Approximate Methods for Solving Chance Constrained Linear Programs in Probability Measure Space

A risk-aware decision-making problem can be formulated as a chance-constrained linear program in probability measure space. Chance-constrained linear program in probability measure space is intractable, and no numerical method exists to solve this problem. This paper presents numerical methods to solve chance-constrained linear programs in probability measure space for the first time. We propose two solvable optimization problems as approximate problems of the original problem. We prove the uniform convergence of each approximate problem. Moreover, numerical experiments have been implemented to validate the proposed methods.


Introduction
Let X ⊂ R n be a compact set with the infinity norm defined by x ∞ = max i=1,...,n |x i |, x ∈ X .Denote D > 0 such that D := sup{ x − x ′ ∞ : x, x ′ ∈ X } for the diameter of X .In this paper, we assume that X can be specified as X = {x ∈ R n : g(x) ≤ 0 ng } where g : R n → R ng is a continuously differentiable constraint function.We have the following assumption on g throughout the paper.
Assumption 1 Cottle Constraint Qualification (CCQ) holds at any points in X .Namely, for any x ∈ X , there is a d ∈ R n such that ∇g(x) ⊤ d < 0 ng (1) holds.
Let B(X ) be Borel σ-algebra on metric space X .This paper uses B(•) to denote the Borel σ-algebra on a metric space.Notice that (X , B(X )) is a Borel space.Let µ be a Borel probability measure on B(X ).Let M (X ) be the space of Borel probability measures on metric space X .Let δ be a random vector with support ∆ ⊆ R s and P{•} be the probability measurable defined on Borel σ-algebra B(∆) on ∆.Let p(δ) be the probability density function associated with P{•}.Given a scalar function J : X → R, and a vector-valued function h : X × ∆ → R m , a chance-constrained linear program in probability measure space is formulated as: where α ∈ (0, 1) is a given probability level and F (x) is defined by Here, I{y} presents the indicator function written as I{y} = 1, if y ≤ 0, 0, if y > 0.
Note that F (x) is the probability of having h(x, δ) ≤ 0 for given x.Throughout the paper, we assume the following conditions on J(x) and h(x, δ).
For each x ∈ X , the following is assumed to be true: Besides, suppose that h(x, δ) has a continuous probability density function for every x ∈ X ; f.There exists L > 0 such that h(x, δ) − h(x ′ , δ) ∞ ≤ L x − x ′ ∞ , ∀x, x ′ ∈ X and ∀δ ∈ ∆, and |J(x) − J(x ′ )| ≤ L x − x ′ ∞ , ∀x, x ′ ∈ X .In fact, according to the content of pp.78-79 of [20], we can obtain the continuity of F (x) from Assumption 2.

Motivation
The motivation for addressing chance-constrained linear programs in probability measure space is from seeking an optimal stochastic policy for the optimal control problem with chance constraints, which is vital for the deployment of reliable autonomous systems by control algorithms that are robust to model misspecifications and for external disturbances [5,13,31].The optimal control problem with chance constraints aims at maximizing a reward function or minimizing a cost function with the constraints that the system state should locate in the safe area with a required probability.The deterministic policy has a fixed value in the decision domain at every time index.In contrast, the stochastic policy provides a probability measure on the decision domain at every time index.The deterministic policy can be regarded as a particular case of the stochastic policy by concentrating the probability measure on a fixed value in the decision domain.The existing techniques for addressing optimal control problems with chance constraints do not touch the essential parts of the problem and may require application-specific assumptions.For example, [13,19] enforces pointwise chance constraints that ensure the independent satisfaction of each chance constraint at each time step, which leads to a more conservative solution.In general, joint chance constraints are desired, which requires all chance constraints to be satisfied jointly at all times.However, it is challenging to tackle the joint chance-constrained optimal control problem since the distribution of the state trajectory needs to be considered fully.It is possible to address the joint chance-constrained optimal control problem by using Boole's inequality [5,26,34] or performing robust optimization within the bounded model parameters obtained by specifying a confident set [21].However, these two methods are conservative.More investigations from the viewpoint of optimization theory should be addressed to enhance new breakthroughs for optimal control with chance constraints.Obtaining open-loop stochastic optimal policies under chance constraints can be essentially written as a chance-constrained linear program in probability measure space [36].Open-loop stochastic policies mean that the stochastic policies only depend on the initial state.Unfortunately, there is still no research on solving chance-constrained linear programs in probability measure space to our knowledge.Investigating the chance-constrained linear programs in probability measure space is vital, which can give more insights into optimal control with chance constraints.

Related Works
Optimization with finite chance constraints in finite-dimensional vector space is generally challenging due to the non-convexity of the feasible set and intractable reformulations [30,12].The existing research has two major streams: (1) give assumptions that the constraint functions or the distribution of random variables have some special structure, for example, linear or convex constraint functions [25], finite sample space of random variables [23], elliptically symmetric Gaussian-similar distributions [1], or (2) extract samples [8,9,22,27,10,35,28,32] or use smooth functions [15] to approximate the chance constraints.For sample-based methods, the most famous approach in the control field is scenario approach [8,9,10,31,11].Scenario approach generates a deterministic optimization problem as the approximation of the original one by extracting samples from the sample space of random variables.The probability of the feasibility of the approximate solution rapidly increases to one as the sample number increases.However, the convergence of the optimality of the approximate solution is not discussed.In another sample-based method, the sample-average approach [22,15,32,28], both feasibility and optimality of the approximate solution are presented.However, neither scenario approach nor sample-average approach can be directly used to solve chance-constrained linear programs in probability measure space since the deduction of the convergence of either scenario approach or sample-average approach assumes that the dimension of the decision variable must be finite.
Optimization with chance/robust constraints in finite-dimensional vector space is also intensively studied, in which the number of chance constraints is infinite [2,3,14,4].In [2], the generalized differentiation of the probability function of infinite constraints is investigated.The optimality condition with an explicit formulation of subdifferentials is given.In [3], the variational tools are ap-plied to formulate generalized differentiation of chance/robust constraints.The method of getting the explicit outer estimations of subdifferentials from data is also established.An adaptive grid refinement algorithm is developed to solve the optimization with chance/robust constraints in [4].However, the above research on optimization with chance/robust constraints in finite-dimensional vector space can prove convergence only when the dimension of the decision variable is finite.
Recently, chance constraints in infinite dimensions have attracted a lot of attention.In [29,16,17], some essential properties, such as convexity and semicontinuity, are generalized into the chance constraints in infinite dimensions.However, the results in [29] assume that the random variable should have a log-concave density to ensure the semicontinuity.In [17], the continuity of the probability function as chance constraints is proved under the assumption of continuous random distributions.The properties of chance constraints in infinite dimensions are crucial to constructing the optimality condition and implementing convergence analysis for optimization with chance constraints in infinite dimensions.In [16], chance-constrained optimization of elliptic partial differential equation systems is addressed by inner-outer approximation.It proves that the inner and outer approximation converges to the original problem and can provide approximate solutions with ensured convergence.However, the proof of the convergence requires the assumption that the state domain is convex.Besides, it concerns the specific problem in partial differential equation systems.

Overview of Proposed Method and Contributions
This paper extends the sample-based approximation method to solve chanceconstrained linear programs in probability measure space.We show the relationship between chance-constrained optimization in finite-dimensional vector space and chance-constrained linear program in probability measure space.By solving a chance-constrained linear program in probability measure space, we can obtain a stochastic policy to improve the expectation of the optimal value further.We also show that the optimal objective values of the chance-constrained linear program in probability measure space and chanceconstrained optimization in finite-dimensional vector space are equal if the constraints involved with random variables are required to be satisfied with probability 1. Namely, in this case, by concentrating the probability measure on an optimal solution of chance-constrained optimization in finite-dimensional vector space, we can obtain an optimal measure for the chance-constrained linear program in probability measure space.Besides, a sample approximate problem and a Gaussian mixture model approximate problem of problem P α are proposed, by solving which the approximate solution of P α can be obtained.The convergences of both approximate problems are investigated.Numerical examples are implemented to validate the proposed methods.
Chance-constrained linear program in probability measure space involves chance constraints in infinite dimensions.Our work differs from the [29,17] in that our purpose is to provide numerical methods for solving chance-constrained linear programs in probability measure space.The properties of chance constraints in infinite dimensions are essential for convergence analysis.
The rest of this paper is organized as follows: Section 2 presents two approximate problems of P α and gives the main results on the convergence for each approximate problem.The proofs of the main results are presented in Section 3. Section 4 presents the results of two numerical examples, which show the effectiveness of our proposed methods.Section 5 concludes the whole paper.

Main Results
This section introduces two approximate problems of P α .We also present the convergence for each approximate problem.The proofs are presented in Section 3.

Chance Constrained Optimization in Finite Space
Chance-constrained optimization Q α is an optimization problem with chance constraints in a finite-dimension vector space.The problem is written as where α ∈ (0, 1) is a given probability level.Let X α := {x ∈ X : F (x) ≥ 1 − α} be the feasible domain of Q α .Denote Jα := min{J(x) : x ∈ X α } for the optimal objective value of Q α and X α := {x ∈ X α : J(x) = Jα } for the optimal solution set of Q α .We have the following assumptions over Q α throughout the paper.
Assumption 3 There exists a globally optimal solution x of Q α such that for any ε > 0 there is x ∈ X such that 0 The existence of chance constraints gives rise to several difficulties.First, the structural properties of h(x, δ) might not be passed to F (x) ≥ 1 − α.The feasible set X α can be equivalently obtained as where .., m are all linear in x for every δ ∈ ∆, the feasible set X α may not be convex due to the infinite union operations.Second, it is difficult to obtain a tractable analytical function F (x) to describe the constraint or find a numerically efficient way to compute it.In most applications, p(δ) is unknown, and only samples of δ are available.We briefly review the samplebased approximation method presented in [22,27,28].Let D N = {δ (1) , ..., δ (N ) } be a set of samples randomly extracted from ∆ where N ∈ N. Suppose the sample extraction is independently and identically distributed.Then, D N can be regarded as a random variable from the augmented sample space ∆ N with probability measure P N {•} defined on the Borel σ-algebra B(∆ N ).Giving D N , ǫ ∈ [0, α), and γ > 0, a sample average approximate problem of Q α , defined by Qǫ,γ (D N ), is written as: The feasible region of Qǫ,γ (D N ) is defined by Denote Jǫ,γ (D N ) := min{J(x) : x ∈ Xǫ,γ (D N )} for the optimal objective function value of Qǫ,γ (D N ) and Xǫ,γ (D N ) := {x ∈ Xǫ,γ (D N ) : J(x) = Jǫ,γ (D N )} for the optimal solution set of Qǫ,γ (D N ).We can regard Jǫ,γ (D N ) as a function Jǫ,γ : ∆ N → R for given ǫ and γ.Since D N is a random variable from ∆ N , Jǫ,γ (D N ) is consequently a random variable.The sets Xǫ,γ (D N ) and Xǫ,γ (D N ) also depend on D N and can be regarded as Xǫ,γ : and Xǫ,γ : ∆ N → B(X ).Xǫ,γ (D N ) and Xǫ,γ (D N ) are called random sets [24].
According to Lemma 1, we can obtain the solution of Q α with probability 1 when N → ∞, ǫ → α, γ → 0. A natural question arises: can we use the solution of Q α to obtain an optimal probability measure for P α ?Let xα ∈ X α be an optimal solution of Q α .Notice that we have {x α } ∈ B(X ) and thus it is possible to define a probability measure µ xα which satisfies that Thus, µ xα is a feasible solution for P α with objective value as Jx,α .However, µ xα is not sure to locate in A α .Only when α = 0, we have µ xα ∈ A α .Notice that it is not ensured that the set X α is a Borel measurable set.However, it is possible to find a subset X m α ⊆ X α that is Borel measurable.A particular example is to choose X m α = {x α } where xα ∈ X α is one element in the optimal solution set.In this paper, without loss of generality, we assume that X α is Borel measurable for all α ∈ [0, 1].Besides, we also assume that X 0 = ∅.Then, X α = ∅ holds for all α ∈ [0, 1].The above content is formally summarized in Theorem 1.
The proof of Theorem 1 is given in Section 3.1.
Remark 1 Theorem 1 implies that deterministic policy is optimal for robust optimal control where α = 0.

Sample-based Approximation
Let X in be the set of all interior points of X .By using Hit-and-Run algorithm [33] and Billiard Walk algorithm [18], uniform samples can be generated from X in .For a positive integer S ∈ N, let C S := {x (1) , ..., x (S) } be a set of uniform samples independently extracted from X in .The set C S is an element of the augmented space X in S .Since each element x (i) , i = 1, ..., S in C S is extracted independently, we define a S-fold probability P S uni (= P uni × ... × P uni , S times) in X in S .Here, P uni is the probability measure of uniform distribution on X in .
With C S and D N , we can obtain a sample approximate problem of P α defined by Pα (C S , D N ): where Let μα ∈ Ãα (C S , D N ) be an optimal measure.The optimal value Jα (C S , D N ) depends on C S and D S , and thus it can be regarded as a function Jα : The deduction of the convergences of Jα (C S , D N ) and Ãα (C S , D N ) requires another assumption on P α .We state the assumption after a brief introduction of weak convergence.
Define a space of continuous R-valued functions by It is able to define a metric on C (X , R) by where f ∞ is defined as The metric τ (•, •) turns C (X , R) into a complete metric space.
The weak convergence of probability measures is defined as follows [7].
Since X is compact, M (X ) can be proved to be weakly compact by Riesz representation theorem [7].Therefore, giving any sequence of {µ k } ∞ k=0 ⊂ M (X ), there is a subsequence which converges weakly to some µ ∈ M (X ) in the sense of Definition 1.By Assumption 2, we have that J(x) and F (x) are continuous with respect to x.Therefore, if {µ k } ∞ k=0 converges weakly to µ, (9) also holds for J(x) or F (x).We give the following assumption on Problem P α .
Assumption 4 There exists a globally optimal solution µ * ∈ A α of Problem P α such that for any δ > 0 there is µ ∈ M (X ) such that X F (x)dµ > 1 − α and W(µ, µ * ) ≤ δ, where W(µ, µ * ) is defined by As S, N → ∞, the convergence analysis on Jα (C S , D N ) and Ãα (C S , D N ) is summarized in Theorem 2.

Gaussian Mixture Model-based Approximation
Another option of approximation is to constrain the choice of µ in M θ (X ) ⊆ M (X ).Here, M θ (X ) is defined as where the probability density function p θ (x) is written as Here, ω i ∈ [0, 1], ∀i = 1, .., L, L i=1 ω i = 1, and φ(x, m i , Σ i ) is multivariate Gaussian distribution written by The notation θ denotes the parameter vector, including all the unknown parameters in ω i , m i , Σ i , ∀i = 1, ..., L. Denote the dimension of θ as n θ .The feasible domain of θ is denoted by Then, given a data set D N and the number of Gaussian distributions L, we can obtain a Gaussian mixture model-based approximate problem of P α defined by Pα (L, D N ): Denote the feasible set of Pα (L, D N ) as and the optimal objective value as Besides, the optimal solution set is The optimal objective value Ĵα (L, D N ) depends on the number of used Gaussian models and the data set D N .Since data set D N is essentially random variable with support ∆ N , Ĵα (L, D N ) is also a random variable.The set Θα (L, D N ) is a random set.As L, N → ∞, optimality and feasibility of using the optimal solution of Pα (L, D N ) are summarized in Theorem 3.
Theorem 3 Consider Problem P α with α > 0. Suppose Assumptions 1, 2, 3, and 4 hold.As L, N → ∞, we have with probability 1. Besides, let θ ∈ Θα (L, D N ) be an optimal solution of Pα (L, D N ).The corresponding probability density function is p θ(x) and the obtained probability measure is The proof of Theorem 3 is given in Section 3.3.
Notice that X 0 is a Borel measurable set.Let µ * 0 (•) ∈ A 0 be an optimal probability measure for P 0 and suppose µ * 0 (X 0 ) < 1 for deriving the contradiction.Thus, µ * 0 (X \ X 0 ) > 0. The corresponding objective function is Theorem 4 For given sample sets C S and D N , define two functions of µ ∈ U S as I{h(x (i) , δ (j) )}.
Then, Gα (µ, C S , D N ) uniformly converges to Gα (µ, C S ) on U S w.p.1, i.e., Proof (Theorem 4).For any given x (i) , I{h(x (i) , δ)} is a measurable function of δ.According to the strong Law of Large Numbers (LLN) [6], we have Thus, for every µ ∈ U S , we have Uniform convergence is ensured since the set U S is compact.
Nextly, we show that Jα (C S , D N ) and Ãα (C S , D N ) converge to Jα (C S ) and Ȃα (C S ), respectively, with probability 1 as N → ∞.
Theorem 5 Consider Problem P α with α > 0. Assume that there exists a Proof (Theorem 5).The set U S is a compact set.The objective function ) is a constant value within [0, 1] for a fixed x (i) , which makes the constraint function Gα (µ, C S ) a linear function of µ ∈ U S .Therefore, Pα (C S ) is a linear program.Due to the assumption that there exists t=1 be a subsequence converging to μ.By Theorem 4, we have Therefore, Gα (μ, C S ) ≥ 1 − α and μ is feasible for problem Pα (C S ) which implies Since this is true for an arbitrary point of {μ k } ∞ k=1 in the compact set U S , we have Besides, we know that there exists a globally optimal solution of Pα (C S ), µ * , such that for any ε > 0 there is µ ∈ U such that 0 < µ − µ * ≤ ε and Gα (µ, C S ) > 1 − α.Namely, there exists a sequence {μ t } ∞ t=1 ⊆ U that converges to an optimal solution µ * such that Gα (μ t , C S ) > 1 − α for all t ∈ N. Notice that Gα (μ t , C S , D N k ) converges to Gα (μ t , C S ) w.p.1.Then, for any fixed t, ∃K(t) such that Gα (μ t , C S , D N k ) ≥ 1 − α for every k ≥ K(t) w.p.1.We can assume K(t) < K(t + 1) for every t and define the sequence {μ k } ∞ k=K(1) by setting μk = μt for all k and t with K(t) ≤ k < K(t + 1).Then, With ( 20) and ( 21), we conclude that Jα (C S , D N ) → Jα (C S ) w.p.1 as N → ∞.
The proof of Ãα (C S , D N ) → Ȃα (C S ) can be referred to Theorem 5.3 of [30].
Nextly, we show that Jα (C S ) converges to Jα with probability 1 as S increases.
Proof (Theorem 6).The outline of the proof of Theorem 6 is summarized as follows: A. Prove that the limit of lower bound of Jα (C S ) is larger than Jα by (23); B. Prove that the limit of upper bound of Jα (C S ) is smaller than Jα by (38); B1.Find a sequence {µ k } ∞ k=1 converges weakly to an optimal solution µ * of P α ; B2.Show that X F (x)dµ k (x) and X J(x)dµ k (x) can be approximated by using discrete probability measure on C S , which refers to ( 34) and ( 35); B3.Show that optimal discrete probability measure on C S for Pα (C S ) has a smaller objective value than the discrete probability measure for approximating any µ k in B2.Then, we obtain (38).
Then, we give the details of the proof.
For any discrete probability measure µ S ∈ Fα (C S ), we have Thus, µ S ∈ M α (x).Then, it holds that Furthermore, with probability 1, we have Assumption 4 implies that there exists a sequence {µ k } ∞ k=1 ⊆ M (X ) that converges weakly to an optimal solution µ * such that for all k ∈ N. Since {µ k } ∞ k=1 converges weakly to µ * , we have Notice that Jα = X J(x)dµ * (x) by (3).For any given

Sample in
Fig. 1 The intuitive explanation of the relationship between C S and Ck

S
Let Ck S := {x k , ..., x( S) k } be a sample set obtained by sampling from X according to probability measure µ k .By Law of Large Numbers (p.457 of [30]), for any f ∈ C (X , R), as S → ∞, with probability 1, we have Since J(•) and F (•) are also elements in C (X , R), (26) also holds by replacing f (•) by either J(•) or F (•). Namely, for any ε1 , there exists Sl (ε J ) such that, if S ≥ Sl (ε J ), with probability 1, the followings hold: On the other hand, according to Lemma 2, as S → ∞, for any s ∈ {1, ..., S} and εr > 0, with probability 1, there exists a sample x (is) ∈ C S := {x (1) , ..., x (S) } such that With a little abuse of notation, let x (is) be the closest sample to x(s) k , namely, Define a set I S := {i 1 , ..., i S } as the set of index corresponding to x (is) .Without loss of generality, we assume that x (is) = x (js) if i s = j s, i s, j s ∈ I S .The intuitive explanation of the relationship between C S and Ck S is illustrated in Figure 1.
it only needs to solve a linear program.In this example, since it is one dimension, the required sample number for obtaining good samples in Sample or approximating probability integration in GMM is few.It can achieve acceptable accuracy with only 50 samples.However, if the dimension of x increases, the "Curse of Dimensionality" will emerge.We will show it in the second example.

Quadrotor System Control
The second example considers a quadrotor system control problem in turbulent conditions.The control problem is expressed as follows: where A, B(m), d(x t , ϕ) are written by , and ∆t is the sampling time, the state of the system is denoted as x t = [p x,t , v x,t , p y,t , v y,t ] ∈ R 4 , the control input of the system is u t = {u x,t , u y,t } within U := {u t ∈ R 2 : −10 ≤ u x,t ≤ 10, −10 ≤ u yt ≤ 10}, and the state and control trajectories are denoted as x = (x t ) T t=1 and u = (u t ) T −1 t=1 .The system starts from an initial point x 0 = [−0.5,0, −0.5, 0].The system is expected to reach the destination set X goal = {x ∈ R 4 | (p x − 10, p y − 10) ≤ 2} at time T = 10 while avoiding two polytopic obstacles O shown in Fig. 3  For the cost function, we adopt Results are shown in Fig. 3 for different methods by setting α as 15%.Fig. 3 shows 5,000 Monte-Carlo (MC) simulations of the quadrotor system using the open-loop policy computed using Dirac-Delta (ǫ = α, γ = 0.01, N = 2000), Sample (S = 5.1 × 10 6 , N = 2000), and GMM (L = 6, N = 2000).When using Dirac-Delta, the algorithm gives a deterministic control policy that satisfies the desired success probability 1 − α.When using Sample, or GMM, the algorithm gives a stochastic control policy that satisfies the desired success probability 1−α.The control inputs that generate trajectories passing through the riskier middle corridor between the obstacles are selected randomly for the stochastic control policies.The costs by using Sample and GMM are reduced by 8.2% and 7.9% compared to using Dirac-Delta.This shows that our approach can compute a better policy that solves the problem than a deterministic policy.
A more comprehensive comparison between the GMM-based and samplebased approximations is plotted in Figure 4. Five cases are considered with different sample numbers for extracting the control input.Figure 4 (a) shows that the two algorithms similarly reduce the optimal objective function value.Figure 4 (b) shows each case's used sample number S of decision variables.By comparing Figure 4 (a) and (b), we can see that enough samples are required to ensure the performance of the approximations.As shown in Figure 4 (c), the computation time increases dramatically as the sample number increases.In this comparison, for GMM, we choose L = 6, and the probability integration is approximated by using the same samples of Sample.The computation time of GMM is even longer than Sample.One way to decrease the computation time of GMM is to develop fast algorithms for probability integration.We leave this for future work.In this example, the dimension of the decision variable is 20.If the dimension increases, the required sample number will increase, and the computation time will consequently increase for Sample and GMM.We leave the issue of the "Curse of Dimensionality" for future work.

Conclusions
In conclusion, the chance-constrained linear program in probability measure space has been addressed using sample approximation or function approximation.We establish optimization problems in finite vector space as approximate problems of chance-constrained linear programs in probability measure space.By solving the approximate problems, we can obtain the approximate solution of the chance-constrained linear program in probability measure space.Numerical examples have been implemented to validate the performance of the proposed method.Future work will be focused on the following points: -To implement sample approximation method Pα (C S , D N ), samples of decision variable are required.As the dimension of the decision variable increases, the required sample number for a good approximation will also increase, bringing the issue of the "Curse of dimensionality."To overcome the issue of the "Curse of Dimensionality," it is important to develop efficient sampling algorithms to get "good but small samples" to ensure good approximation performance and mitigate the computation burden; -For Gaussian mixture model-based approximation method Pα (L, D N ), the remaining issue is how to approximate the probability integration by fast algorithms when the problem is with complex cost function and constrained functions in high dimension space.
as the feasible set of Pα (C S , D N ).Denote the optimal objective function value as Jα (C S , D N ) := min{ S i=1 J(x (i) )µ(i) : µ ∈ F α (C S , D N )}.Denote the optimal solution set for Pα (C S , D N ) as Ãα (C S , D N ) := {µ ∈ F α (C S , D N ) : α and thus Ȃα (C S ) is nonempty.Since Gα (µ, C S , D N ) converges to Gα (µ, C S ) w.p.1 by Theorem 4, there exists N 0 large enough such that Gα (µ, C S , D N0 ) ≥ 1 − α w.p.1.Because Gα (µ, C S , D N0 ) is a linear function of µ and U S is compact, the feasible set of Pα (C S , D N0 ) is compact as well, and hence Ãα (C S , D N0 ) is nonempty w.p.1 for all The dynamics are parametrized by uncertain parameter vector δ t = [m, ϕ] ⊤ , where m > 0 represents the system's mass and ϕ > 0 is an uncertain drag coefficient.The parameter vector δ of the system is uncorrelated random variables such that (m − 0.75)/0.5 ∼ Beta(2, 2) and (ϕ − 0.4)/0.2∼ Beta(2,5), where Beta(a, b) denotes a Beta distribution with shape parameters (a, b).ω t ∈ R 4 is the uncertain disturbance at time step t, which obeys multivariate normal distribution with zero means and covariance matrix

Fig. 3
Fig. 3 Solutions from different methods for the tolerable failure probability threshold α = 15%.Blue trajectories from Monte-Carlo (MC) simulations denote feasible trajectories that reach the goal set X goal and avoid obstacles O. Red trajectories violate constraints: (a) Dirac-Delta (MC = 11.6% represents that the violation probability is 11.6% in the MC simulations); (b) Sample (MC = 12.8% that the violation probability is 12.8% in the MC simulations); (c) GMM (MC = 11.2%represents that the violation probability is 11.2% in the MC simulations).

Fig. 4
Fig. 4 The statistics of the control performance: (a) Reduction of cost; (b) Required samples; (c) Computation time.