A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function *f *: R^{n }*→** *R and a feasible set Ω *⊆*R* ^{n}*), and (ii) a dynamical system (a map

*g*: R

^{n }*→*R

*). Its goal is to minimize f over the set S of initial conditions that forever remain in under g. The focus of this paper is on the case where the mathematical program is a linear program and the dynamical system is either a known linear map, or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to S. Our inner approximations are optimized with respect to the objective function f and their semidefinite characterization|which has a semidefinite constraint of fixed size - is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations to S from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case.*

^{n}By:* Amir Ali Ahmadi, Oktay Günlük*

Published in: RC25677 in 2018

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