Asymptotic Analysis of Sample Average Approximation for Stochastic Optimization Problems with Joint Chance Constraints via Conditional Value at Risk and Difference of Convex Functions

Abstract

Conditional Value at Risk (CVaR) has been recently used to approximate a chance constraint. In this paper, we study the convergence of stationary points, when sample average approximation (SAA) method is applied to a CVaR approximated joint chance constrained stochastic minimization problem. Specifically, we prove under some moderate conditions that optimal solutions and stationary points, obtained from solving sample average approximated problems, converge with probability one to their true counterparts. Moreover, by exploiting the recent results on large deviation of random functions and sensitivity results for generalized equations, we derive exponential rate of convergence of stationary points. The discussion is also extended to the case, when CVaR approximation is replaced by a difference of two convex functions (DC-approximation). Some preliminary numerical test results are reported.

Appendix

In this Appendix, we strengthen [13, Theorem 3.1] by weakening a boundedness condition imposed on the random function.

Theorem 8.1

Let ϕ:ℝⁿ×Ξ→ℝ be a real valued lower semicontinuous function, ξ:Ω→Ξ⊂ℝ^k a random vector defined on probability space $(\varOmega,\mathcal{F},P)$ and $\psi(x):=\mathbb {E}[\phi(x,\xi)]$. Let $\mathcal {X}\subset \mathbb {R}^{n}$ be a compact subset of ℝⁿ. Assume: $\mathrm{(a)}$ for every $x\in \mathcal {X}$ the moment generating function $M_{x}(t):=\mathbb {E}\{e^{t[\phi(x,\xi)-\psi(x)]} \}$ is finite valued for all t in a neighborhood of zero. $\mathrm{(b)}$ ψ(x) is continuous on $\mathcal {X}$, (c) ϕ(x,ξ) is bounded by an integrable function L(ξ) and the moment generating function $\mathbb {E}[e^{(L(\xi)-\mathbb {E}[L(\xi )])t} ]$ is finite valued for t close to 0. Then the following statements hold.

• (i) If ϕ(⋅,ξ) is almost H-clam from above at every point $x\in \mathcal {X}$ with modulus κ _x (ξ) and order γ _x , and the moment generating function $\mathbb {E}[e^{\kappa_{x}(\xi)t}]$ is finite valued for t close to 0, then for every ϵ>0, there exist positive constants c(ϵ) and β(ϵ), independent of N, such that

  $$ \operatorname {Prob}\Bigl\{\sup_{x\in \mathcal {X}}\bigl(\psi_N(x)-\psi(x) \bigr)\geq \epsilon \Bigr\}\leq c(\epsilon)e^{-N\beta(\epsilon)}. $$

  (39)

• (ii) If ϕ(⋅,ξ) is almost H-clam from below at every point $x\in \mathcal {X}$ with modulus κ _x (ξ) and order γ _x , and the moment generating function $\mathbb {E}[e^{\kappa_{x}(\xi)t}]$ is finite valued for t close to 0, then for every ϵ>0, there exist positive constants c(ϵ) and β(ϵ), independent of N, such that

  $$ \operatorname {Prob}\Bigl\{\sup_{x\in \mathcal {X}}\bigl(\psi_N(x)-\psi(x)\bigr)\leq- \epsilon \Bigr\}\leq c(\epsilon)e^{-N\beta(\epsilon)}. $$

  (40)

• (iii) If ϕ(⋅,ξ) is almost H-clam at every point $x\in \mathcal {X}$ with modulus κ _x (ξ) and order γ _x , and the moment generating function $\mathbb {E}[e^{\kappa_{x}(\xi)t}]$ is finite valued for t close to 0, then for every ϵ>0, there exist positive constants c(ϵ) and β(ϵ), independent of N, such that

  $$ \operatorname {Prob}\Bigl\{\sup_{x\in \mathcal {X}}\bigl|\psi_N(x)-\psi(x)\bigr|\geq\epsilon \Bigr\}\leq c(\epsilon)e^{-N\beta(\epsilon)}. $$

  (41)

Due to the limitation of the length of the paper, we omit the proof which can be found in [31].

Note that the exponential convergence is derived for the case when ξ satisfies a continuous distribution. In the case when ξ satisfies a discrete distribution, the concept of almost H-calmness is no longer applicable. However, the uniform exponential convergence may be established in an entirely different way for a class of random function which is uniformly bounded over a considered compact set. We leave this to interested readers.