Abstract-This paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms. This includes the wide class of NP-hard control problems representable by means of parameter-dependent linear matrix inequalities (LMIs). It is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem (the scenario problem) whose solution is approximately feasible for the original (usually infinite) set of constraints, i.e., the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased. We provide an explicit and efficient bound on the number of samples required to attain a-priori specified levels of probabilistic guarantee of robustness. A rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial-time solution, if robustness is intended in the proposed risk-adjusted sense.

Abstract. Many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or 'instance' parameter. Two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty: robust optimization and chance-constrained optimization. Robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances. In chance-constrained optimization a probability distribution is instead assumed on the uncertain parameters, and the constraints are enforced up to a pre-specified level of probability. Unfortunately however, both approaches lead to computationally intractable problem formulations.In this paper, we consider an alternative 'randomized' or 'scenario' approach for dealing with uncertainty in optimization, based on constraint sampling. In particular, we study the constrained optimization problem resulting by taking into account only a finite set of N constraints, chosen at random among the possible constraint instances of the uncertain problem. We show that the resulting randomized solution fails to satisfy only a small portion of the original constraints, provided that a sufficient number of samples is drawn. Our key result is to provide an efficient and explicit bound on the measure (probability or volume) of the original constraints that are possibly violated by the randomized solution. This volume rapidly decreases to zero as N is increased.

Random convex programs (RCPs) are convex optimization problems subject to a finite number N of random constraints. The optimal objective value J * of an RCP is thus a random variable. We study the probability with which J * is no longer optimal if a further random constraint is added to the problem (violation probability, V *). It turns out that this probability rapidly concentrates near zero as N increases. We first develop a theory for RCPs, leading to explicit bounds on the upper tail probability of V *. Then we extend the setup to the case of RCPs with r a posteriori violated constraints (RCPVs): a paradigm that permits us to improve the optimal objective value while maintaining the violation probability under control. Explicit and nonasymptotic bounds are derived also in this case: the upper tail probability of V * is upper bounded by a multiple of a beta distribution, irrespective of the distribution on the random constraints. All results are derived under no feasibility assumptions on the problem. Further, the relation between RCPVs and chanceconstrained problems (CCP) is explored, showing that the optimal objective J * of an RCPV with the generic constraint removal rule provides, with arbitrarily high probability, an upper bound on the optimal objective of a corresponding CCP. Moreover, whenever an optimal constraint removal rule is used in the RCPVs, then appropriate choices of N and r exist such that J * approximates arbitrarily well the objective of the CCP.

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The purpose of this work is to give a contribution to the understanding of the COVID-19 contagion in Italy. To this end, we developed a modified Susceptible-Infected-Recovered-Deceased (SIRD) model for the contagion, and we used official data of the pandemic for identifying the parameters of this model. Our approach features two main non-standard aspects. The first one is that model parameters can be time-varying, allowing us to capture possible changes of the epidemic behavior, due for example to containment measures enforced by authorities or modifications of the epidemic characteristics and to the effect of advanced antiviral treatments. The time-varying parameters are written as linear combinations of basis functions and are then inferred from data using sparse identification techniques. The second non-standard aspect resides in the fact that we consider as model parameters also the initial number of susceptible individuals, as well as the proportionality factor relating the detected number of positives with the actual (and unknown) number of infected individuals. Identifying the model parameters amounts to a non-convex identification problem that we solve by means of a nested approach, consisting in a one-dimensional grid search in the outer loop, with a Lasso optimization problem in the inner step.

This note presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With states, uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as ((+)). With unstructured uncertainty in all system matrices, the worst-case complexity reduces to ().