New results on the scenario design approach M.C. Campi G.C. Calaﬁore Abstract. The scenario optimization method developed in [1] is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimiza- tion problems arising in systems and control design, that would otherwise be hard to tackle via standard deterministic techniques. In this note, we further explore some aspects of the scenario methodology, and present two results pertaining to the tightness of the sample complexity bounds. We also state a new theorem that enables the user to make a-priori probabilistic claims on the scenario solution, with one level of probability only. Keywords: Scenario design, Robust control, Randomized algorithms, Probabilistic robustness, Robust convex opti- mization. I. PRELIMINARIES Techniques based on uncertainty randomization recently gained increasing favor among both control theoreticians and practitioners, the ﬁrsts being appealed by the solid foundations of these methods, rooting in the theory of probability, optimization and stochastic processes, and the seconds being attracted by their relative simplicity of prac- tical implementation. An up-to-date description of this body of techniques, along with applications to control analysis and design problems and many pointers to the literature, can be found in the texts [3], [12]. Among these techniques, the so- called scenario design method developed in [1] permits to solve effectively control design problems that can be cast in the form of a convex optimization program with uncertain constraints. A signiﬁcant class of control problems indeed fall in this framework, see for instance the discussion and examples in [1]. First, we brieﬂy review the essential points of the scenario optimization approach of [1] in order to prepare the terrain for our further discussion. Scenario optimization This work was supported by FIRB funds of Italian Ministry of University and Research (MIUR) and by MIUR project Identiﬁcation and Adaptive Control of Industrial Systems. Marco C. Campi is with the Dipartimento di Elettronica per l’Automazione, Universit` a di Brescia, Via Branze 38, 25123 Brescia – Italy. marco.campi@ing.unibs.it Giuseppe C. Calaﬁore (corresponding author) is with the Dipartimento di Automatica e Informatica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino – Italy. giuseppe.calafiore@polito.it Consider an uncertain convex optimization problem min θ∈Θ c T θ subject to: (1) f (θ, δ) ≤ 0,δ ∈ Δ, where θ ∈ Θ ⊆ R n θ is the decision variable, Θ is convex and closed, δ ∈ Δ ⊆ R n δ is an uncertain parameter, c ∈ R n θ is a given vector, and f (θ, δ):Θ × Δ → [−∞, ∞] is continuous and convex in θ, for any ﬁxed value of δ ∈ Δ. A robust solution associated to (1) is obtained when f (θ, δ) ≤ 0 is required to hold ∀δ ∈ Δ, while different scales of robustness can be achieved by imposing that f (θ, δ) ≤ 0 holds for only a fraction of the δ’s in Δ. The scenario optimization described below is a technology to attain at low computational cost a solution that is robust to a level as speciﬁed by the user. If “Prob” is a probability measure on Δ, the scenario solution ˆ θ N for (1) is the optimal solution of the following convex program min θ∈Θ c T θ subject to: (2) f (θ, δ (i) ) ≤ 0,i =1,...,N, where δ (i) , i = 1,...,N , are independent samples, identically distributed according to Prob. Note that the optimal solution ˆ θ N of this program is a random variable that depends on the random extractions (δ (1) ,...,δ (N) ). Let ǫ ∈ (0, 1), β ∈ (0, 1) be given (small) probability levels. A key result in [1] (Theorem 1 and Corollary 1) states that if N ≥ N (ǫ, β) samples are taken in (2), where N (ǫ, β) is some explicitly given function (see (4) below), then it holds that Prob N {(δ (1) ,...,δ (N) ) ∈ Δ N : V ( ˆ θ N ) ≤ ǫ} ≥ 1 − β, (3) being V (θ) a measure of violation of the constraints in (1) for a given θ, i.e. V (θ) . = Prob{δ ∈ Δ: f (θ, δ) > 0}. In other words, with high probability 1 − β, the scenario solution is feasible for all the constraints in (1), except possibly for those in a set having probability measure smaller than ǫ, that is, this solution is “almost robustly feasible.” A Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007 FrC11.5 1-4244-1498-9/07/$25.00 ©2007 IEEE. 6184