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Solver Platform SDK - Linear Programming

Linear Programming

The LP/Quadratic Solver in the Solver Platform SDK solves linear programming problems up to 8,000 variables, with world-class performance.  It uses an advanced implementation of the Primal and Dual Simplex methods.  It automatically selects the best Simplex method, pricing method, pivoting strategy and the like, and uses robust methods to automatically handle degenerate models.

An automatic Presolve method can eliminate many unnecessary variables and constraints, and tighten bounds on other variables and constraints.

Sparse Matrix Storage

Sparse matrix storage methods take advantage of sparsity in large LP models, where subsets of the constraints typically depend on only a small subset of the variables. For example, an LP coefficient matrix for a problem with 10,000 variables and 10,000 constraints would take about 800 megabytes for matrix storage using dense storage methods, but if this problem has the sparsity typical of larger models, it would take about 24 megabytes using the sparse storage methods in the LP/Quadratic Solver.  (To learn more, click on Problem Size and Sparsity, covered in our Solver Tutorial.)

Matrix Factorization

Large LP models require hundreds of thousands to millions of floating-point arithmetic calculations to solve. Because of the finite precision inherent in computer arithmetic, small numerical errors occur in these calculations. Using conventional matrix representation and Simplex method iterations, these errors typically have a cumulative effect, leading to a numerically unstable problem and possibly large errors in the "solution."

The LP/Quadratic Solver uses a factorization of the LP matrix, called the LU decomposition, which remains numerically stable. This factorization is updated using advanced numerical methods (various matrix updates and dynamic Markowitz refactorization) that are much faster than a full update on each Simplex iteration. These methods enable the LP/Quadratic Solver to maintain accuracy, find the optimal solution, and do so much faster than conventional methods.

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