Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem spe- cified by two pieces of input: (i) a mathematical program (an objective function f: R" -> R and a feasible set QCR") and (it) a dynamical system (a map g: R"->"). Its goal is to minimize f over the set SCQ of initial conditions that forever remain in Q under g. The focus of this paper is on the case where the mathematical program is a linear program and where 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 approxi- mations are optimized with respect to the objective function f, and their semidefinite characterization-which has a semidefinite constraint of fixed size is obtained by apply- ing polar duality to convex sets that are invariant under (multiple) linear maps. We charac- terize 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 polyno- mial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radtus 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.