How can you efficiently solve large-scale stochastic programming problems?

1 Decomposition methods

One of the most common and effective ways to tackle large-scale stochastic programming problems is to decompose them into smaller and easier subproblems that can be solved separately or in parallel. Decomposition methods can exploit the structure of the problem, such as the separability of the objective function, the independence of the constraints, or the sparsity of the coefficients. Some of the well-known decomposition methods for stochastic programming are Benders decomposition, L-shaped method, progressive hedging, and stochastic dual dynamic programming.

2 Sampling methods

Another way to reduce the complexity of large-scale stochastic programming problems is to use sampling methods, which generate a subset of scenarios or realizations of the uncertain parameters from the original probability distribution. Sampling methods can reduce the dimensionality and variability of the problem, but they also introduce sampling errors and require careful selection of the sample size and the sampling technique. Some of the common sampling methods for stochastic programming are Monte Carlo sampling, Latin hypercube sampling, scenario reduction, and scenario tree generation.

3 Approximation methods

A third way to deal with large-scale stochastic programming problems is to use approximation methods, which replace the original problem with a simpler or more tractable one that can be solved more efficiently. Approximation methods can involve simplifying the objective function, relaxing the constraints, linearizing the nonlinear terms, or discretizing the continuous variables. Some of the popular approximation methods for stochastic programming are linearization, convexification, piecewise linearization, and discretization.

4 Software and tools

To implement and solve large-scale stochastic programming problems, you will need some software and tools that can handle the stochastic modeling, the decomposition, the sampling, and the approximation techniques. There are many options available, ranging from general-purpose optimization solvers to specialized stochastic programming packages. Some of the widely used software and tools for stochastic programming are PySP, SMI, SMPS, GAMS, AMPL, Pyomo, CPLEX, Gurobi, and Xpress.

5 Applications and examples

Large-scale stochastic programming problems arise in many domains and applications, such as energy, finance, logistics, agriculture, and health care. They can help decision makers to optimize their plans and policies under various sources of uncertainty, such as demand, supply, prices, weather, and regulations. Some of the illustrative examples of large-scale stochastic programming problems are unit commitment, portfolio optimization, vehicle routing, crop planning, and epidemic control.

6 Tips and best practices

When attempting to solve large-scale stochastic programming problems, there are several tips and best practices to consider. Firstly, it is important to choose a stochastic programming formulation that matches the problem characteristics and objectives. Secondly, analyzing the structure and properties of the problem can help to apply suitable decomposition, sampling, and approximation methods. Thirdly, software and tools that can handle the stochastic programming features should be used. Additionally, testing and validating your solutions with different scenarios and sensitivity analysis is recommended. Lastly, comparing and benchmarking your solutions with other methods and approaches is beneficial.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?