Randomized Solutions to Convex Programs with Multiple Chance Constraints

TL;DR

The scenario-based optimization approach provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty ( ``scenarios'').

Abstract

The scenario-based optimization approach (`scenario approach') provides an intuitive way of approximating the solution to chance-constrained optimization programs, based on finding the optimal solution under a finite number of sampled outcomes of the uncertainty (`scenarios'). A key merit of this approach is that it neither assumes knowledge of the uncertainty set, as it is common in robust optimization, nor of its probability distribution, as it is usually required in stochastic optimization. Moreover, the scenario approach is computationally efficient as its solution is based on a deterministic optimization program that is canonically convex, even when the original chance-constrained problem is not. Recently, researchers have obtained theoretical foundations for the scenario approach, providing a direct link between the number of scenarios and bounds on the constraint violation probability. These bounds are tight in the general case of an uncertain optimization problem with a single chance constraint. However, this paper shows that these bounds can be improved in situations where the constraints have a limited `support rank', a new concept that is introduced for the first time. This property is typically found in a large number of practical applications---most importantly, if the problem originally contains multiple chance constraints (e.g. multi-stage uncertain decision problems), or if a chance constraint belongs to a special class of constraints (e.g. linear or quadratic constraints). In these cases the quality of the scenario solution is improved while the same bound on the constraint violation probability is maintained, and also the computational complexity is reduced.

Figures (5)

• Figure 3.1: Illustration of Definition \ref{['Def:SupConstr']} in $\mathbb{R}^{2}$. The arrow indicates the optimization direction, the bold lines are the support constraints of the respective configuration.
• Figure 3.2: Illustration of Example \ref{['Exa:SupportRank']} in $\mathbb{R}^{3}$. The arrows indicate the dimension of the unconstrained subspace, equal to $3$ minus the respective support rank$\alpha$, $\beta$, or $\gamma$.
• Figure 3.3: Illustration of Example \ref{['Exa:SuppDim']}. The plot shows a projection on the $x_{1}, x_{2}$-plane for $x_{3}=-1$. The unit box $\mathbb{X}$ is depicted by a dotted line. Two (possible) samples are shown for the linear constraint $i=1$ ($x_{1}\geq\delta_{1}$) and for the V-shaped constraint $i=2$ ($x_{2}\geq\bigl|x_{1}+\delta_{2}\bigr|-1$).
• Figure 5.1: Illustration of Example \ref{['Exa:Monotonicity']}. Non-bold constraints are generated by the multi-sample $\omega^{(i)}\in\Delta^{K_{i}}$ of chance constraint $i=1,2$; bold constraints are generated by the uncertainty $\delta\in\Delta$. In (b) a feasible point is made infeasible without affecting the optimum, which is not possible in the case of (a).
• Figure 6.1: Illustration of the numerical example for $n=2$. The point $\delta\in\Delta$ appears at random in $\mathbb{R}^{2}$, according to some unknown distribution; the points drawn here are $166$ i.i.d. samples of $\delta$. The objective is to construct the smallest product of two closed intervals ('2-cuboid'), drawn here as the shaded rectangle, such that the probability of failing to contain the realization of $\delta$ is smaller than $\varepsilon_{1}$ and $\varepsilon_{2}$ in dimension $1$ and $2$, respectively.

Theorems & Definitions (18)

• Remark 1.1: Problem Formulation
• Example 1.3: Multi-Stage Decision Problems
• Definition 3.1: Support Constraint
• Definition 3.3: Support Dimension
• Proposition 3.4: Probability Bound
• Example 3.5
• Definition 3.6: Unconstrained Subspace, Support Rank
• Proposition 3.7: Well-Definedness of Unconstrained Subspace
• Lemma 3.8: Support Rank
• Remark 3.9: Support Rank versus Support Dimension