arXiv subjects

Dominique Orban

Publications and source records attributed to Dominique Orban.

At least 19 recordsLinked to original sources

arXiv2026

Quasi-Newton and Krylov Methods for the Solution of Nonconvex Trust-Region Subproblems

We study the solution of symmetric positive-definite linear systems by way of families of full- and limited-memory methods. Our contributions are threefold. We first derive new relationships between the conjugate-gradient method (CG) and quasi-Newton methods of the Broyden class that refine existing results, and clarify when those methods generate the same iterates and enjoy quadratic termination. We extend this perspective to the limited-memory BFGS (LBFGS) method. Next, we examine how DIOM, a limited-memory variant of the full orthogonalization Krylov method (FOM), is akin to LBFGS in that it provides a memory lever that is critical in practical performance. Finally, we generalize LBFGS and DIOM to the computation of trust-region steps for unconstrained, potentially nonconvex, optimization. We report numerical experience on positive-definite linear systems and unconstrained optimization problems. The results show that memory is a key algorithmic lever: LBFGS and DIOM are consistently more robust than CG and often achieve comparable accuracy with fewer Hessian-vector products. They emerge as viable alternatives to CG when high accuracy is desirable or when operations with the Hessian are at a premium. The limited-memory SR1 (LSR1) method can be competitive in full-memory form, but its limited-memory variant suffers from discarded curvature information.

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arXiv2026

A Spectral Preconditioner for the Conjugate Gradient Method with Iteration Budget

We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral preconditioners map a subset of eigenvalues to a positive cluster via a scaling parameter, and leave the remainder of the spectrum unchanged, in hopes to reduce the number of iterations to convergence. We formulate the design of the spectral preconditioners as a constrained optimization problem. The optimal cluster placement is defined to minimize the error in energy norm at a fixed iteration. This optimality criterion provides new insight into the design of efficient spectral preconditioners when PCG is stopped short of convergence. We propose practical strategies for selecting the scaling parameter, hence the cluster position, that incur negligible computational cost. Numerical experiments highlight the importance of cluster placement and demonstrate significant improvements in terms of error in energy norm, particularly during the initial iterations.

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arXiv2025

An Inexact Modified Quasi-Newton Method for Nonsmooth Regularized Optimization

We introduce iR2N, a modified proximal quasi-Newton method for minimizing the sum of a smooth function $f$ and a lower semi-continuous prox-bounded function $h$, allowing inexact evaluations of $f$, its gradient, and the associated proximal operators. Both $f$ and $h$ may be nonconvex. iR2N is particularly suited to settings where proximal operators are computed via iterative procedures that can be stopped early, or where the accuracy of $f$ and $\nabla f$ can be controlled, leading to significant computational savings. At each iteration, the method approximately minimizes the sum of a quadratic model of $f$, a model of $h$, and an adaptive quadratic regularization term ensuring global convergence. Under standard accuracy assumptions, we prove global convergence in the sense that a first-order stationarity measure converges to zero, with worst-case evaluation complexity $O(\epsilon^{-2})$. Numerical experiments with $\ell_p$ norms, $\ell_p$ total variation, and the indicator of the nonconvex pseudo $p$-norm ball illustrate the effectiveness and flexibility of the approach, and show how controlled inexactness can substantially reduce computational effort.

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arXiv2025

MadNCL: A GPU Implementation of Algorithm NCL for Large-Scale, Degenerate Nonlinear Programs

We present a GPU implementation of Algorithm NCL, an augmented Lagrangian method for solving large-scale and degenerate nonlinear programs. Although interior-point methods and sequential quadratic programming are widely used for solving nonlinear programs, the augmented Lagrangian method is known to offer superior robustness against constraint degeneracies and can rapidly detect infeasibility. We introduce several enhancements to Algorithm NCL, including fusion of the inner and outer loops and use of extrapolation steps, which improve both efficiency and convergence stability. Further, NCL has the key advantage of being well-suited for GPU architectures because of the regularity of the KKT systems provided by quadratic penalty terms. In particular, the NCL subproblem formulation allows the KKT systems to be naturally expressed as either stabilized or condensed KKT systems, whereas the interior-point approach requires aggressive reformulations or relaxations to make it suitable for GPUs. Both systems can be efficiently solved on GPUs using sparse \ldlt factorization with static pivoting, as implemented in NVIDIA cuDSS. Building on these advantages, we examine the KKT systems arising from NCL subproblems. We present an optimized GPU implementation of Algorithm NCL by leveraging MadNLP as an interior-point subproblem solver and utilizing the stabilized and condensed formulations of the KKT systems for computing Newton steps. Numerical experiments on various large-scale and degenerate NLPs, including optimal power flow, COPS benchmarks, and security-constrained optimal power flow, demonstrate that MadNCL operates efficiently on GPUs while effectively managing problem degeneracy, including MPCC constraints.

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arXiv2025

Dual-Regularized Riccati Recursions for Interior-Point Optimal Control

We derive closed-form extensions of the sequential and parallel Riccati recursions for solving dual-regularized linear-quadratic regulator (LQR) problems, with $O(N)$ sequential time and $O(\log(N))$ parallel time, respectively. We show that these subproblems arise when using regularized primal-dual interior-point methods to solve smooth, constrained, non-convex, discrete-time optimal control problems via multiple-shooting, even in the presence of stagewise equality or inequality constraints, and without imposing any rank requirements on constraint Jacobians. We prove that, when certain inertia conditions on the Newton-KKT matrix are met, each nonzero primal step is a descent direction of an augmented barrier-Lagrangian merit function. We characterize these inertia conditions in terms of the positive-definiteness of the dual-regularized Riccati pivots (a weaker condition than the standard LQR positive-definiteness requirements), thereby yielding inexpensive certificates of the required inertia. We provide MIT-licensed implementations of our methods in C++ and in JAX, as well as a full formalization of our results in Lean. We benchmark our algorithm against leading optimal control and nonlinear programming solvers on complex trajectory optimization problems, establishing competitive performance on moderate problems and substantial gains as the horizon length, problem dimension, and constraint count increase.

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arXiv2025

Primal-Dual iLQR for GPU-Accelerated Learning and Control in Legged Robots

This paper introduces a novel Model Predictive Control (MPC) implementation for legged robot locomotion that leverages GPU parallelization. Our approach enables both temporal and state-space parallelization by incorporating a parallel associative scan to solve the primal-dual Karush-Kuhn-Tucker (KKT) system. In this way, the optimal control problem is solved in $\mathcal{O}(\log^2(n)\log{N} + \log^2(m))$ complexity, instead of $\mathcal{O}(N(n + m)^3)$, where $n$, $m$, and $N$ are the dimension of the system state, control vector, and the length of the prediction horizon. We demonstrate the advantages of this implementation over two state-of-the-art solvers (acados and crocoddyl), achieving up to a 60\% improvement in runtime for Whole Body Dynamics (WB)-MPC and a 700\% improvement for Single Rigid Body Dynamics (SRBD)-MPC when varying the prediction horizon length. The presented formulation scales efficiently with the problem state dimensions as well, enabling the definition of a centralized controller for up to 16 legged robots that can be computed in less than 25 ms. Furthermore, thanks to the JAX implementation, the solver supports large-scale parallelization across multiple environments, allowing the possibility of performing learning with the MPC in the loop directly in GPU. The code associated with this work can be found at https://github.com/iit-DLSLab/mpx.

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arXiv2024

Nonsmooth exact penalty methods for equality-constrained optimization: complexity and implementation

Penalty methods are a well known class of algorithms for constrained optimization. They transform a constrained problem into a sequence of unconstrained \emph{penalized} problems in the hope that approximate solutions of the latter converge to a solution of the former. If Lagrange multipliers exist, exact penalty methods ensure that the penalty parameter only need increase a finite number of times, but are typically scorned in smooth optimization for the penalized problems are not smooth. This led researchers to consider the implementation of exact penalty methods inconvenient. Recent advances in proximal methods have led to increasingly efficient solvers for nonsmooth optimization. We study a general exact penalty algorithm and use it to show that the exact $\ell_2$-penalty method for equality-constrained optimization can, in fact, be implemented efficiently by solving the penalized problem using a proximal-type algorithm. We study the convergence of our algorithm and establish a worst-case complexity bound of $\mathcal{O}(\epsilon^{-2})$ to bring a stationarity measure below $\epsilon > 0$ under the Mangarasian-Fromowitz constraint qualification and Lipschitz continuity of the objective gradient and constraint Jacobian. While the Lipschitz continuity of the objective gradient is not required for convergence in view of recent works, it is used in our analysis to derive the complexity bound. In a degenerate scenario where the penalty parameter grows unbounded, the complexity becomes $\mathcal{O}(\epsilon^{-8})$, which is worse than another bound found in the literature. Finally, we report numerical experience on small-scale problems from a standard collection and compare our solver with an augmented-Lagrangian and an SQP method. Our preliminary implementation is superior to the augmented Lagrangian in terms of robustness and efficiency, and is competitive with the SQP method.

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arXiv2024

An Efficient Scaled spectral preconditioner for sequences of symmetric positive definite linear systems

We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but also with consideration of its use within the conjugate gradient (CG) method. We propose three different strategies for selecting a scaling parameter, which aims to position the eigenvalues of the preconditioned matrix in a way that reduces the energy norm of the error, the quantity that CG monotonically decreases at each iteration. Our focus is on accelerating convergence especially in the early iterations, which is particularly important when CG is truncated due to computational cost constraints. Numerical experiments provide in data assimilation confirm that the scaled spectral preconditioner can significantly improve early CG convergence with negligible computational cost.

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arXiv2024

A Proximal Modified Quasi-Newton Method for Nonsmooth Regularized Optimization

We develop R2N, a modified quasi-Newton method for minimizing the sum of a $\mathcal{C}^1$ function $f$ and a lower semi-continuous prox-bounded $h$. Both $f$ and $h$ may be nonconvex. At each iteration, our method computes a step by minimizing the sum of a quadratic model of $f$, a model of $h$, and an adaptive quadratic regularization term. A step may be computed by a variant of the proximal-gradient method. An advantage of R2N over trust-region (TR) methods is that proximal operators do not involve an extra TR indicator. We also develop the variant R2DH, in which the model Hessian is diagonal, which allows us to compute a step without relying on a subproblem solver when $h$ is separable. R2DH can be used as standalone solver, but also as subproblem solver inside R2N. We describe non-monotone variants of both R2N and R2DH. Global convergence of a first-order stationarity measure to zero holds without relying on local Lipschitz continuity of $\nabla f$, while allowing model Hessians to grow unbounded, an assumption particularly relevant to quasi-Newton models. Under Lipschitz-continuity of $\nabla f$, we establish a tight worst-case complexity bound of $O(1 / \epsilon^{2/(1 - p)})$ to bring said measure below $\epsilon > 0$, where $0 \leq p < 1$ controls the growth of model Hessians. The latter must not diverge faster than $|\mathcal{S}_k|^p$, where $\mathcal{S}_k$ is the set of successful iterations up to iteration $k$. When $p = 1$, we establish the tight exponential complexity bound $O(\exp(c \epsilon^{-2}))$ where $c > 0$ is a constant. We describe our Julia implementation and report numerical experience on a classic basis-pursuit problem, an image denoising problem, a minimum-rank matrix completion problem, a nonlinear support vector machine and an inverse nonlinear problem.

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arXiv2024

Complexity of trust-region methods in the presence of unbounded Hessian approximations

We extend traditional complexity analyses of trust-region methods for unconstrained, possibly nonconvex, optimization. Whereas most complexity analyses assume uniform boundedness of the model Hessians, we work with potentially unbounded model Hessians. Boundedness is not guaranteed in practical implementations, in particular ones based on quasi-Newton updates such as PSB, BFGS and SR1. We examine two regimes of Hessian growth: one bounded by a power of the number of successful iterations, and one bounded by a power of the number of iterations. This allows us to formalize and address the intuition of Powell [IMA J. Numer. Ana. 30(1):289-301,2010], who studied convergence under a special case of our assumptions, but whose proof contained complexity arguments. Specifically, for $0 \leq p < 1$, we establish sharp $O([(1-p)\epsilon^{-2}]^{1/(1-p)})$ evaluation complexity to find an $\epsilon$-stationary point when model Hessians are $O(|\mathcal{S}_{k-1}|^p)$, where $|\mathcal{S}_{k-1}|$ is the number of iterations where the step was accepted, up to iteration $k-1$. For $p = 1$, which is the case studied by Powell, we establish a sharp $O(\exp(c_1\epsilon^{-2}))$ evaluation complexity for a certain constant $c_1 > 0$. This is far better than the double exponential bound that \citet{powell-2010} suspected, and is far worse than other bounds surmised elsewhere in the literature. We establish similar sharp bounds when model Hessians are $O(k^p)$, where $k$ is the iteration counter, for $0 \leq p < 1$. When $p = 1$, the complexity bound depends on the parameters of the family, but reduces to $O((1 - \log(\epsilon))\exp(c_2\epsilon^{-2}))$ for a certain constant $c_2 > 0$ for the special case of the standard trust-region method. As special cases, we derive novel complexity bounds for (strongly) convex objectives under the same growth assumptions.

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arXiv2024

An efficient algorithm for solving linear equality-constrained LQR problems

We consider finite-horizon linear-quadratic regulator (LQR) problems with arbitrary stagewise linear equality constraints. We present a two-part reduction to an unconstrained LQR problem. First, every stage is represented by an affine relation between its endpoint states. Composition of adjacent relations eliminates their shared state and is associative. An associative suffix scan therefore computes, for every node, an affine parameterization $x_i=T_i z_i+t_i$ of the states from which the remaining horizon is feasible. Second, once these parameterizations are known, every stage can be transformed independently. The original mixed constraints and the requirement that the successor lie in its feasible domain yield an affine control map $u_i=Y_i z_i+Z_i v_i+y_i$. Substitution produces an ordinary unconstrained LQR problem in the reduced state and control variables $(z_i,v_i)$, with smaller or equal dimensions, possibly varying by stage. We prove equivalence of the two problems and preservation of the standard convexity assumptions. The reduction has linear work and logarithmic depth in the horizon. Finally, we show that multipliers can be recovered by applying the same affine-relation contraction and expansion to the original KKT stationarity equations. We validate work-efficient sequential and logarithmic-depth parallel implementations against dense KKT solutions and original KKT residuals. An optimized C++/CUDA implementation exploits the varying reduced dimensions and provides a substantial end-to-end GPU speedup over sequential C++ in our experiments. We release the JAX and C++/CUDA packages under the MIT License.

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arXiv2024

Primal-Dual iLQR

We introduce a new algorithm for solving unconstrained discrete-time optimal control problems. Our method follows a direct multiple shooting approach, and consists of applying the SQP method together with an $\ell_2$ augmented Lagrangian primal-dual merit function. We use the LQR algorithm to efficiently solve the primal-dual Newton-KKT system. As our algorithm is a specialization of NPSQP, it inherits its generic properties, including global convergence, fast local convergence, and the lack of need for second order corrections or dimension expansions, improving on existing direct multiple shooting approaches such as acados, ALTRO, GNMS, FATROP, and FDDP. The solutions of the LQR-shaped subproblems posed by our algorithm can be be parallelized to run in time logarithmic in the number of stages, states, and controls. Moreover, as our method avoids sequential rollouts of the nonlinear dynamics, it can run in $O(1)$ parallel time per line search iteration. Therefore, this paper provides a practical, theoretically sound, and highly parallelizable (for example, with a GPU) method for solving nonlinear discrete-time optimal control problems. An open-source JAX implementation of this algorithm can be found on GitHub (joaospinto/primal_dual_ilqr).

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arXiv2024

An interior-point trust-region method for nonsmooth regularized bound-constrained optimization

We develop an interior-point method for nonsmooth regularized bound-constrained optimization problems. Our method consists of iteratively solving a sequence of unconstrained nonsmooth barrier subproblems. We use a variant of the proximal quasi-Newton trust-region algorithm TR of arXiv:2103.15993v3 to solve the barrier subproblems, with additional assumptions inspired from well-known smooth interior-point trust-region methods. We show global convergence of our algorithm with respect to the criticality measure of arXiv:2103.15993v3. Under an additional assumption linked to the convexity of the nonsmooth term in the objective, we present an alternative interior-point algorithm with a slightly modified criticality measure, which performs better in practice. Numerical experiments show that our algorithm performs better than the trust-region method TR, the trust-region method with diagonal hessian approximations TRDH of arXiv:2309.08433, and the quadratic regularization method R2 of arXiv:2103.15993v3 for two out of four tested bound-constrained problems. On those two problems, our algorithm obtains smaller objective values than the other solvers using fewer objective and gradient evaluations. On the two other problems, it performs similarly to TR, R2 and TRDH.

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arXiv2023

Complexity of trust-region methods with unbounded Hessian approximations for smooth and nonsmooth optimization

We develop a worst-case evaluation complexity bound for trust-region methods in the presence of unbounded Hessian approximations. We use the algorithm of arXiv:2103.15993v3 as a model, which is designed for nonsmooth regularized problems, but applies to unconstrained smooth problems as a special case. Our analysis assumes that the growth of the Hessian approximation is controlled by the number of successful iterations. We show that the best known complexity bound of $\epsilon^{-2}$ deteriorates to $\epsilon^{-2/(1-p)}$, where $0 \le p < 1$ is a parameter that controls the growth of the Hessian approximation. The faster the Hessian approximation grows, the more the bound deteriorates. We construct an objective that satisfies all of our assumptions and for which our complexity bound is attained, which establishes that our bound is sharp. To the best of our knowledge, our complexity result is the first to consider potentially unbounded Hessians and is a first step towards addressing a conjecture of Powell [38] that trust-region methods may require an exponential number of iterations in such a case. Numerical experiments conducted in double precision arithmetic are consistent with the analysis.

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arXiv2023

A Multi-Precision Quadratic Regularization Method for Unconstrained Optimization with Rounding Error Analysis

We propose a multi-precision extension of the Quadratic Regularization (R2) algorithm that enables it to take advantage of low-precision computations, and by extension to decrease energy consumption during the solve. The lower the precision in which computations occur, the larger the errors induced in the objective value and gradient, as well as in all other computations that occur in the course of the iterations. The Multi-Precision R2 (MPR2) algorithm monitors the accumulation of rounding errors with two aims: to provide guarantees on the result of all computations, and to permit evaluations of the objective and its gradient in the lowest precision possible while preserving convergence properties. MPR2's numerical results show that single precision offers enough accuracy for more than half of objective evaluations and most of gradient evaluations during the algorithm's execution. However, MPR2 fails to converge on several problems of the test set for which double precision does not offer enough precision to ensure the convergence conditions before reaching a first order critical point. That is why we propose a practical version of MPR2 with relaxed conditions which converges for almost as many problems as R2 and potentially enables to save about 50 % time and 60% energy for objective evaluation and 50 % time and 70% energy for gradient evaluation.

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arXiv2023

MinAres: An Iterative Solver for Symmetric Linear Systems

We introduce an iterative solver named MINARES for symmetric linear systems $Ax \approx b$, where $A$ is possibly singular. MINARES is based on the symmetric Lanczos process, like MINRES and MINRES-QLP, but it minimizes $\|Ar_k\|$ in each Krylov subspace rather than $\|r_k\|$, where $r_k$ is the current residual vector. When $A$ is symmetric, MINARES minimizes the same quantity $\|Ar_k\|$ as LSMR, but in more relevant Krylov subspaces, and it requires only one matrix-vector product $Av$ per iteration, whereas LSMR would need two. Our numerical experiments with MINRES-QLP and LSMR show that MINARES is a pertinent alternative on consistent symmetric systems and the most suitable Krylov method for inconsistent symmetric systems. We derive properties of MINARES from an equivalent solver named CAR that is to MINARES as CR is to MINRES, is not based on the Lanczos process, and minimizes $\|Ar_k\|$ in the same Krylov subspace as MINARES. We establish that MINARES and CAR generate monotonic $\|x_k - x_{\star}\|$, $\|x_k - x_{\star}\|_A$ and $\|r_k\|$ when $A$ is positive definite.

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arXiv2023

The Indefinite Proximal Gradient Method

We introduce a variant of the proximal gradient method in which the quadratic term is diagonal but may be indefinite, and is safeguarded by a trust region. Our method is a special case of the proximal quasi-Newton trust-region method of arXiv:2103.15993v3. We provide closed-form solution of the step computation in certain cases where the nonsmooth term is separable and the trust region is defined in infinity norm, so that no iterative subproblem solver is required. Our analysis expands upon that of arXiv:2103.15993v3 by generalizing the trust-region approach to problems with bound constraints. We provide an efficient open-source implementation of our method, named TRDH, in the Julia language in which Hessians approximations are given by diagonal quasi-Newton updates. TRDH evaluates one standard proximal operator and one indefinite proximal operator per iteration. We also analyze and implement a variant named iTRDH that performs a single indefinite proximal operator evaluation per iteration. We establish that iTRDH enjoys the same asymptotic worst-case iteration complexity as TRDH. We report numerical experience on unconstrained and bound-constrained problems, where TRDH and iTRDH are used both as standalone and subproblem solvers. Our results illustrate that, as standalone solvers, TRDH and iTRDH improve upon the quadratic regularization method R2 of arXiv:2103.15993v3 but also sometimes upon their quasi-Newton trust-region method, referred to here as TR-R2, in terms of smooth objective value and gradient evaluations. On challenging nonnegative matrix factorization, binary classification and data fitting problems, TRDH and iTRDH used as subproblem solvers inside TR improve upon TR-R2 for at least one choice of diagonal approximation.

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arXiv2023

A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares

We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squar es term $f(x) = \tfrac{1}{2} \|F(x)\|_2^2$ and a nonsmooth term $h$. Both $f$ and $h$ may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of $h$ using a first-order method such as the proximal gradient method. We establish global convergence to a first-order stationary point of both a trust-region and a regularization variant of the Levenberg-Marquardt method under the assumptions that $F$ and its Jacobian are Lipschitz continuous and $h$ is proper and lower semi-continuous. In the worst case, both methods perform $O(ε^{-2})$ iterations to bring a measure of stationarity below $ε\in (0, 1)$. We report numerical results on three examples: a group-lasso basis-pursuit denoise example, a nonlinear support vector machine, and parameter estimation in neuron firing. For those examples to be implementable, we describe in detail how to evaluate proximal operators for separable $h$ and for the group lasso with trust-region constraint. In all cases, the Levenberg-Marquardt methods perform fewer outer iterations than a proximal-gradient method with adaptive step length and a quasi-Newton trust-region method, neither of which exploit the least-squares structure of the problem. Our results also highlight the need for more sophisticated subproblem solvers than simple first-order methods.

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