Exploiting structure of chance constrained programs via submodularity

Abstract

We introduce a novel approach to reduce the computational effort of solving mixed-integer convex chance constrained programs through the scenario approach. Instead of reducing the number of required scenarios, we directly minimize the computational cost of the scenario program. We exploit the problem structure by efficiently partitioning the constraint function and considering a multiple chance constrained program that gives the same probabilistic guarantees as the original single chance constrained problem. We formulate the problem of finding the optimal partition, a partition achieving the lowest computational cost, as an optimization problem with nonlinear objective and combinatorial constraints. By using submodularity of the support rank of a set of constraints, we propose a polynomial-time algorithm to find suboptimal solutions to this partitioning problem and we give approximation guarantees for special classes of cost metrics. We illustrate that the resulting computational cost savings can be arbitrarily large and demonstrate our approach on two case studies from production and multi-agent planning.

Chance-constrained models for transactive energy management of interconnected microgrid clusters

“…Stochastic programming and CCP are two popular tools that can be employed as probabilistic optimization methods to manage random uncertainties (Marino et al, 2018). Chance-constrained optimization is stochastic programming that uses probabilistic measures over the constraints with uncertainty parameters (Frick et al, 2019). Indeed, for risk-based decision making, this approach is deemed as a typical model for stochastic programming.…”

A two‐layer chance‐constrained optimization model for a thickening‐dewatering process with uncertain variables

“…[ 17 ] To solve this problem, many scholars had studied from the perspective of reducing the number of scenarios [ 27–30 ] and reducing the minimum scenario calculation cost. [ 31 ] In addition to the above methods, there is the power series expansions (PSE)‐based function [ 32,33 ] ; this PSE function does not require the simulation of the nonlinear process model for each sampled realization of the uncertain variables and describes those distributions using a deterministic approach (i.e., using their mean and standard deviation, , where is a constant coefficient). Those can be chosen from (which means only expected value is used) up to (which means that you cover 99.99% of the Gaussian distribution).…”