-art GPT architectures, and remains memory-efficient. We provide theoretical justification by showing that, combined with the convergence guarantees of the FW algorithm, we obtain an approximate solution to the original combinatorial problem upon rounding the relaxed solution to integrality.

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arXiv2025

Norm-induced Cuts: Outer Approximation for Lipschitzian Constraint Functions

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method's correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.

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arXiv2025

Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes

We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $μ$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results.

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arXiv2025

The Pivoting Framework: Frank-Wolfe Algorithms with Active Set Size Control

We propose the pivoting meta algorithm (PM) to enhance optimization algorithms that generate iterates as convex combinations of vertices of a feasible region $C\subseteq \mathbb{R}^n$, including Frank-Wolfe (FW) variants. PM guarantees that the active set (the set of vertices in the convex combination) of the modified algorithm remains as small as $\mathrm{dim}(C)+1$ as stipulated by Carathéodory's theorem. PM achieves this by reformulating the active set expansion task into an equivalent linear program, which can be efficiently solved using a single pivot step akin to the primal simplex algorithm; the convergence rate of the original algorithms are maintained. Furthermore, we establish the connection between PM and active set identification, in particular showing under mild assumptions that PM applied to the away-step Frank-Wolfe algorithm or the blended pairwise Frank-Wolfe algorithm bounds the active set size by the dimension of the optimal face plus $1$. We provide numerical experiments to illustrate practicality and efficacy on active set size reduction.

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arXiv2025

Improved algorithms and novel applications of the FrankWolfe.jl library

Frank-Wolfe (FW) algorithms have emerged as an essential class of methods for constrained optimization, especially on large-scale problems. In this paper, we summarize the algorithmic design choices and progress made in the last years of the development of FrankWolfe.jl, a Julia package gathering high-performance implementations of state-of-the-art FW variants. We review key use cases of the library in the recent literature, which match its original dual purpose: first, becoming the de-facto toolbox for practitioners applying FW methods to their problem, and second, offering a modular ecosystem to algorithm designers who experiment with their own variants and implementations of algorithmic blocks. Finally, we demonstrate the performance of several FW variants on important problem classes in several experiments, which we curated in a separate repository for continuous benchmarking.

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arXiv2025

Scalable DC Optimization via Adaptive Frank-Wolfe Algorithms

We consider the problem of minimizing a difference of (smooth) convex functions over a compact convex feasible region $P$, i.e., $\min_{x \in P} f(x) - g(x)$, with smooth $f$ and Lipschitz continuous $g$. This computational study builds upon and complements the framework of Maskan et al. [2025] by integrating advanced Frank-Wolfe variants to reduce computational overhead. We empirically show that constrained DC problems can be efficiently solved using a combination of the Blended Pairwise Conditional Gradients (BPCG) algorithm [Tsuji et al., 2022] with warm-starting and the adaptive error bound from Maskan et al. [2025]. The result is a highly efficient and scalable projection-free algorithm for constrained DC optimization.

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arXiv2025

Conditional Gradient Methods

The purpose of this survey is to serve both as a gentle introduction and a coherent overview of state-of-the-art Frank--Wolfe algorithms, also called conditional gradient algorithms, for function minimization. These algorithms are especially useful in convex optimization when linear optimization is cheaper than projections. The selection of the material has been guided by the principle of highlighting crucial ideas as well as presenting new approaches that we believe might become important in the future, with ample citations even of old works imperative in the development of newer methods. Yet, our selection is sometimes biased, and need not reflect consensus of the research community, and we have certainly missed recent important contributions. After all the research area of Frank--Wolfe is very active, making it a moving target. We apologize sincerely in advance for any such distortions and we fully acknowledge: We stand on the shoulder of giants.

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arXiv2025

A Unified Toolbox for Multipartite Entanglement Certification

We present a unified framework for multipartite entanglement characterization based on the conditional gradient (CG) method, incorporating both fast heuristic detection and rigorous witness construction with numerical error control. Our method enables entanglement certification in quantum systems of up to ten qubits and applies to arbitrary entanglement structures. We demonstrate its power by closing the gap between entanglement and separability bounds in white noise robustness benchmarks for a class of bound entangled states. Furthermore, the framework extends to entanglement robustness under general quantum noise channels, providing accurate thresholds in cases beyond the reach of previous algorithmic methods. These results position CG methods as a powerful tool for practical and scalable entanglement analysis in realistic experimental settings.

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arXiv2025

Second-order Conditional Gradient Sliding

Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. We present the \emph{Second-Order Conditional Gradient Sliding} (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly. When the feasible region is a polytope the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires $\mathcal{O}(\log(\log 1/\varepsilon))$ first-order and Hessian oracle calls and $\mathcal{O}(\log (1/\varepsilon) \log(\log1/\varepsilon))$ linear minimization oracle calls to achieve an $\varepsilon$-optimal solution. This algorithm is useful when the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information of the function, although possible, is costly.

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arXiv2025

Secant Line Search for Frank-Wolfe Algorithms

We present a new step-size strategy based on the secant method for Frank-Wolfe algorithms. This strategy, which requires mild assumptions about the function under consideration, can be applied to any Frank-Wolfe algorithm. It is as effective as full line search and, in particular, allows for adapting to the local smoothness of the function, such as in Pedregosa et al 2018, but comes with a significantly reduced computational cost, leading to higher effective rates of convergence. We provide theoretical guarantees and demonstrate the effectiveness of the strategy through numerical experiments.

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arXiv2025

Approximating Latent Manifolds in Neural Networks via Vanishing Ideals

Deep neural networks have reshaped modern machine learning by learning powerful latent representations that often align with the manifold hypothesis: high-dimensional data lie on lower-dimensional manifolds. In this paper, we establish a connection between manifold learning and computational algebra by demonstrating how vanishing ideals can characterize the latent manifolds of deep networks. To that end, we propose a new neural architecture that (i) truncates a pretrained network at an intermediate layer, (ii) approximates each class manifold via polynomial generators of the vanishing ideal, and (iii) transforms the resulting latent space into linearly separable features through a single polynomial layer. The resulting models have significantly fewer layers than their pretrained baselines, while maintaining comparable accuracy, achieving higher throughput, and utilizing fewer parameters. Furthermore, drawing on spectral complexity analysis, we derive sharper theoretical guarantees for generalization, showing that our approach can in principle offer tighter bounds than standard deep networks. Numerical experiments confirm the effectiveness and efficiency of the proposed approach.

cs.LG↗

arXiv2025

Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem. Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

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arXiv2025

Fast convergence of Frank-Wolfe algorithms on polytopes

We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound.

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arXiv2025

Adaptive Open-Loop Step-Sizes for Accelerated Convergence Rates of the Frank-Wolfe Algorithm

Recent work has shown that in certain settings, the Frank-Wolfe algorithm (FW) with open-loop step-sizes $η_t = \frac{\ell}{t+\ell}$ for a fixed parameter $\ell \in \mathbb{N},\, \ell \geq 2$, attains a convergence rate faster than the traditional $O(t^{-1})$ rate. In particular, when a strong growth property holds, the convergence rate attainable with open-loop step-sizes $η_t = \frac{\ell}{t+\ell}$ is $O(t^{-\ell})$. In this setting there is no single value of the parameter $\ell$ that prevails as superior. This paper shows that FW with log-adaptive open-loop step-sizes $η_t = \frac{2+\log(t+1)}{t+2+\log(t+1)}$ attains a convergence rate that is at least as fast as that attainable with fixed-parameter open-loop step-sizes $η_t = \frac{\ell}{t+\ell}$ for any value of $\ell \in \mathbb{N},\,\ell\geq 2$. To establish our main convergence results, we extend our previous affine-invariant accelerated convergence results for FW to more general open-loop step-sizes of the form $η_t = g(t)/(t+g(t))$, where $g:\mathbb{N}\to\mathbb{R}_{\geq 0}$ is any non-decreasing function such that the sequence of step-sizes $(η_t)$ is non-increasing. This covers in particular the fixed-parameter case by choosing $g(t) = \ell$ and the log-adaptive case by choosing $g(t) = 2+ \log(t+1)$. To facilitate adoption of log-adaptive open-loop step-sizes, we have incorporated this rule into the {\tt FrankWolfe.jl} software package.

math.OC↗

arXiv2025

Mixed-Integer Optimization for Loopless Flux Distributions in Metabolic Networks

Constraint-based metabolic models can be used to investigate the intracellular physiology of microorganisms. These models couple genes to reactions, and typically seek to predict metabolite fluxes that optimize some biologically important metric. Classical techniques, like Flux Balance Analysis (FBA), formulate the metabolism of a microbe as an optimization problem where growth rate is maximized. While FBA has found widespread use, it often leads to thermodynamically infeasible solutions that contain internal cycles (loops). To address this shortcoming, Loopless-Flux Balance Analysis (ll-FBA) seeks to predict flux distributions that do not contain these loops. ll-FBA is a disjunctive program, usually reformulated as a mixed-integer program, and is challenging to solve for biological models that often contain thousands of reactions and metabolites. In this paper, we compare various reformulations of ll-FBA and different solution approaches. Overall, the combinatorial Benders' decomposition is the most promising of the tested approaches with which we could solve most instances. However, the model size and numerical instability pose a challenge to the combinatorial Benders' method.

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arXiv2025

Has the Creativity of Large-Language Models peaked? An analysis of inter- and intra-LLM variability

Following the widespread adoption of ChatGPT in early 2023, numerous studies reported that large language models (LLMs) can match or even surpass human performance in creative tasks. However, it remains unclear whether LLMs have become more creative over time, and how consistent their creative output is. In this study, we evaluated 14 widely used LLMs -- including GPT-4, Claude, Llama, Grok, Mistral, and DeepSeek -- across two validated creativity assessments: the Divergent Association Task (DAT) and the Alternative Uses Task (AUT). Contrary to expectations, we found no evidence of increased creative performance over the past 18-24 months, with GPT-4 performing worse than in previous studies. For the more widely used AUT, all models performed on average better than the average human, with GPT-4o and o3-mini performing best. However, only 0.28% of LLM-generated responses reached the top 10% of human creativity benchmarks. Beyond inter-model differences, we document substantial intra-model variability: the same LLM, given the same prompt, can produce outputs ranging from below-average to original. This variability has important implications for both creativity research and practical applications. Ignoring such variability risks misjudging the creative potential of LLMs, either inflating or underestimating their capabilities. The choice of prompts affected LLMs differently. Our findings underscore the need for more nuanced evaluation frameworks and highlight the importance of model selection, prompt design, and repeated assessment when using Generative AI (GenAI) tools in creative contexts.

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