On the Sample Size of Randomized MPC for ChanceConstrained Systems with Application to Building Climate Control Xiaojing Zhang, Sergio Grammatico, Georg Schildbach, Paul Goulart and John Lygeros Abstract— We consider Stochastic Model Predictive Control (SMPC) for constrained linear systems with additive disturbance, under affine disturbance feedback (ADF) policies. One approach to solve the chance-constrained optimization problem associated with the SMPC formulation is randomization, where the chance constraints are replaced by a number of sampled hard constraints, each corresponding to a disturbance realization. The ADF formulation leads to a quadratic growth in the number of decision variables with respect to the prediction horizon, which results in a quadratic growth in the sample size. This leads to computationally expensive problems with solutions that are conservative in terms of both cost and violation probability. We address these limitations by establishing a bound on the sample size which scales linearly in the prediction horizon. The new bound is obtained by explicitly computing the maximum number of active constraints, leading to significant advantages both in terms of computational time and conservatism of the solution. The efficacy of the new bound relative to the existing one is demonstrated on a building climate control case study. I. I NTRODUCTION Model Predictive Control (MPC) is a powerful methodology for control design for systems in which state and input constraints are present [1]. By predicting the future behavior of the plant, MPC is able to incorporate feedforward information in the control design. Such information may come, for example, in the form of predictions or reference tracking. At each sampling time, MPC requires to solve a finite horizon optimal control problem (FHOCP), and to implement the first element of the optimal control sequence. One way of designing controllers is to assume an exact model of the plant and perfect evolution of the states according to that model. In practice, however, disturbances arise from both model uncertainty and prediction errors. The latter, for example, is a challenging problem in building climate control [2], [3], where the goal is to control the comfort dynamics in a building. In this case, disturbances appear due to the uncertainty in weather and rooms occupancy. In general, within the MPC literature two main methods have been studied to address uncertainty: robust and stochastic MPC. Robust MPC computes a control law that guarantees constraint satisfaction for all possible disturbance realizations [4]. Although successful in many cases, this strategy may lead to conservative controllers that exhibit poor performance The authors are with the Automatic Control Laboratory, Department of Electrical Engineering and Information Technology, Swiss Federal Institute of Technology Zurich (ETH Zurich), 8092 Zurich, Switzerland. {xiaozhan, grammatico, schildbach, pgoulart, lygeros}@control.ee.ethz.ch. This research was partially funded by a RTD grant from Swiss NanoTera.ch under the project HeatReserves. in practice. This is due to the need to over-approximate the uncertainty set to obtain a tractable problem formulation. These limitations are partially overcome by adopting the notion of Stochastic MPC (SMPC) [5], where the constraints of the FHOCP are interpreted probabilistically via chance constraints, allowing for a (small) constraint violation probability. Unfortunately, chance constrained FHOCPs are in general non-convex and require the computation of multidimensional integrals. Hence, SMPC is computationally intractable for many applications. Randomized MPC (RMPC) [6] is a new method to approximate SMPC problems. It is computationally tractable without being limited to specific probability distributions. At every time step, the chance constrained FHOCP is solved via the scenario approach [7]–[9], which is a randomized technique for solving convex chance constrained optimization problems. The basic idea is to replace the chance constraints with a finite number of sampled constraints, which correspond to independently sampled disturbance realizations. The sample size is chosen so that, with high confidence, the violation probability of the solution of the sampled FHOCP remains small. An important challenge for the practical application of RMPC is its requirement for a large sample size, even for relatively small systems. A large sample size generates equally many constraints, so that the resulting sampled FHOCP becomes computationally expensive to solve, even if it is convex. Moreover, the bound on the sample complexity established in [8], [9] is tight for the class of “fully supported” problems, yet many MPC problems do not fall into this category. This typically leads to conservative solutions, both in terms of cost and empirical violation probability. The authors in [10] try to alleviate this conservatism by using a combination of randomized and robust optimization, which can be successfully applied towards SMPC in openloop control. However, when the same approach is applied to closed-loop policies using affine disturbance feedback (ADF) [11], the solution usually becomes more conservative than the standard RMPC approach. Unfortunately, in case of ADF policies, the latter is expensive to solve in practice, because the number of decision variables grows quadratically in the prediction horizon. This results in a quadratic growth of the sampled constraints, making RMPC with ADF almost impossible to solve in practice, even if the problem remains convex. In this paper we attempt to overcome these limitations by establishing a novel upper bound on the sample size for RMPC based on the ADF policy. Instead of a quadratic growth of the number of samples, our bound on the sample size grows linearly in the prediction horizon. This is achieved by exploiting structural properties of the constraints in the sampled FHOCP. We apply this improved bound to a building control problem, where the sample size is reduced significantly. We restrict to RMPC for linear systems, while we refer to [12] for the case of nonlinear control-affine systems, based on the non-convex scenario-approach results in [13], [14]. In Section II we formulate the Stochastic MPC problem. Section III summarizes the standard RMPC approach. We describe the proposed methodology in Section IV, whereas Section V discusses numerical results for the building control case study. Conclusions are drawn in Section VI. The proofs are given in Appendix. II. MPC P ROBLEM D ESCRIPTION A. Dynamics, constraints and control objective We consider the following discrete-time affine system subject to additive disturbance x+ = Ax + Bu + V v + Ew, (1) where x ∈ Rnx is the state vector, x+ ∈ Rnx the successor state, u ∈ Rnu the input vector, v ∈ Rnv a vector that is known a priori, and A, B, V and E are matrices of appropriate dimensions. The vector w ∈ Rnw models the stochastic disturbance. If N is the prediction horizon and wk , for k ∈ {0, . . . , N −1}, the disturbance at the kth step, we define the “full-horizon” disturbance as w := [w0 , . . . , wN −1 ]. We assume that w is defined by a probability measure P. The distribution itself need not be known explicitly, but we require that independent samples can be drawn according to this distribution. In practical applications, the samples could arise from historical data. Note that the wk at different times inside w need not be independent and identically distributed (i.i.d.). We assume that at each prediction step k ∈ {1, . . . , N }, the state xk is subject to polyhedral constraints which may be violated with a probability no greater than k ∈ (0, 1). They can be expressed as P[F xk ≤ f ] ≥ 1 − k , (2) PN −1 k=0 `(xk , uk ) + `f (xN ), In its most general setting, the control problem consists of finding a control policy Π := {µ0 , . . . , µN −1 }, with µk = µk (x0 , . . . , xk ) ∈ Rnu , which minimizes the cost in (3) subject to the constraints in (2) and dynamics in (1). The intuition is to take measurements of the past and current states into account when computing future control inputs. However, since it is generally intractable to optimize over the function space of state feedback policies, a common approximation is the affine disturbance feedback (ADF) policy [11] of the form Pk−1 uk := hk + j=0 Mk,j wj , (4) where one optimizes over all Mk,j ∈ Rnu ×nw and hk ∈ Rnu . It is shown in [11] that (4) is equivalent to affine state feedback, but gives rise to a convex problem. If x is the initial state and we define wk := [w0 , . . . , wk−1 ] as the restriction of w over its first k components, hk := [h0 , . . . , hk−1 ], and vk := [v0 , . . . , vk−1 ], then the state xk can be expressed as xk = Ak x + Mk wk + Bk hk + Vk vk + Ek wk , (5) for suitable matrices Mk , Bk , Vk , and Ek . Note that due to causality, the matrix Mk is strictly block lower triangular, while Bk , Vk , and Ek are block lower triangular. C. Stochastic MPC formulation The Stochastic MPC (SMPC) problem is obtained by combining (1) – (4). Therefore, at each sampling time, we solve the following chance constrained FHOCP min E[J(M, h)] (6) M,h s.t. P [F xk ≤ f ] ≥ 1 − k ∀k ∈ {1, . . . , N }, where E is the expectation associated to P and xk is as in (5). The matrix M ∈ RN nu ×N nw is a strict block lower triangular matrix collecting all Mk,j and h := [h0 , . . . , hN −1 ]. Note that the problem in (6) has multiple chance constraints (“multi-stage problem”), where each constraint must be satisfied with a predefined probability of k . Remark 1 (Input Constraints): To ease the presentation of our results we consider only state constraints. Nevertheless, our results can handle input constraints as well, provided they are interpreted in a probabilistic sense. III. R ANDOMIZED MPC where F ∈ Rnf ×nx , f ∈ Rnf , and nf is the number of state constraints. Note that each stage is viewed as one (joint) chance constraint. The control objective is to minimize a performance function of the form J(u0 , . . . , uN −1 ) = B. Affine disturbance feedback policy (3) where ` : Rnx × Rnu → R and `f : Rnx → R are strictly convex functions, and {xk }N k=0 satisfies the dynamics in (1) under control inputs u0 , . . . , uN −1 . In general, the chance constraints turn the FHOCP in (6) into a non-convex and computationally intractable problem, making SMPC impractical to implement. RMPC is one method to obtain a tractable approximation, based on the scenario approach [7]–[9]. In its original formulation, RMPC uses one joint chance constraint for all constraints along the horizon [6]. The authors in [15] propose an RMPC formulation to cope with multiple chance constraints, as discussed next. (1) (S ) For each stage k ∈ {1, . . . , N }, let {wk , . . . , wk k } be a collection of samples, obtained by first drawing Sk independent full-horizon samples according to P and then restricting them to their first k components. The idea of the scenario approach is to replace the kth chance constraint in (6) with Sk hard constraints, each corresponding to a sample (i) wk , with i ∈ {1, . . . , Sk }, [7]–[9]. Hence, we solve the sampled FHOCP min E[J(M, h)] (7) M,h s.t. (i) F xk ≤ f ∀i ∈ {1, . . . , Sk }, ∀k ∈ {1, . . . , N }, (i) where xk is the ith predicted state. It is obtained by (i) substituting the ith sample wk into (5). In the interest of space, we do not describe all technical details of the scenario approach; instead, the interested reader is referred to [7] for typical assumptions (convexity, uniqueness of optimizer, i.i.d. sampling), and to [14] for measure-theoretic technicalities about the well-definedness of the probability integrals. The main challenge in RMPC is to establish the required sample sizes Sk , so that the solution of (7) is feasible for (6) with high confidence. Based on the results of [8], [9], it was shown in [15] that if Sk satisfies Pζk −1 Sk j Sk −j ≤ βk , (8) j=0 j k (1 − k ) where βk ∈ (0, 1) is the confidence parameter, then the solution of (7) is feasible for the kth constraint in (6) with confidence at least 1 − βk . The parameter ζk is the socalled support dimension (s-dimension) [15, Definition 4.1], and upper bounds the number of support constraints [7, Definition 4] of the kth chance constraint. We call a sampled constraint a support constraint if its removal changes the optimizer. An explicit lower bound on the sample size was established in [16] as e . (9) ζk − 1 + ln β1k Sk ≥ 1k e−1 Thus, for fixed k and βk , it follows that Sk ∼ O(ζk ), so that problems with a lower ζk require fewer samples. Unfortunately, explicitly computing ζk is difficult in general, and usually an upper bound has to be computed. As reported in [10, Section III], the standard bound k(k − 1) =: dk , (10) 2 always holds, where dk is the number of decision variables up to stage k. Hence using the standard bound, Sk scales as O(k 2 nu nw ), which is quadratic along the horizon k 1 . This quadratic growth makes the application of RMPC challenging because of the number of sampled constraints that need to be stored in the computer memory and processed when solving the optimization problem. Moreover, if the sample size is chosen larger than necessary, the obtained solution ζk ≤ knu + nu nw PN 1 If the total number of samples is denoted by S := k=1 Sk , then S grows as O(N 3 nu nw ) using the standard bound in (10). To simplify discussion, we compare the sample sizes Sk of the individual stages. Hence, if we say “quadratic growth” along the horizon, we refer to Sk rather than S. From Faulhaber’s formula, the total number of samples has a growth rate that is 1 order higher compared to the individual stages. becomes conservative in terms of violation probability and consequently in terms of cost. Section IV presents tighter bounds on the s-dimension ζk by exploiting structural properties of the constraint functions. IV. R EDUCING THE S AMPLE S IZE We here provide two methods to upper bound the stagewise s-dimensions ζk for the FHOCP in (7). The first bound is obtained by exploiting structure in the decision space. The second bound exploits structure in the uncertainty space and scales linearly in k. In general, the first bound performs well for small k, whereas the second is better for larger k. Hence, the minimum among them should be taken to obtain the tightest possible bound. A. Structure in the decision space One way of bounding the stage-wise s-dimensions ζk is to exploit structural properties of the constraint function in (2) with respect to the decision space. To this end, we recall the so-called support rank (s-rank) ρk from [15, Definition 4.6]. Definition 1 (s-rank): For k ∈ {1, . . . , N }, let Lk be the largest linear subspace of the decision space RdN , with dN (i) as in (10), that remains unconstrained by F xk ≤ f for all (i) sampled instances wk almost surely. Then the s-rank of the kth chance constraint is defined as ρk := dN − dim(Lk ). Note that dN is the total number of decision variables of the sampled FHOCP in (7). We know from [15, Theorem 4.7] that the s-rank upper bounds the s-dimension, i.e. ζk ≤ ρk . The next statement establishes an explicit bound on the srank for the FHOCP in (7). Proposition 1: For all k ∈ {1, . . . , N }, the s-rank ρk of the sampled FHOCP in (7) satisfies k(k − 1) + min{rank(F ), knu }. 2 Proposition 1 consists of an improvement upon the standard bound in (10), but still scales quadratically along the horizon. ρk ≤ n u n w B. Structure in the uncertainty space Another way of obtaining a bound on ζk is to find an upper bound on the number of active constraints, which we define for the kth chance constraint as follows. (i) Definition 2 (Active Constraint): The sample wk ∈ (1) (S ) {wk , . . . , wk k } is called an active sample for the kth stage and generates an active constraint if, at the optimal (i) (i) solution (M? , h? ) of (7), F xk = f , where xk is the (i) trajectory generated by (M? , h? ) and wk according to (5). For any stage k ∈ {1, . . . , N }, let the set Ak ⊆ {1, . . . , Sk } index the active samples and |Ak | denote its cardinality. From [8, pag. 1219], we have that the s-dimension ζk is bounded by the number of active contraints as follows. Lemma 1: For all k ∈ {1, . . . , N }, ζk ≤ |Ak |. For constraint functions of general structure, it is not easy to determine |Ak |. For FHOCP in (7), however, such a bound can be found under the following assumption. Assumption 1 (Probability Measure): The random variable w is defined on a probability space with an absolutely continuous probability measure P. Note that a probability measure is absolutely continuous if and only if it admits a probability density function. Under this assumption, the next proposition provides a bound on the number of active constraints. Proposition 2: For all k ∈ {1, . . . , N }, the number of active constraints of the sampled FHOCP in (7) almost surely satisfies |Ak | ≤ nf knw . Unlike the bounds dk and ρk , |Ak | does not depend on the number of decision variables at all, but rather on the dimension of the uncertainty affecting the kth state. Moreover, it shows that the s-dimension scales at most linearly in the prediction horizon as O(knw ), as opposed to O(k 2 nw nu ) when using dk or ρk . Moreover, Proposition 2 suggests that for the same probabilistic guarantees, plants subject to fewer uncertainties (smaller nw ) require a smaller sample size compared to plants affected by higher dimensional uncertainties. Remark 2: Note that Assumption 1 does not allow for “concentrated” probability masses. While it might seem restrictive in general, this is not the case for many practical applications. For example in building climate control, the main disturbances are due to the uncertainty in future temperatures, solar radiation, and occupancy, which can be assumed to arise from continuous distributions. Indeed, even occupancy can take continuous values. This is because the system model is discretized, but people enter and leave a room at arbitrary times. We refer to [17] for similar sample-size bounds where Assumption 1 is released. C. Combining the bounds In general, the bounds ρk and |Ak | will not be the same since they depend on the dimension of the decision and uncertainty space, respectively. To obtain the tightest possible bound, the minimum among both should be taken when upper bounding the s-dimension. Theorem 1: For all k , βk ∈ (0, 1), with k ∈ {1, . . . , N }, if Sk satisfies (8) with o n k(k − 1) + min{rank(F ), knu } min nf knw , nu nw 2 in place of ζk , with confidence no smaller than 1 − βk , the optimal solution of the sampled FHOCP in (7) is feasible for each chance constraint in (6). Note that for k = 1, ρk ≤ |Ak |, while for larger k we have |Ak | ≤ ρk because |Ak | grows linearly in k compared to ρk . V. A PPLICATION TO B UILDING C LIMATE C ONTROL In this section we consider a case study in building climate control, where we regulate the room temperature of an office room. In the spirit of [3], we use historical data or scenarios in ensemble forecasting to construct samples for weather prediction uncertainty, without having to know the exact distribution. We use the reduced model presented in [18]. The system dynamics are affine in the form of (1). The vector xk = [xk,1 , xk,2 , xk,3 ]> ∈ R3 is the state vector, where xk,1 is the room temperature, xk,2 the temperature of the inside wall, and xk,3 the temperature of the outside wall. The weather and occupancy prediction is modeled by the vector vk = [vk,1 , vk,2 , vk,3 ]> ∈ R3 where vk,1 is the outside air temperature, vk,2 the solar radiation, and vk,3 the occupancy. We investigate three cases in which we allow the uncertainty w to have different dimensions varying between nw ∈ {1, 2, 3}. For nw = 1 we only assume uncertainty in vk,1 , for nw = 2 uncertainty in vk,1 and vk,2 , and for nw = 3 uncertainty in all three predicted components. The control objective is to keep room temperature above 21◦ C with minimum energy cost. Four constrained inputs uk ∈ R4 represent actuators commonly found in Swiss office buildings: a radiator heater, cooled ceiling, floor heating system, and mechanical ventilation for additional heating/cooling purposes. TheP minimum energy requirement N is modeled by a linear cost E[ k=0 c> uk ]. The probabilistic state constraint can be expressed as P [xk,1 ≥ 21] ≥ 1 − k for every k = 1, . . . , N . For the purpose of illustration and to keep computation short, we select N = 8, k = 0.2 and βk = 0.1 for all k ∈ {1, . . . , N }. In the following, we compare the performance of the sampled FHOCP using the new bound from Theorem 1 with rank(F ) = 1 and nf = 1, versus the standard bound based on (10). We focus on the following three questions: sample size, computational complexity, and conservatism of the solution. A. Sample size Table I lists the sample complexity of nw = 1, 2, 3. The sample sizes Sk are obtained by numerical inversion of (8). It can be seen from Table I that the new bound dramatically improves upon the standard one of (10). Indeed, for this particular example the total number of samples S based on Theorem 1 is almost an order of magnitude lower than the one obtained using the standard bound. TABLE I P C OMPARISON OF THE TOTAL NUMBER OF SAMPLES S := 8k=1 Sk , FOR nw ∈ {1, 2, 3}. S= P k Sk nw = 1 new standard nw = 2 new standard nw = 3 new standard 277 491 694 2729 4500 6254 TABLE II AVERAGE CPU TIMES TO SOLVE RMPC FOR nw = 1, 2, 3 FOR DIFFERENT HORIZON LENGTH N . 0.12 0.1 pr e de fine d ne w bound s t andar d bound Solver times N N N N N =8 = 16 = 24 = 32 = 40 nw = 1 new standard 61 ms 750 ms 5.7 s 26 s 1.5 min 560 ms 16 s 3.2 min # # nw = 2 new standard 450 ms 4.2 s 36 s 5.5 min 12 min 2.0 s 1.3 min # # # nw = 3 new standard 480 ms 11 s 2.5 min 27 min 75 min 4.1 s 3.4 min # # # βk 0.08 0.06 0.04 0.02 0 1 2 3 4 B. Computational complexity The difference in the sample size also influences the computation time and memory usage when solving the sampled FHOCP in (7). Table II reports the average solver times and Table III the required memory allocation to formulate the sampled problem for different values of the prediction horizon N ∈ {8, 16, 24, 32, 40}. We note that prediction horizons of 40 hours are common in building control. In fact, [3] has observed that longer horizons improve the performance of the MPC controller. The timings in Table II are taken on a server running a 64-bit Linux operating system, equipped with 16-core hyperthreaded Intel Xeon processor at 2.6 GHz and 128 GB memory (RAM). We use the solver CPLEX interfaced via MATLAB 2013a. TABLE III R EQUIRED MEMORY ALLOCATION FOR CONSTRAINT MATRICES IN RMPC FOR nw = 1, 2, 3. Memory N N N N N nw = 1 new standard nw = 2 new standard 5 6 7 8 time [h] #: Out of memory error from MATLAB. nw = 3 new standard = 8 6.2 MB 60 MB 21 MB 187 MB 44 MB 383 MB = 16 81 MB 1.4 GB 287 MB 5.0 GB 614 MB 10.6 GB = 24 373 MB 9.8 GB 1.3 GB 36 GB 2.87 GB 77.5 GB = 32 1.1 GB 39 GB 4.0 GB 146 GB 8.77 GB 320 GB = 40 2.6 GB 116 GB 9.6 GB 437 GB 21 GB 963 GB Table II shows that our new bound dramatically reduces the computational times required to solve the sampled program. Moreover, due to the linear scaling of the sample size in the prediction horizon, we are now able to solve problems that could not be managed previously due to memory problems, indicated by “#” in Table II. The reason for that can be explained by Table III. For N = 32, for example, the sampled FHOCP based on the standard bound can not be solved, even for nw = 1. On the other hand, the sampled FHOCP based on the new bound remains manageable in practice even for nw = 3, allowing the problem to be solved on most hardware. C. Conservatism of the solution A poor bound on the sample size not only results in an excessive number of samples, but also introduces conservatism into the solution, both in terms of violation probability Fig. 1. Predefined βk (“red”), empirical estimate of βk using the new bounds (“black”), and empirical estimate of βk using the standard bounds (“blue”) for nw = 1. TABLE IV C OMPARISON OF EMPIRICAL COST FOR nw ∈ {1, 2, 3}. Cost N =8 nw = 1 new standard 100.87 102.24 nw = 2 new standard 103.92 107.70 nw = 3 new standard 103.94 107.95 and cost. Fig. 1 depicts the empirical estimate of βk over the prediction horizon for the case nw = 1. We observe that the empirical confidence level using the new bound is much closer to the predefined value of βk = 0.1 than the standard bound. Since the new bound results in more frequent violations (but remains smaller than a predefined acceptance level), it also results in a lower cost, as displayed in Table IV. From there we also see that the solution based on the new sample size results in lower cost, allowing us to save more energy in building control in all three cases. VI. C ONCLUSION In this paper, we have proposed new bounds on the sample sizes for RMPC problems with additive uncertainty and polyhedral constraints. The obtained bound results in a sample size that scales linearly in the prediction horizon instead of quadratically, as for previous bounds. This leads to less conservative solutions and dramatically reduces the computational cost. The building control case study has demonstrated that previously computationally infeasible problems can now be solved. A PPENDIX Proof of Proposition 1 Let k ∈ {1, . . . , N }. The part “ρk ≤ nu nw k(k−1) + 2 knu =: dk ” is the standard bound in (10) and follows immediately from causality of the system in (5), where the decision N −2 variables hk , . . . , hN −1 ; {Mk,j }k−1 j=0 , . . . , {MN −1,j }j=0 do not constrain the kth stage. For the second part, note that the kth constraint can be expressed as gk (M , h ) := k k F Ak x + Mk wk + Bk hk + Vk vk + Ek wk − f ≤ 0, where (Mk , hk ) are decision variables. Clearly, for all ˜ k ∈ hk : F Bk hk = 0 the constraint gk (Mk , h ˜k ) ≤ 0 h remains the same for all Mk . Hence, dim(Lk ) is at least dim(null(F Bk )) = knu − rank(F Bk ), where the equality is due to the Rank-Nullity Theorem. Thus, ρk is at most + rank(F ). dk − (knu − rank(F Bk )) ≤ nu nw k(k−1) 2 Proof of Proposition 2 (1) (S ) Let k ∈ {1, . . . , N }, and ωk := {wk , . . . , wk k } ⊂ knw R be a collection of Sk i.i.d. samples. Furthermore, let the set Ak [ωk ] ⊆ {1, . . . , Sk } index the active samples of the kth stage. Then, let us first prove the following supporting statement. Claim 1: For F ∈ R1×nx and f ∈ R, we almost surely have |Ak [ωk ]| ≤ knw . Proof: We define p(Mk ) := F (Mk +Ek ) and q(hk ) := F (Ak x + Bk hk + Vk vk ) − f , so that the ith sampled constraint at step k reads (i) p(Mk )wk + q(hk ) ≤ 0, ∀i ∈ {1, . . . , Sk }. (11) 2 Let us assume that p(Mk ) 6= 0 for any Mk . This implies that, for any pair (Mk , hk ), (11) can be interpreted as a halfspace in Rknw , separated by a halfplane of the form H(Mk , hk ) := wk ∈ Rknw p(Mk )wk + q(hk ) = 0 . From linear algebra we know that knw points in “general position” uniquely define a hyperplane in Rknw . By Assumption 1 it follows that, with probability one, any (1) (kn ) ¯k ,...,w ¯k w } collection of knw drawn samples ω ¯ := {w uniquely defines a hyperplane. Since hyperplanes are affine sets of dimension (knw − 1), their measure with respect to Rknw is zero. Therefore, the probability of another sample (kn +1) ¯k w w lying on the hyperplane defined by ω ¯ is zero. Since the argument holds for any hyperplane H defined by any (Mk , hk ), it also holds for the particular hyperplane H(M?k , h?k ) associated with the solution (M? , h? ) of (7). By inspection of (11) and the definition of an active sample (Definition 2), it can be seen that all active samples lie on the hyperplane H(M?k , h?k ). This concludes the proof of the claim. Let now F ∈ Rnf ×nx and f ∈ Rnf . It follows from the above claim that for each row of the constraint, with probability one we have at most knw active samples. Therefore, with nf constraints, with probability one we immediately get at most nf knw active samples. This concludes the proof. 2 For any M it can be verified that the condition p(M ) 6= 0 is satisfied k k whenever F E 6= 0. For most practical systems, the latter is satisfied because F E = 0 would imply that the constraint is not affected by uncertainty. 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