Abstract
In this paper, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time $k$, $x_k^\star$ and $u_k^\star$, with respect to the reference signal at time $i$, $d_i$, will have exponential decay in terms of $|k-i|$ with a decay rate $\rho$ independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al., 2017 can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.
Sen Na
Postdoc in Statistics and ICSI
Sen Na is a postdoctoral scholar in the Statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods in biology, neuroscience, and engineering.
Mihai Anitescu
Professor in Statistics and CAM
Mihai Anitescu is a Professor in the Statistics and CAM departments at the University of Chicago, and is also a senior computational mathematician in the Mathematics and Computer Science Division at Argonne. He works on a variety of topics on control, optimization, and computational statistics.