arXiv subjects

Anton Pozharskiy

Publications and source records attributed to Anton Pozharskiy.

11 recordsLinked to original sources

arXiv2026

CCOpt: an Open-Source Solver for Large-Scale Mathematical Programs with Complementarity Constraints

This paper presents the Julia package CCOpt, built on top of the interior-point solver MadNLP. CCOpt implements a suite of algorithms for Mathematical Programs with Complementarity Constraints (MPCCs). The solver additionally comes with interfaces for use in Matlab, Python, and C++. MPCCs have recently gained renewed attention in engineering optimization, as complementarity provides a powerful modeling tool for nonsmooth functions and logical conditions. These problems are inherently challenging since their nonlinear programming reformulations violate classical regularity conditions at all feasible points, complicating both theoretical analysis and numerical treatment. Consequently, specialized algorithms are required to handle this degeneracy, and several approaches have been proposed. We implement a toolbox of methods, including relaxation and penalty approaches, as well as a crossover to recently proposed active-set methods. Our solver is based on nonlinear interior-point algorithms that couple the relaxation or penalty parameter with the barrier parameter, yielding substantial speedups compared to standard implementations. Both monotone and nonmonotone strategies for updating this joint parameter update are proposed and investigated. In addition, we propose regularization techniques that improve the conditioning of the KKT system for small relaxation parameters, enhancing robustness and computational efficiency. The implementation is validated on the classical MacMPEC benchmark, large-scale problems in security-constrained optimal power flow, optimal control of nonsmooth systems, as well as on quadratic programs with complementarity constraints arising in model predictive control. This benchmarking reveals an algorithmically driven improvement of often an entire order of magnitude over other methods, including commercial solvers.

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arXiv2026

Real-Time Algorithms for Model Predictive Control of Hybrid Dynamical Systems

Model predictive control (MPC) of hybrid dynamical systems is challenging because the associated optimization problem is nonsmooth and the resulting feedback law is discontinuous. This paper develops real-time MPC algorithms for nonlinear hybrid systems modeled as dynamical complementarity systems. The resulting optimal control problems are formulated as mathematical programs with complementarity constraints (MPCCs). We show that the solution map of parametric MPCCs is discontinuous, and that standard nonlinear-programming-based approaches may become infeasible when the hybrid system switches. To address this, we introduce three real-time hybrid MPC schemes whose feedback phase solves a quadratic program with complementarity constraints per sample, yielding local discontinuous piecewise affine approximations of the MPC feedback law. Moreover, we derive continuity and differentiability results for parametric MPCCs, and establish conditions under which the approximation error of our new hybrid MPC algorithms remains uniformly bounded despite solution discontinuities. The algorithms are demonstrated on a robotic manipulation example, where contact sequences are discovered online.

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arXiv2025

Real-Time Non-Smooth MPC for Switching Systems: Application to a Three-Tank Process

Real-time model predictive control of non-smooth switching systems remains challenging due to discontinuities and the presence of discrete modes, which complicate numerical integration and optimization. This paper presents a real-time feasible non-smooth model predictive control scheme for a physical three-tank process, implemented without mixed-integer formulations. The approach combines Filippov system modeling with finite elements and switch detection for time discretization, leading to a finite-dimensional optimal control problem formulated as a mathematical program with complementarity constraints. The mathematical program is solved via a homotopy of smooth nonlinear programs. We introduce modeling adjustments that make the three-tank dynamics numerically tractable, including additional modes to avoid non-Lipschitz points and undefined function values. Hardware experiments demonstrate efficient handling of switching events, mode-consistent tracking across reference changes, correct boundary handling, and constraint satisfaction. Furthermore, we investigate the impact of model mismatch and show that the tracking performance and computation times remain within real-time limits for the chosen sampling time. The complete controller is implemented using the non-smooth optimal control framework NOSNOC

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arXiv2025

An Augmented Lagrangian Method on GPU for Security-Constrained AC Optimal Power Flow

We present a new algorithm for solving large-scale security-constrained optimal power flow in polar form (AC-SCOPF). The method builds on Nonlinearly Constrained augmented Lagrangian (NCL), an augmented Lagrangian method in which the subproblems are solved using an interior-point method. NCL has two key advantages for large-scale SC-OPF. First, NCL handles difficult problems such as infeasible ones or models with complementarity constraints. Second, the augmented Lagrangian term naturally regularizes the Newton linear systems within the interior-point method, enabling to solve the Newton systems with a pivoting-free factorization that can be efficiently parallelized on GPUs. We assess the performance of our implementation, called MadNCL, on large-scale corrective AC-SCOPFs, with complementarity constraints modeling the corrective actions. Numerical results show that MadNCL can solve AC-SCOPF with 500 buses and 256 contingencies fully on the GPU in less than 3 minutes, whereas Knitro takes more than 3 hours to find an equivalent solution.

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arXiv2025

Towards Solutions of Manipulation Tasks via Optimal Control of Projected Dynamical Systems

We introduce a modeling framework for manipulation planning based on the formulation of the dynamics as a projected dynamical system. This method uses implicit signed distance functions and their gradients to formulate an equivalent gradient complementarity system. The optimal control problem is then solved via a direct method, discretized using finite-elements with switch detection. An extension to this approach is provided in the form of a friction formulation commonly used in quasi-static models. We show that this approach is able to generate trajectories for problems including multiple pushers, friction, and non-convex objects modeled as unions of convex ellipsoids with reasonable computational effort.

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arXiv2024

First-Order Sweeping Processes and Extended Projected Dynamical Systems: Equivalence, Time-Discretization and Numerical Optimal Control

Constrained dynamical systems are systems such that, by some means, the state stays within a given set. Two such systems are the (perturbed) Moreau sweeping process and the recently proposed extended Projected Dynamical System (ePDS). We show that under certain conditions solutions to the ePDS correspond to the solutions of a dynamic complementarity system, similar to the one equivalent to ordinary PDS. We then show that the perturbed sweeping process with time varying set can, under similar conditions, be reformulated as an ePDS. In this paper, we leverage these equivalences to develop an accurate discretization method for perturbed first-order Moreau sweeping processes via the finite elements with switch detection method. This allows the efficient optimal control of systems governed by ePDS and perturbed first-order sweeping processes.

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arXiv2024

Finite Elements with Switch Detection for Numerical Optimal Control of Nonsmooth Dynamical Systems with Set-Valued Heaviside Step Functions

This paper develops high-accuracy methods for numerically solving optimal control problems subject to nonsmooth differential equations with set-valued step functions. A notable subclass of these systems are Filippov systems. The set-valued step functions are here written as the solution map of a linear program. Using the optimality conditions of this problem we rewrite the initial nonsmooth system into a equivalent dynamic complementarity systems (DCS). We extend the Finite Elements with Switch Detection (FESD) method [Nurkanović et al., 2024], initially developed for Filippov systems transformed via Stewart's reformulation into DCS [Stewart, 1990], to the class of nonsmooth systems with set-valued step functions. The key ideas are to start with a standard Runge-Kutta method for the obtained DCS and to let the integration step sizes to be degrees of freedom. Next, we introduce additional conditions to enable implicit but exact switch detection and to remove possible spurious degrees of freedom if no switches occur. The theoretical properties of the method are studied. Its favorable properties are illustrated on numerical simulation and optimal control examples. All methods introduced in this paper are implemented in the open-source software package NOSNOC.

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arXiv2024

Solving mathematical programs with complementarity constraints arising in nonsmooth optimal control

This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call nosbench. The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes' relaxation [Scholtes, 2002] with IPOPT [Wächter and Biegler, 2006] as NLP solver manages to solve 73.8 % of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.

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arXiv2024

Finite Elements with Switch Detection for Numerical Optimal Control of Projected Dynamical Systems

The Finite Elements with Switch Detection (FESD) method is a highly accurate direct transcription method for optimal control of several classes of nonsmooth dynamical systems. This paper extends the FESD method to Projected Dynamical Systems (PDS) and first-order sweeping processes with time-independent sets. This method discretizes an equivalent dynamic complementarity system and exploits the particular structure of the discontinuities present in these systems. In the FESD method, allowing integration step sizes to be degrees of freedom, and introducing additional complementarity constraints, enables the exact detection of nonsmooth events. In contrast to the standard fixed-step Runge-Kutta methods, this approach allows for the recovery of full-order integration accuracy and the correct computation of numerical sensitivities. Numerical examples illustrate the effectiveness of the proposed method in an optimal control context. This method and the examples are included in the open-source software package nosnoc.

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arXiv2023

Finite Elements with Switch Detection for Direct Optimal Control of Nonsmooth Systems with Set-Valued Step Functions

This paper extends the Finite Elements with Switch Detection (FESD) method [Nurkanović et al., 2022] to optimal control problems with nonsmooth systems involving set-valued step functions. Logical relations and common nonsmooth functions within a dynamical system can be expressed using linear and nonlinear expressions involving step functions. A prominent subclass of these systems are Filippov systems. The set-valued step function can be expressed by the solution map of a linear program, and using its KKT conditions allows one to transform the initial system into an equivalent dynamic complementarity system (DCS). Standard Runge-Kutta (RK) methods applied to DCS have only first-order accuracy. The FESD discretization makes the step sizes degrees of freedom and adds further constraints that ensure exact switch detection to recover the high-accuracy properties that RK methods have for smooth ODEs. We use the novel FESD method for the direct transcription of optimal control problems. All methods and examples in this paper are implemented in the open-source software package NOSNOC.

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arXiv2023

FESD-J: Finite Elements with Switch Detection for Numerical Optimal Control of Rigid Bodies with Impacts and Coulomb Friction

The Finite Elements with Switch Detection (FESD) is a high-accuracy method for the numerical simulation and solution of optimal control problems subject to discontinuous ODEs. In this article, we extend the FESD method [Nurkanović et al., 2022] to the dynamic equations of multiple rigid bodies that exhibit state jumps due to impacts and Coulomb friction. This new method is referred to as FESD with Jumps (FESD-J). Starting from the standard Runge-Kutta equations, we let the integration step sizes be degrees of freedom. Additional constraints are introduced to ensure exact switch detection and to remove spurious degrees of freedom if no switches occur. Moreover, at the boundaries of each finite element, we impose the impact equations in their complementarity form, at both the position and velocity level. They compute the normal and tangential impulses in case of contact making. Otherwise, they are reduced to the continuity conditions for the velocities. FESD-J treats multiple contacts, where each contact can have a different coefficient of restitution and friction. All methods introduced in this paper are implemented in the open-source software package NOSNOC. We illustrate the use of FESD-J in both simulation and optimal control examples.

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