Distributionally Robust Joint Chance Constrained Problem under Moment Uncertainty

Abstract

We discuss and develop the convex approximation for robust joint chance constraints under uncertainty of first- and second-order moments. Robust chance constraints are approximated by Worst-Case CVaR constraints which can be reformulated by a semidefinite programming. Then the chance constrained problem can be presented as semidefinite programming. We also find that the approximation for robust joint chance constraints has an equivalent individual quadratic approximation form.

1. Introduction

Chance constraints, also called probabilistic constraints in the literature, have a long history in stochastic programming and are the direct way to treat stochastic data uncertainty. With a large class problem involved, it can be formulated in the following form: min_{𝐱∈ℝ𝑛}𝐜^𝑇𝐱s.t.ℙ_𝝃𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃≤0,∀𝑖=1,2,…,𝑚≥1−𝜖,𝐱∈𝜒, (1) where 𝝃 is undetermined vector, 𝐱∈𝜒 is the decision vector, 𝜒∈ℝ^𝑛 is a bounded, convex closed set which can be represented by a set of additional deterministic semidefinite constraints, and 𝐜∈ℝ^𝑛 is the deterministic cost vector. The chance constraint in the above problem requires all the 𝑚 uncertainty-affected constraints to be jointly feasible with probability at least 1−𝜖, where 𝜖 is a desired safety factor given by the decision-maker. Problem (1) can be classified as an individual chance constrained problem when 𝑚=1 and a joint chance constrained problem when 𝑚≥2.

This problem has been considered by Charnes et al. [1], Miller and Wagnet [2], and Prékopa [3]. Due to the feasible set of problems, (1) is typically nonconvex and sometimes even disconnected; at the same time, the full and accurate information about the probability distribution ℙ_𝝃 cannot be required; the problem has not found interest and wide application in theory and practice for a long time.

One interesting issue on the chance constrained problem is to determine the distributional condition under which the problem can be reformulated as tractable convex programming. Alizadeh and Goldfarb [4], Calafiore and EI Ghaoui [5], Erdoğan and Iyengar [8], and Zymler et al. [6] showed that the chance constraint can be reformulated as tractable convex and cone constraints under some special exact information, respectively. However, the computation of chance constrained problems for general case is intractable. Nemirovski and Shapiro [7] pointed out that computing the probability of a weighted sum of uniformly distributed variables being nonpositive is already NP-hard. The intractability of a chance constrained problem using exact information has spurred recent interest in robust optimization in which data uncertainties are controlled in several types of uncertainty sets [9, 10]. Moreover, robust optimization generally needs segmental information on probability distributions such as known supports and covariances. Zymler et al. [6] showed an exact LMIS reformulation for chance constrained problem which can be computed by solving a tractable SDP under known first- and second-order moments information. Usually, in practice, one only has limited information about the probability distribution driving the uncertain parameters, involved in the decision-making process. It implies that we cannot obtain the exact moments information. In this paper, we extend the framework of Zymler et al. [6] to the case of inexact known moments information. We use the following two constraints parameterized by 𝜂≥0 which rely on empirical estimates of the mean 𝝁₀ and covariance matrix 𝚺₀ of the random vector to construct the distributional information: 𝝁₀−𝜼≤𝔼[𝝃]≤𝝁₀𝚺+𝜼,₀+𝝁₀−𝜼𝝁₀−𝜼^𝑇⪯𝔼𝝃𝝃^𝑇⪯𝚺₀+𝝁₀+𝜼𝝁₀+𝜼^𝑇. (2) It describes how likely 𝝃 is to be close to 𝝁₀ controlled by the vector 𝜼. At the same time, the parameter 𝜼 provides natural means of quantifying one’s confidence in 𝝁₀ and 𝚺₀. In what follows, we consider the latest problem in this paper under the distributional set 𝔻ℝ^𝐿,𝝁₀,𝚺₀=,𝜼ℙ∈𝒫∶ℙ𝝃∈ℝ^𝐿=1andsatisfies(2), (3) where 𝔻 is the set of all probability distributions on the measurable space (ℝ^𝐿,𝔹) and 𝔹 is the Borel 𝜎-algebra on ℝ^𝐿. To this end, let 𝒫 denote the set of all probability measures corresponding to 𝔻.

Now let us consider the following distributionally robust chance constrained program: inf_ℙ∈𝒫ℙ𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃≤0,∀𝑖=1,2,…,𝑚≥1−𝜖. (4)

It is easy to verify that the feasible set of the above inequality is a subset of the feasible set of problem (1). This yields the following distributionally robust chance constrained program: min_{𝐱∈ℝ𝑛}𝑐^𝑇𝑥s.t.inf_ℙ∈𝒫ℙ𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃≤0,∀𝑖=1,2,…,𝑚≥1−𝜖,𝐱∈𝜒, (5) which constitutes a conservative approximation for problem (1) in the sense that it has the same objective functions but a smaller feasible set.

In this paper, we discuss approximations for distributionally robust joint chance constraints under inexact information of first- and second-order moments, which extends the framework of Zymler et al. [6] from exact known first- and second-order moments. We prove that it can be approximated by a Worst-Case CVaR constraint which can be represented as a semidefinite programming. Then, we show that distributionally robust joint chance constrained problem has an equivalent quadratic approximation form. The advantage of the new framework lies on the limited information about distribution and the tractable convex approximation.

2. Distributionally Robust Joint Chance Constraints for LP

Let 𝑔(𝐱,𝝃) be the chance constraint with the decision vector 𝐱 and the random vector 𝝃. Now, we consider the general robust individual chance constraint inf_ℙ∈𝒫ℙ𝑔(𝐱,𝝃)≤0≥1−𝜖, (6) whose feasible set is denoted by Π_𝑔={𝑥∈ℝ^𝑛∶inf_ℙ∈𝒫ℙ(𝑔(𝑥,𝝃)≤0)≥1−𝜖}. Shapiro et al. [11] showed that the feasible set is convex if the probability distribution function is 𝛼-concave and 𝑔(𝐱,𝝃) are quasiconcave jointly in both arguments. Unfortunately, the above chance constraint is not necessarily convex in the decision variables 𝐱.

It is well-known that CVaR method, popularized by Rockafellar and Uryasev [12], is the tightest convex approximation to the general individual probabilistic constraint (see, e.g., [7, 13]). Then, using the CVaR method, we get the tractability and convex approximation of the individual above chance constraint. The corresponding conditional value-at-value is defined as follows: ℙ-CVaR_𝜖𝑔(𝐱,𝝃)=inf_𝛽∈ℝ1𝛽+𝜖𝔼_ℙ𝑔(𝐱,𝝃)−𝛽⁺. (7)

Next, we show that CVaR can be used to construct convex approximation for general chance constraint. By definition (7), we have ℙ𝑔(𝐱,𝝃)≤ℙ-CVaR_𝜖𝑔(𝐱,𝝃)≥1−𝜖. (8) Then ℙ-CVaR_𝜖𝑔(𝐱,𝝃)≤0⟹ℙ𝑔(𝐱,𝝃)≤0≥1−𝜖. (9) Thus, from (9), we obtain sup_ℙ∈𝒫ℙ-CVaR_𝜖𝑔(𝐱,𝝃)≤0⟹inf_ℙ∈𝒫ℙ𝑔(𝐱,𝝃)≤0≥1−𝜖. (10)

Therefore, the worst-case constraint on the left hand side constitutes a conservative approximation for the distributionally robust chance constraint on the right hand side. The above discussion makes us define a feasible set as follows: Θ_𝑔={𝐱∈ℝ^𝑛∶sup_ℙ∈𝒫 ℙ-CVaR_𝜖(𝑔(𝐱,𝝃))≤0}.

Theorem 1. The feasible set Θ_𝑔 constitutes a conservative approximation for Π_𝑔, in which Θ_𝑔⊆Π_𝑔.

Lemma 2. For any fixed 𝑥, let 𝑔(𝑥,𝝃)∶ℝ^𝑛×ℝ^𝐿→ℝ be a measure function and F-integrable for all 𝐹∈𝔻(ℝ^𝐿,𝝁₀,𝚺₀,𝜼). We define the worst-case expectation problem as follows: Ψ(𝐱,𝜼)=sup_{𝐹∈𝔻ℝ𝐿}max0,𝑔(𝐱,𝝃)𝑑𝐹(𝝃). (11) Consequently, the problem can be rewritten into the following: inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0𝑠.𝑡.𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝝃1⪰0,_𝑇𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0×𝝃1≥𝑔(𝐱,𝝃),∀𝝃∈ℝ^𝐿, (12) where Ω₊=𝚺₀+𝝁₀+𝜼𝝁₀+𝜼^𝑇𝝁₀𝝁+𝜼_𝑇0+𝜼^𝑇1,Ω₋=𝚺₀+𝝁₀−𝜼𝝁₀−𝜼^𝑇𝝁₀𝝁−𝜼_𝑇0−𝜼^𝑇1. (13)

Proof. The worst-case expectation Ψ(𝐱,𝜼) can equivalently be unfolded as sup_{𝐹∈𝒟ℝ𝐿}max0,𝑔(𝐱,𝝃)𝑑𝐹(𝝃)s.t._ℝ𝐿(𝝃)𝝁𝑑𝐹=1,₀−𝜼≤_ℝ𝐿𝝃𝑑𝐹(𝝃)≤𝝁₀𝚺+𝜼,₀+𝝁₀−𝜼𝝁₀−𝜼^𝑇⪯_ℝ𝐿𝝃𝝃^𝑇𝑑𝐹(𝝃)⪯𝚺₀+𝝁₀+𝜼𝝁₀+𝜼^𝑇. (14) We can formulate the Lagrangian dual problem of the above problem which takes the following form: inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}𝑦₀+𝐲^𝑇₁𝝁₀+𝜼−𝐲^𝑇₂𝝁₀+𝐘−𝜼₁,𝚺_𝟎+𝝁₀+𝜼𝝁₀+𝜼^𝑇−𝐘₂,𝚺_𝟎+𝝁₀−𝜼𝝁₀−𝜼^𝑇s.t.𝑦₀+𝐲₁−𝐲₂^𝑇𝐲₁+𝝃𝝃^𝑇,𝐘₁−𝐘₂≥max0,𝑔(𝑥,𝝃),∀𝝃∈ℝ^𝐿,𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂⪰0, (15) where 𝑦₀∈ℝ, 𝐲₁,𝐲₂∈ℝ^𝐿, and 𝐘₁,𝐘₂∈𝕊^𝐿 are the dual variables for the constraints, respectively. It is obvious that the conditions on 𝜼 and 𝚺₀ are sufficient to ensure that the Dirac measure 𝛿_𝝁0 lies in the relative interior of the feasible set of the above original problem. It implies that Ψ(𝐱,𝜼) must be equal to the optimal value of the above dual problem. Rewriting the above model into the form of LMIS, we have inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0𝝃1s.t.^𝑇𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0×𝝃1≥max𝑔(𝐱,𝝃),0,∀𝝃∈ℝ^𝐿,𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂⪰0. (16)
Actually, we complete this proof when we divide the first inequality into two parts.

We define the feasible set Π^JCC of the distributionally robust joint chance constraint as Π^JCC=𝑥∈ℝ^𝑛∶inf_ℙ∈𝒫ℙ𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇.𝝃≤0,∀𝑖=1,2,…,𝑚≥1−𝜖 (17) A popular approximation for Π^JCC is based on Bonferroni’ inequality, but this method can be overly conservative even if 𝜖 is divided among the 𝑚 individual chance constraints, and Chen et al. [14] give an example which highlights this shortcoming. In order to mitigate the potential overconservatism of the Bonferroni approximation, we proposed the following approximation based on a combined inequality: inf_ℙ∈𝒫ℙmax_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃≤0≥1−𝜖. (18)

The problem becomes an individual chance constraint which can be conservatively approximated by a Worst-Case CVaR constraint. We define the approximated set as follows: Θ^JCC=𝐱∈ℝ^𝑛∶sup_ℙ∈𝒫ℙ-CVaR_𝜖×max_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃≤0, (19)which is a tight approximation for Π^JCC. The following theorem proves that the set Θ^JCC has a tractable reformation in terms of LMIS and therefore promises to get out a convex approximation for Π^JCC.

Theorem 3. The feasible set Θ^𝐽𝐶𝐶 can be written asΘ^𝐽𝐶𝐶=∃𝑦₀∈ℝ,𝐲₁,𝐲₂∈ℝ^𝐿,𝐘₁,𝐘₂∈𝕊^𝐿,𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂1⪰0,𝛽+𝜖Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0≤0𝐱∈ℝ^𝑛:𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0−𝟎12𝐰_𝑖(𝐱)12𝐰^𝑇_𝑖(𝐱)𝑤⁰_𝑖(𝐱).−𝛽⪰0,∀𝑖=1,2,…,𝑚. (20)

Proof. We find that the constraint 𝐱∈Θ^ICC is coinciding with Υ(𝐱)≤0, where Υ(𝐱)=sup_ℙ∈𝒫ℙ-CVaR_𝜖max_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃=sup_ℙ∈𝒫inf_𝛽∈ℝ1𝛽+𝜖𝔼_ℙmax_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃−𝛽⁺. (21)
Actually, we know that 𝛽+(1/𝜖)𝔼_ℙ[(max_{𝑖=1,2,…,𝑚}{𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃}−𝛽)⁺] is real valued, convex in 𝛽 and linear in ℙ, where 𝒫 is weakly compact [15]. Then, interchanging the sup_ℙ∈𝒫 and inf_𝛽∈ℝ operators leads to an equivalent formulation of the Worst-Case CVaR problem: inf_𝛽∈ℝ1𝛽+𝜖sup_ℙ∈𝒫𝔼_ℙmax_{𝑖=1,2,…,𝑚}{𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃}−𝛽⁺. (22) Firstly, we deal with the subordinate maximization problem in (22); we can derive an SDP reformulation for the worst-case expectation problem sup_ℙ∈𝒫𝔼_ℙmax_{𝑖=1,2,…,𝑚}{𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃}−𝛽⁺. (23)
For any fixed 𝑥 and 𝛽, as shown in Lemma 2, the above subordinate maximization problem in (22) can be rewritten into inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0s.t.𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝝃1⪰0,_𝑇𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝝃1≥max_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃−𝛽,∀𝝃∈ℝ^𝐿. (24) The last inequality constraint in the above problem can be expanded into 𝑚 simpler inequality. Representing the subordinate worst-case expectation problem in (22), we get inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}1𝛽+𝜖Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0s.t.𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0−𝟎12𝐰_𝑖(𝐱)12𝐰^𝑇_𝑖(𝐱)𝑤⁰_𝑖(𝐱)−𝛽⪰0,∀𝑖=1,2,…,𝑚. (25) Thus, we can easily get the exact representation of Θ^JCC. This completes the proof.

Remark 4. When 𝑚=1, we set 𝜼=0, which implies that we have the exact first- and second-order moments information for the individual chance constrained problem. We can get an exact approximation for Π^JCC: Θ^JCC=∃𝛽×𝐌∈ℝ×𝕊^𝐿+1,𝐱∈ℝ^𝑛1∶𝐌⪰0,𝛽+𝜖⟨Ω₊𝟎1,𝐌⟩≤0,𝐌−2𝐰(𝐱)12𝐰^𝑇(𝐱)𝑤⁰(𝐱),−𝛽⪰0 (26) where we denote 𝐌=^{𝐘1−𝐘2(1/2)(𝐲1−𝐲2)}_{(1/2)(𝐲1−𝐲2)𝑇𝑦0}∈𝕊^𝐿. This representation is exactly the one in [6]; then we can get the exact result of problem (5) (where 𝑚=1) by computing the above problem.

By now, we can compute the feasible set Θ^JCC by solving a tractable SDP. We appreciate that many modern methods can be used to solve such a convex problem to any precision 𝜖 in polynomial time and we can use YALMIP [16] to solve it in practice.

Consider the robust individual chance constraint (18) which represents the robust joint chance constraint (17). In convex programming, we use the max piecewise linear function to approximate a convex function frequently; there, we use a quadratic function to approximate the max function in the chance constraint (18) inversely. Note that 𝔥(𝝃)=𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰ that satisfies (i)𝔥(𝝃)≥max_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃,∀𝝃∈ℝ^𝐿,⟺𝔥(𝝃)≥𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃,∀𝝃∈ℝ^𝐿,𝑖=1,2,…,𝑚,(ii)inf_ℙ∈𝒫ℙ𝔥(𝝃)=𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰≤0≥1−𝜖. (27)(28) For better argumentation, we define Θ^JCC_𝑄=𝐱∈ℝ^𝑛:∃𝐇∈𝕊^𝐿,𝐡∈ℝ^𝐿,ℎ⁰𝔥(𝝃)∈ℝ,=𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰satisfies(27)and(28). (29)

Actually, it is easy to find that the feasible set Θ^JCC_𝑄 constitutes a conservative approximation for Θ^JCC, which means Θ^JCC_𝑄⊆Θ^JCC.

Theorem 5. The feasible set Θ_𝑄^𝐼𝐶𝐶 can be written asΘ_𝑄^𝐼𝐶𝐶=∃𝑦₀,ℎ⁰∈ℝ,𝐲₁,𝐲₂,𝐡∈ℝ^𝐿,𝐘₁,𝐘₂,𝐇∈𝕊^𝐿,𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂1⪰0,𝛽+𝜖Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0≤0𝐱∈ℝ^𝑛∶𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0−𝐇12𝐡12𝐡^𝑇ℎ₀𝐇1−𝛽⪰0,2𝐡−𝐰_𝑖(𝐱)12𝐡−𝐰_𝑖(𝐱)^𝑇ℎ⁰−𝑤⁰_𝑖(𝐱).⪰0 (30)Moreover, we find Θ_𝑄^𝐽𝐶𝐶=Θ^𝐼𝐶𝐶.

Proof. By a similar discussion as before, we get that the robust quadratic chance constraint (28) is equivalent to the Worst-Case CVaR constraint: sup_ℙ∈𝒫ℙ-CVaR_𝜖𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰=inf_𝛽∈ℝ1𝛽+𝜖sup_ℙ∈𝒫𝔼_ℙ𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰−𝛽⁺≤0. (31) We know that the above inequality can be reformulated as sup_ℙ∈𝒫ℙ-CVaR_𝜖𝝃^𝑇𝐇𝝃+𝝃^𝑇𝐡+ℎ⁰=inf_{𝑦0,𝐲1,𝐲2,𝐘1,𝐘2}1𝛽+𝜖Ω₊,𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−Ω₋,𝐘₂12𝐲₂12𝐲^𝑇₂0s.t.𝐲₁≥0,𝐲₂≥0,𝐘₁⪰0,𝐘₂𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0𝐘⪰0,₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0−𝐇12𝐡12𝐡^𝑇ℎ₀−𝛽⪰0. (32) Note that the constraints in (27) are equivalent to 𝐇12𝐡−𝐰_𝑖(𝐱)12𝐡−𝐰_𝑖(𝐱)^𝑇ℎ⁰−𝑤⁰_𝑖(𝐱)⪰0,∀𝑖=1,2,…,𝑚. (33) Thus, we can get the tractable form of Θ^JCC_𝑄 in Theorem 5. The LMIs in Theorem 5 can be represented as 𝐘₁12𝐲₁12𝐲^𝑇₁𝑦₀−𝐘₂12𝐲₂12𝐲^𝑇₂0+⪰𝐡𝟎𝟎𝟎𝛽𝐇𝐡_𝑇ℎ⁰⪰𝟎12𝐰_𝑖(𝐱)12𝐰_𝑖(𝐱)^𝑇𝑤⁰_𝑖(𝐱). (34) Finally, we get Θ^JCC_𝑄=Θ^ICC with vanishing the middle matrix which is formed by the components of 𝔥(𝝃).

This theorem presents that the approximation of a distributionally robust joint chance constraint by a Worst-Case CVaR constraint is equivalent to the approximation of the max function implied by the joint chance constraint by a quadratic majorant.

Let us consider the case that 𝝃 have finite support information which can be present as the matrix 𝜻=(𝝃₁,𝝃₂,…,𝝃_𝑁). Let the measures 𝐩=(𝑝₁,…,𝑝_𝑁) and 𝐩=(𝑝₁,…,𝑝_𝑁) be two known vectors which give the bounds of the probability measure with 𝐩<𝐩. The ambiguity distributional set can be defined as 𝔻₁ℝ^𝐿,𝝁₀=,𝜼𝐩∶𝐩^𝑇𝐞=1,𝐩≤𝐩≤𝐩,𝝁₀−𝜼≤𝜻𝐩≤𝝁₀+𝜼. (35)

Consider the minimax problem (22) with the above ambiguity distributional set; we have the following results.

Theorem 6. Θ^𝐽𝐶𝐶 has the following tractable reformulation in terms of LMIs:Θ^𝐽𝐶𝐶=∃𝜆₀∈ℝ,𝝀₁,𝝀₂,𝝀₃,𝝀₄∈ℝ^𝐿,𝝀₁≥0,𝝀₂≥0,𝝀₃≥0,𝝀₄≥0,𝐱∈ℝ^𝑛∶𝐮−𝜆₀𝐞+𝝀₁−𝝀₂+𝜻𝑇𝝀₃−𝝀₄1≥0,𝛽+𝜖𝜆₀+𝐩^𝑇𝝀₂−𝐩^𝑇𝝀₁+𝝁₀+𝜼^𝑇𝝀₄−𝝁₀−𝜼^𝑇𝝀₃𝑢≤0,_𝑖≥0,𝑢_𝑖≥𝑤⁰_𝑗(𝐱)+𝐰_𝑗(𝐱)^𝑇𝝃_𝑖,−𝛽,𝑖=1,…,𝑁,𝑗=1,…,𝑚 (36)where 𝐞 denotes the vector of ones.

Proof. Consider the subordinate maximization problem in (22); we can derive a reformulation for the worst-case expectation problem: sup_ℙ∈𝔻1𝔼_ℙmax_{𝑖=1,2,…,𝑚}{𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃}−𝛽⁺=sup_ℙ∈𝔻1^𝑁_𝑖=1𝑝_𝑖max_{𝑖=1,2,…,𝑚}𝑤⁰_𝑖(𝐱)+𝐰_𝑖(𝐱)^𝑇𝝃_𝑖−𝛽⁺. (37)
For any fixed 𝑥 and 𝛽, we can formulate the Lagrangian dual problem of the above linear programming. Actually, the standard duality theory guarantees that there is no duality gap between the above problem and its dual problem as min_{𝜆0,𝝀1,𝝀2,𝝀3,𝝀4}1𝛽+𝜖𝜆₀+𝐩^𝑇𝝀₂−𝐩^𝑇𝝀₁+𝝁₀+𝜼^𝑇𝝀₄−𝝁₀−𝜼^𝑇𝝀₃s.t.𝐮−𝜆₀𝐞+𝝀₁−𝝀₂+𝜻𝑇𝝀₃−𝝀₄𝑢≥0,_𝑖≥0,𝑢_𝑖≥𝑤⁰_𝑗(𝐱)+𝐰_𝑗(𝐱)^𝑇𝝃_𝑖−𝛽,𝑖=1,…,𝑁,𝑗=1,…,𝑚. (38)

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank Professor Nan-jing Huang for handling the review of this paper and two referees for helpful suggestions and comments. This work was supported by the Fundamental Research Funds for the Central Universities (2014NZYQN49).