Nonlinear Problems
Xpress NonLinear will solve nonlinear problems. In this context, a nonlinear problem is one in which there are nonlinear relationships between variables or where there are nonlinear terms in the objective function. There is no such thing as a nonlinear variable — all variables are effectively the same — but there are nonlinear constraints and formulae. A nonlinear constraint contains terms which are not linear. A nonlinear term is one which is not a constant and is not a variable with a constant coefficient. A nonlinear constraint can contain any number of nonlinear terms.
Xpress NonLinear will also solve linear problems — that is, if the problem presented to Xpress NonLinear does not contain any nonlinear terms, then Xpress NonLinear will still solve it, using the normal optimizer library.
The solution mechanism used by Xpress-SLP is Successive (or Sequential) Linear Programming. This involves building a linear approximation to the original nonlinear problem, solving this approximation (to an optimal solution) and attempting to validate the result against the original problem. If the linear optimal solution is sufficiently close to a solution to the original problem, then the SLP is said to have converged, and the procedure stops. Otherwise, a new approximation is created and the process is repeated. Xpress-SLP has a number of features which help to create good approximations to the original problem and therefore help to produce a rapid solution.
When license, Xpress NonLinear may also utilize Knitro to solve nonlinear problems.
Note that although the solution is the result of an optimization of the linear approximation, there is no guarantee that it will be an optimal solution to the original nonlinear problem. It may be a local optimum — that is, it is a better solution than any points in its immediate neighborhood, but there is a better solution rather further away. However, a converged SLP solution will always be (to within defined tolerances) a self-consistent — and therefore practical — solution to the original problem.
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