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We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of “solvable cases”, most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (“ellitopic” and “co-ellitopic”, respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding “robust operator norm of uncertain matrix with box uncertainty”, that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude $\le 1$, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of “uncertain-but-bounded” noise varying in a box.

In this paper we first present a novel operator extrapolation (OE) method for solving deterministic variational inequality (VI) problems. Similar to the gradient (operator) projection method, OE updates one single search sequence by solving a single projection subproblem in each iteration. We show that OE can achieve the optimal rate of convergence for solving a variety of VI problems in a much simpler way than existing approaches. We then introduce the stochastic operator extrapolation (SOE) method and establish its optimal convergence behavior for solving various stochastic VI problems. In particular, SOE achieves the optimal complexity for solving a fundamental problem, i.e., stochastic smooth and strongly monotone VI, for the first time in the literature. We also present a stochastic block operator extrapolation method to further reduce the iteration cost for the OE method applied to large-scale deterministic VIs with a certain block structure. Numerical experiments have been conducted to demonstrate the potential advantages of the proposed algorithms. In fact, all these algorithms are applied to solve generalized monotone variational inequality problems whose operator is not necessarily monotone. We will also discuss optimal OE-based policy evaluation methods for reinforcement learning in a companion paper. [ABSTRACT FROM AUTHOR]

This paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete-time stochastic jump linear systems. The approximation error, which is captured by means of the stochastic [L.sub.2] gain, is bounded from above by twice the sum of singular numbers associated to the truncated states, similar to the case of linear time-invariant systems. A two step model reduction algorithm for hidden Markov models is also developed. The first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden Markov models and a subclass of stochastic jump linear systems. In a second step the positivity constraints, which reflect the hidden Markov model structure, are enforced by solving a low dimensional optimization problem. Index Terms--Balanced truncation, error bound, finite state machines, hidden Markov models, jump systems, model reduction, reduced order systems, stochastic automata, stochastic hybrid systems, stochastic systems.

This work addresses the finite-horizon robust covariance control problem for discrete time, partially observable, linear systems affected by random zero mean noise and deterministic uncertain-but-bounded disturbances restricted to lie in what is called ellitopic uncertainty set (e.g., finite intersection of centered at the origin ellipsoids/elliptic cylinders). Performance specifications are imposed on the random state-control trajectory via averaged convex quadratic inequalities, linear inequalities on the mean, chance constrained linear inequalities as well as convex-monotone constraints on the covariance matrix. For this problem we develop a computationally tractable procedure for designing affine control policies, in the sense that the parameters of the policy that guarantees the aforementioned performance specifications are obtained as solutions to an explicit convex program. Our theoretical findings are illustrated by a numerical example. [ABSTRACT FROM AUTHOR]