How can you solve linear programming problems faster?

1 Choose the right solver

The first step to solve an LP problem faster is to choose the right solver for your problem. A solver is a software tool that applies a specific algorithm to find the optimal solution. Different solvers have different strengths and weaknesses, depending on the characteristics of your problem, such as its size, sparsity, integrality, and convexity. For example, if your problem is small and dense, you might want to use a simplex-based solver, such as CPLEX or Gurobi. If your problem is large and sparse, you might want to use an interior-point-based solver, such as MOSEK or IPOPT. If your problem has integer variables, you might want to use a branch-and-bound or branch-and-cut solver, such as SCIP or CBC.

2 Simplify your model

The second step to solve an LP problem faster is to simplify your model as much as possible. This can reduce computation time and memory, and make it easier to debug and interpret. To do this, you can remove redundant or irrelevant variables and constraints, combine or aggregate similar variables and constraints, use linear approximations or relaxations for nonlinear terms, and use scaling or normalization to avoid numerical issues. For instance, if you have a quadratic objective function or a quadratic constraint, you can replace it with a linear or piecewise-linear function or constraint that approximates it well enough. Additionally, if you have variables or coefficients that differ by several orders of magnitude, you can divide them by a common factor or transform them into a standard range to improve the stability and accuracy of your solver.

3 Use pre-solving and post-processing

The third step to solve an LP problem faster is to utilize pre-solving and post-processing techniques to optimize your solver's performance. Pre-solving involves applying transformations or reductions to the model before sending it to the solver, while post-processing is the process of making corrections or refinements to the solution after receiving it from the solver. Common pre-solving and post-processing techniques include presolve, dualization, sensitivity analysis, and parametric analysis. Presolve is a built-in feature of most solvers that simplifies or eliminates variables or constraints. Dualization converts the primal problem into a dual problem that might have fewer variables or constraints. Sensitivity analysis evaluates how the optimal solution and objective value change when varying some parameters or coefficients of the model. Parametric analysis solves the problem for a range of values of a parameter or coefficient, allowing you to explore trade-offs and alternatives.

4 Use heuristics and decomposition

The fourth step to solve an LP problem faster is to use heuristics and decomposition methods to cope with hard or large problems that cannot be solved by standard solvers. Heuristics are methods that find good but not necessarily optimal solutions in a short time, by using some rules of thumb or approximations, while decomposition methods break down a complex problem into smaller subproblems. Popular heuristics and decomposition methods include Greedy heuristics, which make the best local choice at each step without considering the global consequences; Metaheuristics, which use general strategies or frameworks to guide the search for a good solution; Lagrangian relaxation, which relaxes some of the constraints of your problem by introducing penalty terms into the objective function; and Benders decomposition, which splits your problem into a master problem and a subproblem. Utilizing these techniques can help you find efficient solutions in a timely manner.

5 Use parallel and distributed computing

The fifth step to solve an LP problem faster is to use parallel and distributed computing techniques, which involve dividing the problem or the solver into smaller parts and assigning them to different processors or machines that work simultaneously or independently. Parallel simplex is a technique that parallelizes the simplex algorithm, which is a widely used solver for LP problems. It updates the simplex tableau by splitting it into smaller blocks and working on them concurrently with different processors or threads. Parallel interior-point is a technique that parallelizes the interior-point algorithm, which is another popular solver for LP problems. It splits the linear system that needs to be solved at each iteration into smaller sub-systems and solves them concurrently using different processors or threads. Parallel branch-and-bound is a technique that parallelizes the branch-and-bound algorithm, which is a common solver for mixed-integer LP problems. It splits the search tree that needs to be explored into smaller sub-trees and explores them concurrently using different processors or threads. Finally, distributed optimization is a technique that distributes your problem or your solver across multiple machines or networks and coordinates them using communication or synchronization protocols, such as exchanging information or solutions through message passing or consensus algorithms.

6 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?