# Primal-dual interior point methods in state and input constrained optimal control.

Topic: Optimal control | All

Thursday 12th May 2022, 9am – 10am (Paris time).

SPEAKER

Paul MALISANI, IFPEN, France

ABSTRACT

Adapting Interior Point Methods (IPMs) to Optimal Control Problems (OCPs) is not straightforward and has not been completely performed yet. Indeed, to be complete, this adaptation requires proving two things. Firstly, that the optimal trajectories of the penalized problem are interior, i.e. strictly satisfy the constraints. Secondly, it requires proving the convergence of the method to a point satisfying the first order conditions of optimality. In other words, to prove the convergence of primal variable (namely state and control variables) and of the dual variables (namely the adjoint state of Pontryagin and the constraints KKT multipliers). In the existing literature, primal and dual convergence results have been proved for input constrained OCPs and only primal convergence has been proved for state and input constrained OCPs.\\

The aim of the presented work is to fill this gap by proving both interiority of the penalized trajectories and convergence of primal and dual variables. To do so, sufficient conditions on penalty functions guaranteeing the interiority of local optimal solutions are exhibited. Then we prove that logarithmic functions satisfy these conditions. Using logarithmic penalties, we also prove that the sequence of the penalty functions derivatives associated with the sequence of local optimal solution of the penalized problems are uniformly bounded. These uniform boundedness properties allow to use some standard compactness argument to prove strong, weak or weak-* convergence, depending on the case, of the derivative of the penalty functions to dual variables. The proof of convergence of the primal variables when interiority is guaranteed has already been proved previous publications.

BIO

Paul Malisani is researcher in optimization for energy systems at the department of Applied Mathematics at IFPEN. Prior to that he was a research engineer at EDF Lab. Between 2009 and 2012, he was Ph.D. candidate at CAS-MINES-ParisTech under the supervision of F. Chaplais and N. Petit.

SPEAKER

Paul MALISANI, IFPEN, France

ABSTRACT

Adapting Interior Point Methods (IPMs) to Optimal Control Problems (OCPs) is not straightforward and has not been completely performed yet. Indeed, to be complete, this adaptation requires proving two things. Firstly, that the optimal trajectories of the penalized problem are interior, i.e. strictly satisfy the constraints. Secondly, it requires proving the convergence of the method to a point satisfying the first order conditions of optimality. In other words, to prove the convergence of primal variable (namely state and control variables) and of the dual variables (namely the adjoint state of Pontryagin and the constraints KKT multipliers). In the existing literature, primal and dual convergence results have been proved for input constrained OCPs and only primal convergence has been proved for state and input constrained OCPs.\\

The aim of the presented work is to fill this gap by proving both interiority of the penalized trajectories and convergence of primal and dual variables. To do so, sufficient conditions on penalty functions guaranteeing the interiority of local optimal solutions are exhibited. Then we prove that logarithmic functions satisfy these conditions. Using logarithmic penalties, we also prove that the sequence of the penalty functions derivatives associated with the sequence of local optimal solution of the penalized problems are uniformly bounded. These uniform boundedness properties allow to use some standard compactness argument to prove strong, weak or weak-* convergence, depending on the case, of the derivative of the penalty functions to dual variables. The proof of convergence of the primal variables when interiority is guaranteed has already been proved previous publications.

BIO

Paul Malisani is researcher in optimization for energy systems at the department of Applied Mathematics at IFPEN. Prior to that he was a research engineer at EDF Lab. Between 2009 and 2012, he was Ph.D. candidate at CAS-MINES-ParisTech under the supervision of F. Chaplais and N. Petit.