MOSEK Solver Engine Version 9.0

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Introduction

The MOSEK Solver Engine extends the power of the Premium Solver Platform to solve linear (LP), quadratic (QP), quadratically constrained (QCP) and second order cone programming (SOCP) problems. The Extended MOSEK Solver Engine provides the additional capability to solve smooth convex nonlinear optimisation (NLP) problems. The Standard MOSEK Solver can solve problems of up to 32,000 variables and 32,000 constraints, while the version does not have any fixed limits on variables and constraints. These solver engines also supports integer variables and 'alldifferent' constraints.

Solve Linear, Quadratic and New Conic Optimisation problems with lightning speed

With the Standard MOSEK Solver, you will be able to solve LP, QP, QCP, SOCP and NLP* problems hundreds to thousands of times larger than what the standard Excel Solver can handle. The MOSEK Solver typically matches the Large-Scale LP/QP Solver and other world-class LP Solvers for linear and mixed-integer linear programming problems (LP/MIP), and it is widely regarded as the world's best solver for QCP and SOCP problems. This solver takes maximum advantage of the new Polymorphic Spreadsheet Interpreter in the Premium Solver Platform to obtain second partial derivatives of the problem functions (the Hessian matrix) at each major iteration or trial solution. For QP and SOCP problems, whereby the Hessian matrices are constant, the MOSEK Solver is as fast as it is on LP problems.

New options and tolerances

The MOSEK Solver offers nearly 40 options and tolerances that you can set to fine-tune the solution for LP, QP, QCP and NLP* problems.  These include the primal and dual feasibility tolerance, complementarity gap tolerance, relative step size to the constraint boundary and central path tolerance.

The MOSEK Solver uses the Premium Solver Platform's Branch and Bound method to handle integer variables and alldifferent constraints. If your problem includes integer constraints, you can obtain a quick solution of the relaxation (by temporarily ignoring the integer constraints) without having to delete these constraints and then enter them later. You can control the number of Branch and Bound subproblems and the number of integer feasible solutions found before the solver stops. Further, you can speed up the solution of problems with integer constraints by supplying an integer cut-off value.

The MOSEK Solver offers several methods for faster solution of LP problems with integer variables. It uses preprocessing and probing techniques to preset values for some binary integer variables based on the settings of others. It automatically recognises, and takes advantage of cliques (also called Special Ordered Sets), tightens bounds on constraints, and reorders the variables to be branched upon. It also uses cut generation techniques to automatically add new constraints, known as knapsack cuts or lifted cover inequalities that reduce the size of the LP feasible region without eliminating any integer solutions. Depending on the model, these methods can save significant amounts of time.

Mosek advanced methods

Sparse matrix storage

Sparse matrix storage methods take advantage of sparsity in large 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. However, if this problem has the sparsity typical of large models, it would occupy only about 24 megabytes using the sparse storage methods.

Matrix factorisation

Large 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, these errors typically have a cumulative effect, leading to a numerically unstable problem and possibly large errors in the solution. The MOSEK Solver Engine computes the Cholesky factorisation of the matrix of normal equations, using advanced methods that minimise fill-in (that preserve sparsity), and maintain matrix conditioning and numerical stability. These methods enable the MOSEK Solver to minimise the effect of floating point roundoff errors, find the optimal solution, and perform much faster than conventional methods.

Comprehensive user guide

In addition to the user guide that comes with the Premium Solver Platform, you will receive the Solver Engine User Guide that describes several field-installable solver engines, including the MOSEK Solver. This guide provides complete information on MOSEK Solver options, the various messages which the MOSEK Solver can return, how to diagnose problems and how to control the MOSEK Solver from VBA in Excel.

Software maintenance

Annual maintenance is required for the first year, and at all times in order to obtain software upgrades and technical support beyond basic installation assistance. The maintenance includes:

• A limited warranty for the functionality and performance of the software product
  
• All software upgrades for the product released during the contract term
  
• Ability to trade in this product for an even more powerful Excel Solver product
  
• Access to protected support pages of our Website
  
• Technical support by phone and email during normal business hours
  
• Up to 15 minutes of consulting assistance arising during the contract term

*NLP capability is only available in the Extended Solver Engine.

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