Chantal Landry

Solving Optimization-Constrained Differential Equations with Discontinuity Points, with Application to Atmospheric Chemistry

Ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time. A particular application arises in the modeling of atmospheric particles. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method for the solution of optimization-constrained differential equations is proposed by coupling an implicit Runge-Kutta method (RADAU5), with numerical techniques for the detection of the events (activation and deactivation of constraints). The computation of the events is based on dense output formulas, continuation techniques, and geometric arguments. Numerical results are presented for the simulation of the time-dependent equilibrium of organic atmospheric aerosol particles, and show the efficiency and accuracy of the approach.

Path-Planning with Collision Avoidance in Automotive Industry

An optimal control problem to find the fastest collision-free trajectory of a robot is presented. The dynamics of the robot is governed by ordinary differential equations. The collision avoidance criterion is a consequence of Farkas’s lemma and is included in the model as state constraints. Finally an active set strategy based on backface culling is added to the sequential quadratic programming which solves the optimal control problem.

A Second Order Scheme for Solving Optimization-Constrained Differential Equations with Discontinuities

A numerical method for the resolution of a system of ordinary differential equations coupled with a mixed constrained minimization problem is presented. This coupling induces discontinuities of some time-dependent variables when inequality constraints are activated or deactivated. The ordinary differential equations are discretized in time and combined with the first order optimality conditions of the optimization problem. We use a second order multistep method based on a predictor-corrector Adams scheme to detect the discontinuities by extrapolation of the trajectories. Optimization features, namely a sensitivity analysis, are exploited to compute the derivatives of the optimization variables and track the discontinuity points. The main difficulty consists in the impossibility of defining an explicit event function to characterize the activation or deactivation of a constraint. The order of convergence of our method is proved when inequality constraints are activated and numerical results for atmospheric organic particles are presented.

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.