Online Least-Squares Optimization with Convex Sample-Constraints

We consider infinite-dimensional convex quadratic optimization problems, with shift-invariant quadratic forms, subject to convex sample-wise constraints. From a systems perspective, the latter correspond to those of designing discrete-time systems which are optimal in frequency-weighted least squares sense subject to sample-wise (convex) constraints on the output signal. In such problems, the optimality conditions do not provide an explicit way of generating the optimal output as a real-time implementable transformation of the input, due to causal instability of the resulting dynamical equations and sequential nature in which problem should be solved (over time). In this talk, I will first show that the optimal system has exponentially fading memory, which suggests the existence of arbitrarily good receding horizon (i.e., finite latency) approximations. Then, I will present an online algorithm for solving the above optimization problem with arbitrary precision. The algorithm is realized as a causally stable nonlinear discrete-time system, which is allowed to look ahead at the input signal over a finite horizon (and is, therefore, of finite latency). The upper bound on approximation error is obtained by extending the classical balanced truncation algorithm for linear systems to a class of nonlinear systems with weakly contractive operators.