arXiv subjects

Panagiotis Patrinos

Publications and source records attributed to Panagiotis Patrinos.

At least 19 recordsLinked to original sources

arXiv2026

Scaled relative graphs for pairs of operators beyond classical monotonicity

We introduce a generalization of the scaled relative graph (SRG) to pairs of operators, enabling the visualization of their relative incremental properties. This novel SRG framework provides the geometric counterpart for the study of nonlinear resolvents based on paired monotonicity conditions. We demonstrate that these conditions apply to linear operators composed with monotone mappings, a class that notably includes NPN transistors, allowing us to compute the response of multivalued, nonsmooth, and highly nonmonotone electrical circuits.

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arXiv2026

Active learning-based Bayesian optimization in the realm of copper slag-blended cement systems

Accelerated mix design optimization is critical for deploying low-carbon supplementary cementitious materials (SCMs) because traditional experimental approaches require extensive testing campaigns. The authors demonstrate that Bayesian optimization (BO) can identify near-optimal blended cement formulations using an AI-driven approach with minimal experimental data. Starting with only 10 initial experiments and a 2:1 data-to-variable ratio reflecting realistic laboratory constraints, the authors optimized copper slag, limestone, Portland cement systems for 32.5N strength class with respect to 2-day compressive strength, cost, and CO2 emissions. Gaussian process surrogate models guided sequential experimentation through Expected Improvement and Upper Confidence Bound acquisition functions. Within 2-6 iterations, BO identified Pareto non-dominated solutions meeting strength requirements (greater than 10 MPa at 2 days, greater than 32.5 MPa at 28 days) while achieving CO2 emissions below 500 kg CO2/ton; consistent with industry decarbonization targets. A conservative update strategy incorporating uncertainty bounds for 28-day strength enabled rapid iteration without waiting 28 days per cycle. Comparative analysis revealed that linear kernels outperformed nonlinear alternatives in predictive accuracy, though radial basis function kernels were preferred for active learning due to superior uncertainty quantification. This work demonstrates BO as a practical decision-support tool for cement research under severe data constraints.

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arXiv2026

A localized consensus-based sampling algorithm

We propose a localized consensus-based method for sampling from non-Gaussian distributions, a task that frequently arises when solving Bayesian inverse problems. Our method arises from an alternative derivation of consensus-based sampling (CBS). Starting from ensemble-preconditioned Langevin dynamics, we replace the potential by its Moreau envelope -- a smoother approximation -- in order to replace the gradient in the Langevin equation with a proximal operator. We then approximate this operator by a weighted mean. In the limit of infinitely smoothing the potential to a quadratic function, this procedure recovers the standard CBS dynamics. In addition, outside this limit, we retrieve a refined variant of polarized CBS. We call the resulting algorithm localized consensus-based sampling, since particles interact more with nearby particles than with faraway ones. Our method is affine-invariant, exact for Gaussian targets in the mean-field limit, and demonstrates improved robustness over polarized CBS in numerical experiments. Like other consensus-based methods, localized CBS is gradient-free and easily parallelizable.

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arXiv2026

A Lasry-Lions envelope approach for mathematical programs with complementarity constraints

We propose a homotopy method for solving mathematical programs with complementarity constraints (CCs). The indicator function of the CCs is relaxed by the Lasry--Lions double envelope, an extension of the Moreau envelope that enjoys an additional smoothness property, making it amenable to fast optimization algorithms. The proposed algorithm mimics the behavior of homotopy methods for systems of nonlinear equations or penalty methods for constrained optimization: it solves a sequence of smooth subproblems that progressively approximate the original problem, using the solution of each subproblem as the starting point for the next one. In the limiting setting, we establish the convergence to Mordukhovich and Clarke stationary points. We also provide a worst-case complexity analysis for computing an approximate stationary point. Preliminary numerical results on a suite of benchmark problems demonstrate the effectiveness of the proposed approach.

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arXiv2026

Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives

In this work, we develop proximal preconditioned gradient methods with a focus on spectral gradient methods providing a proximal extension to the Muon and Scion optimizers. We introduce a family of stochastic algorithms that can handle a wide variety of convex and nonconvex constraints and study its convergence under heavy-tailed noise, through a novel analysis tailored to the geometry of the proposed methods. We further propose a variance-reduced version, which achieves faster convergence under standard noise assumptions. Finally, we show that the polynomial iterations used in Muon are more accurately captured by a nonlinear preconditioner than by the ideal matrix sign, leading to a convergence analysis that more faithfully reflects practical implementations.

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arXiv2026

Newton methods beyond Hessian Lipschitz continuity: A nonlinear preconditioning approach

Newton-type methods are typically analyzed under Lipschitz continuity of the Hessian, an assumption that can fail for objectives with higher-order or polynomial growth. We introduce a class of nonlinearly preconditioned Newton methods that apply Newton's root-finding scheme to a transformed optimality mapping, thereby extending recent nonlinear preconditioning ideas from first-order methods to the second-order setting. The resulting methods are naturally analyzed under Lipschitz continuity of a preconditioned Hessian, a condition that significantly relaxes the classical Hessian Lipschitz continuity assumption. Under this generalized smoothness model, we establish local superlinear and quadratic convergence guarantees, and develop a globalization strategy for the nonregularized method despite the fact that the preconditioned Newton direction need not be a descent direction. We further propose a regularized variant for isotropic preconditioners, and show that it attains an $O(\varepsilon^{-3/2})$ iteration complexity. An adaptive version removes the need to know the smoothness constant and allows inexact subproblem solutions while preserving the same complexity order.

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arXiv2026

Nonlinearly preconditioned gradient flows

We study a continuous-time dynamical system which arises as the limit of a broad class of nonlinearly preconditioned gradient methods. Under mild assumptions, we establish existence of global solutions and derive Lyapunov-based convergence guarantees. For convex costs, we prove a sublinear decay in a geometry induced by some reference function, and under a generalized gradient-dominance condition we obtain exponential convergence. We further uncover a duality connection with mirror descent, and use it to establish that the flow of interest solves an infinite-horizon optimal-control problem of which the value function is the Bregman divergence generated by the cost. These results clarify the structure and optimization behavior of nonlinearly preconditioned gradient flows and connect them to known continuous-time models in non-Euclidean optimization.

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arXiv2026

PANOC-lite: A simpler and more efficient algorithm for composite minimization

This work introduces a simple and efficient linesearch method for composite minimization that accelerates proximal-gradient iterations with fast Newton-type directions. Our algorithm is based on simple operations and only requires the standard proximal-gradient oracle, similar to PANOC and ZeroFPR, provided that the nonsmooth term is convex. Noteworthy improvements include a cheaper backtracking procedure, in the sense that no additional gradients need to be evaluated, and an enlarged range of permitted stepsizes. Global subsequential convergence and local superlinear convergence are established under conventional assumptions by considering a novel merit function which is less expensive to evaluate than alternatives like the forward-backward envelope. Finally, the proposed approach is validated on model predictive control problems with collision avoidance constraints, as well as on the LIBSVM and CUTEst benchmarks.

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arXiv2026

Parametric Nonconvex Optimization via Convex Surrogates

This paper presents a novel learning-based approach to construct a surrogate problem that approximates a given parametric nonconvex optimization problem. The surrogate function is designed to be the minimum of a finite set of functions, given by the composition of convex and monotonic terms, so that the surrogate problem can be solved directly through parallel convex optimization. As a proof of concept, numerical experiments on a nonconvex path tracking problem confirm the approximation quality of the proposed method.

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arXiv2026

On the Regularity of Generalized Conjugate Functions

We investigate regularity properties of generalized conjugate functions induced by a general coupling function and the associated generalized proximal mapping. Our main results provide verifiable conditions ensuring local single-valuedness, continuity, Lipschitz continuity, and differentiability of the generalized proximal mapping, and transfer these properties to generalized conjugates providing explicit derivative formulas. These results are based on a nonsmooth implicit function theorem for generalized equations, relying on graphical localizations and second-order variational tools. Beyond first-order regularity, we also derive conditions under which generalized conjugates are strictly twice differentiable.

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arXiv2026

Cyqlone: A Parallel, High-Performance Linear Solver for Optimal Control

We present Cyqlone, a solver for linear systems with a stage-wise optimal control structure that fully exploits the various levels of parallelism available in modern hardware. Cyqlone unifies algorithms based on the sequential Riccati recursion, parallel Schur complement methods, and cyclic reduction methods, thereby minimizing the required number of floating-point operations, while allowing parallelization across a configurable number of processors. Given sufficient parallelism, the solver run time scales with the logarithm of the horizon length (in contrast to the linear scaling of sequential Riccati-based methods), enabling real-time solution of long-horizon problems. Beyond multithreading on multi-core processors, implementations of Cyqlone can also leverage vectorization using batched linear algebra routines. Such batched routines exploit data parallelism using single instruction, multiple data (SIMD) operations, and expose a higher degree of instruction-level parallelism than their non-batched counterparts. This enables them to significantly outperform BLAS and BLASFEO for the small matrices that arise in optimal control. Building on this high-performance linear solver, we develop CyQPALM, a parallel and optimal-control-specific variant of the QPALM quadratic programming solver. It combines the parallel and vectorized linear algebra operations from Cyqlone with a parallel line search and parallel factorization updates, resulting in order-of-magnitude speedups over the state-of-the-art HPIPM solver. Open-source C++ implementations of Cyqlone and CyQPALM are available at https://github.com/kul-optec/cyqlone

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arXiv2026

Exploiting Parallelism in a QPALM-based Solver for Optimal Control

We discuss the opportunities for parallelization in the recently proposed QPALM-OCP algorithm, a solver tailored to quadratic programs arising in optimal control. A significant part of the computational work can be carried out independently for the different stages in the optimal control problem. We exploit this specific structure to apply parallelization and vectorization techniques in an optimized C++ implementation of the method. Results for optimal control benchmark problems and comparisons to the original QPALM method are provided.

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arXiv2026

Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions

We consider gradient descent with constant stepsizes and derive exact worst-case convergence rates on the minimum gradient norm of the iterates. Our analysis covers all possible stepsizes and arbitrary upper/lower bounds on the curvature of the objective function, thus including convex, strongly convex and weakly convex (hypoconvex) objective functions. Among the challenging parts of the analysis, we note the necessity to exploit dependencies between non-consecutive iterates. While this complicates the proofs to some extent, it enables us to achieve an exact full-range analysis of gradient descent for any constant stepsize (covering, in particular, normalized stepsizes greater than one), whereas the literature contained only conjectured rates of this type. In the nonconvex case, allowing arbitrary bounds on upper and lower curvatures extends existing partial results that are valid only for gradient Lipschitz functions (i.e., where lower and upper bounds on curvature are equal), leading to improved rates for weakly convex functions. From our exact worst-case performance bounds, we deduce the optimal constant stepsize for gradient descent. Leveraging our analysis, we also introduce a new variant of gradient descent based on a unique, fixed sequence of variable stepsizes, demonstrating its superiority in the worst-case over any constant stepsize schedule.

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arXiv2026

Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems

We examine stability properties of primal-dual gradient flow dynamics for composite convex optimization problems with multiple, possibly nonsmooth, terms in the objective function under the generalized consensus constraint. The proposed dynamics are based on the proximal augmented Lagrangian and they provide a viable alternative to ADMM which faces significant challenges from both analysis and implementation viewpoints in large-scale multi-block scenarios. In contrast to customized algorithms with individualized convergence guarantees, we develop a systematic approach for solving a broad class of challenging composite optimization problems. We leverage various structural properties to establish global (exponential) convergence guarantees for the proposed dynamics. Our assumptions are much weaker than those required to prove (exponential) stability of primal-dual dynamics as well as (linear) convergence of discrete-time methods such as standard two-block and multi-block ADMM and EXTRA algorithms. Finally, we show necessity of some of our structural assumptions for exponential stability and provide computational experiments to demonstrate the convenience of the proposed approach for parallel and distributed computing applications.

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arXiv2025

Anisotropic Proximal Point Algorithm

In this paper we study a nonlinear dual space preconditioning approach for the relaxed Proximal Point Algorithm (PPA) with application to monotone and relatively cohypomonotone inclusions, called anisotropic PPA. The algorithm is an instance of Luque's nonlinear PPA wherein the nonlinear preconditioner is chosen as the gradient of a Legendre convex function. Since the preconditioned operator is nonmonotone in general, convergence cannot be shown using standard arguments, unless the preconditioner exhibits isotropy (preserves directions) as in existing literature. To address the broader applicability we show convergence along subsequences invoking a Bregman version of Fejér-monotonicity in the dual space. Via a nonlinear generalization of Moreau's decomposition for operators, we provide a dual interpretation of the algorithm in terms of a forward iteration applied to a $D$-firmly nonexpansive mapping which involves the Bregman resolvent. For a suitable preconditioner, convergence rates of arbitrary order are derived under a mild Hölder growth condition. Finally, we discuss an anisotropic generalization of the proximal augmented Lagrangian method obtained via the proposed scheme. This aligns with Rockafellar's generalized and sharp Lagrangian functions.

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arXiv2025

EM++: A parameter learning framework for stochastic switching systems

This paper proposes a general switching dynamical system model, and a custom majorization-minimization-based algorithm EM++ for identifying its parameters. For certain families of distributions, such as Gaussian distributions, this algorithm reduces to the well-known expectation-maximization method. We prove global convergence of the algorithm under suitable assumptions, thus addressing an important open issue in the switching system identification literature. The effectiveness of both the proposed model and algorithm is validated through extensive numerical experiments.

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arXiv2025

Probabilistic Safety under Arbitrary Disturbance Distributions using Piecewise-Affine Control Barrier Functions

We propose a simple safety filter design for stochastic discrete-time systems based on piecewise affine probabilistic control barrier functions, providing an appealing balance between modeling flexibility and computational complexity. Exact evaluation of the safety filter consists of solving a mixed-integer quadratic program (MIQP) if the dynamics are control-affine (or a mixed-integer nonlinear program in general). We propose a heuristic search method that replaces this by a small number of small-scale quadratic programs (QPs), or nonlinear programs (NLPs) respectively. The proposed approach provides a flexible framework in which arbitrary (data-driven) quantile estimators can be used to bound the probability of safety violations. Through extensive numerical experiments, we demonstrate improvements in conservatism and computation time with respect to existing methods, and we illustrate the flexibility of the method for modeling complex safety sets. Supplementary material can be found at https://mathijssch.github.io/ecc26-supplementary/.

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arXiv2025

The inexact power augmented Lagrangian method for constrained nonconvex optimization

This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems that involve nonlinear equality constraints. In a first part of this work, we conduct a full complexity analysis of the method under a mild regularity condition, leveraging an accelerated first-order algorithm for solving the Hölder-smooth subproblems. Interestingly, this worst-case result indicates that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease of the dual residual. Notably, our analysis does not assume boundedness of the iterates. Thereafter, we present an inexact proximal point method for solving the weakly-convex and Hölder-smooth subproblems, and demonstrate that the combined scheme attains an improved rate that reduces to the best-known convergence rate whenever the augmenting term is a classical squared Euclidean norm. Different augmenting terms, involving a lower power, further improve the primal complexity at the cost of the dual complexity. Finally, numerical experiments validate the practical performance of unconventional augmenting terms.

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