Introduction

This part of the manual is intended to provide a general description of the facilities available for modeling with Xpress NonLinear. It is not an exhaustive list of possibilities, and it does not go into very great depth on some of the more advanced topics. All the functions and formats are given in more detail in the second part of this manual and the Xpress-Mosel Reference Manual (Module mmxnlp section).

Xpress Nonlinear consists of the Xpress Optimizer to solve linear, mixed integer linear, and convex quadratic problems, Xpress-SLP which uses Successive Linear Programming to solve non-linear models, and as an plugin Knitro.

The functionalities of Xpres NonLinear extend those of the Xpress Optimizer. Almost any problem that fits into the problem types supported by the Xpress Optimizer are automatically detected and converted into the appropriate format to take advantage of the power of the optimizer's purpose written algorithms.

Xpress-SLP is in essence, is a technique which involves making a linear approximation of the original problem at a chosen point, solving the linear approximation and seeing how "far away" the solution point is from the original chosen point. If it is "sufficiently close" then the solution is said to have converged and the process stops. Otherwise, a new point is chosen, based on the solution, and a new linear approximation is made. This process repeats (iterates) until the solution converges. Although this process will find a solution which is the optimum for the linear approximation, there is no guarantee that the solution will be the optimum for the original non-linear problem (that is to say: it may not be the best possible solution to the original problem). Such a solution is called a "local optimum", because it is a better solution than any others in the immediate neighbourhood, but may not be better than one a long way away.

The problem of local optima can be thought of as being like trying to find the deepest valley in a range of mountains. You can find a valley relatively easily (just keep going downhill). However, when you reach it, you have no idea whether there is a deeper valley somewhere else, because the mountains block your view. You have found a local optimum, but you do not know whether it is a global optimum. Indeed, in general, there is no way to find the global optimum except an exhaustive search (check every valley in the mountain range).

While Xpress-SLP is most powerful for large or integer nonlinear problems, Knitro which can take advantage of using second order partial derivative information can be more beneficial for hihgly nonlinear models.