arXiv subjects

João Sousa-Pinto

Publications and source records attributed to João Sousa-Pinto.

8 recordsLinked to original sources

arXiv2026

Rake-Compress Riccati Recursions for Parallel Scenario-Tree Model Predictive Control

Scenario-tree model predictive control (MPC) represents future information by a rooted tree and optimizes a nonanticipative policy over that tree. Numerical methods for solving the resulting nonlinear program typically compute their search directions through a sequence of branched linear-quadratic regulator (LQR) subproblems. The standard tree Riccati recursion requires linear work but has a dependency chain proportional to tree height. We present an algebraically exact parallel solver based on rake-compress tree contraction. After independent local control condensation, its two operations act on node and edge data that represent conditional quadratic functions. A rake eliminates a leaf and its parent edge, adding their reduced contribution to the parent-node data. A compress eliminates a unary node and replaces its two adjacent edges by one edge, using the same conditional-value composition as parallel Riccati methods on a chain. Together they contract an arbitrary rooted tree to its root; reversing the contraction recovers every Riccati coefficient, state, control, and multiplier. Given a reusable topology plan, a solve with $N$ nodes and fixed state and control dimensions has $O(N)$ arithmetic work and storage and $O(\log N)$ span, independently of tree height, balance, and maximum out-degree. The formulation allows positive-semidefinite dual regularization, including the unregularized case, and an exact linear-size lifting covers the standard scenario-MPC convention of one control per information node. We prove the contraction identities and equivalence to the Karush-Kuhn-Tucker (KKT) system. Three MIT-licensed JAX packages implement the bidirectional contraction, the dual-regularized LQR solver, and a user-facing primal-dual interior-point solver for tree-structured optimal control.

math.OC↗

arXiv2026

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.

math.OC↗

arXiv2026

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.

math.OC↗

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.

cs.RO↗

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).

math.OC↗

arXiv2019

On the Decidability of Reachability in Linear Time-Invariant Systems

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which control sets are affine subspaces of $\mathbb{R}^n$ is a fundamental result in control theory. Our first result is that reachability is undecidable if the set of controls is a finite union of affine subspaces. We also consider versions of the reachability problem in which (i)~the set of controls consists of a single affine subspace together with the origin and (ii)~the set of controls is a convex polytope. In these two cases we respectively show that the reachability problem is as hard as Skolem's Problem and the Positivity Problem for linear recurrence sequences (whose decidability has been open for several decades). Our main contribution is to show decidability of a version of the reachability problem in which control sets are convex polytopes, under certain spectral assumptions on the transition matrix.

math.OC↗

arXiv2017

On the Polytope Escape Problem for Continuous Linear Dynamical Systems

The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$, whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the trajectory of the unique solution to the differential equation $\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$, $\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation.

cs.CC↗

arXiv2016

Solvability of Matrix-Exponential Equations

We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals $t_{1}, \ldots, t_{k}$ such that \begin{align*} \prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) = C . \end{align*} We show that this problem is undecidable in general, but decidable under the assumption that the matrices $A_{1}, \ldots, A_{k}$ commute. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.

cs.DM↗