## mmnl

The

mmnlmodule extends the Mosel language with a new type for representing nonlinear expressions and constraints and also with some additional subroutines. To use this module the following line must be included in the header of the Mosel model file:uses 'mmnl'The first section presents the new functionality for the Mosel language provided by

mmnl, namely the new type nlctr and a set of subroutines that may be applied to objects of this type.The following sections give detailed documentation of the subroutines (other than mathematical operators) defined by this module.

## New functionality for the Mosel language

## The problem type mpproblem.nl

This module exposes its functionality through an extension to the mpproblem problem type. As a consequence, all routines presented here are executed in the context of the current problem.

## The type nlctr and its operators

The module

mmnldefines the type nlctr to represent nonlinear constraints in the Mosel Language. As shown in the following example (SectionExample: using mmnl for QCQP),mmnlalso defines the standard arithmetic operations that are required for working with objects of this type. By and large, these are the same operations as for linear expressions (type linctr of the Mosel language) with additionally the possibility to multiply or divide by decision variables and to use the exponential notation x^r (assuming that x is of type mpvar). Nonlinear constraints may also be defined by using overloaded versions of Mosel's arithmetic and trigonometric functions on expressions involving decision variables (see SectionProcedures and functionsfor a complete list).## Setting initial values

An important feature in Nonlinear Programming is the possibility to set initial values for decision variables. With

mmnlthis is done by the procedure setinitval. Nonlinear solvers use initial values as starting point for the search. The choice of the initial values may not only have an impact on the time spent by the solver but also, depending on the problem type, on the best (locally optimal) solution found by the solver.The definitions of initial values can be removed with clearinitvals. It is also possible to employ the solution values obtained from the immediately preceding optimization run as initial values to the next by calling the procedure copysoltoinit.

## Example: using

mmnlfor QCQPThe following example shows how to solve a QCQP (Quadratically Constrained Quadratic Programming) problem with the Xpress-MP QCQP solver. To use this solver we need to load the module

mmxprsin addition tommnlsince the modulemmnldoes not include any solver.The problem we wish to solve is a classical NLP test problem (source: http://www.orfe.princeton.edu/~rvdb/ampl/nlmodels/ that determines the shape of a hanging chain by minimizing its potential energy. The objective function is linear and the problem has convex quadratic constraints.

model "catenary" uses "mmxprs", "mmnl" parameters N = 100 ! Number of chainlinks L = 1 ! Difference in x-coordinates of endlinks H = 2*L/N ! Length of each link end-parameters declarations RN = 0..N x: array(RN) of mpvar ! x-coordinates of endpoints of chainlinks y: array(RN) of mpvar ! y-coordinates of endpoints of chainlinks end-declarations forall(i in RN) x(i) is_free forall(i in RN) y(i) is_free ! Objective: minimise the potential energy potential_energy:= sum(j in 1..N) (y(j-1)+y(j))/2 ! Bounds: positions of endpoints ! Left anchor x(0) = 0; y(0) = 0 ! Right anchor x(N) = L; y(N) = 0 ! Constraints: positions of chainlinks forall(j in 1..N) Link_up(j):= (x(j)-x(j-1))^2+(y(j)-y(j-1))^2 <= H^2 ! Setting start values forall(j in RN) setinitval(x(j), j*L/N) forall(j in RN) setinitval(y(j), 0) setparam("XPRS_verbose", true) minimise(potential_energy) writeln("Solution: ", getobjval) forall(j in RN) writeln(strfmt(getsol(x(j)),10,5), " ", strfmt(getsol(y(j)),10,5)) end-modeA QCQP matrix can be exported to a text file (in MPS or LP format) by adding the following lines to your model after the problem definition:

setparam("XPRS_loadnames", true) ! Enable loading of names loadprob(potential_energy) ! Load the problem writeprob("catenary.mat", "") ! Write an MPS matrix ("l" for LP format)Not all problems with quadratic constraints conform with the properties required by QCQP solvers. Xpress-Optimizer therefore performs a convexity check before starting the optimization. This test takes some time and if you know that your problem is convex you may disable it by setting the following parameter before starting the optimization.

setparam("XPRS_ifcheckconvexity", false) ! Disable convexity check## Procedures and functions

The module

mmnloverloads certain mathematical functions of the Mosel language, replacing an argument of type real by the types linctr and nlctr. The return value of these functions is of type nlctr. This means they can be used as operators in the definition of nonlinear constraints as shown in the example of SectionExample: using mmnl for QCQP. The relevant functions are:Since these mathematical operators are fairly self-explanatory, we shall forego any more detailed documentation of these functions.

The following list gives an overview of all other functions and procedures defined by

mmnlfor which we give detailed descriptions later.

clearinitvals Delete all initial value definitions.copysoltoinit Copy solution values to initial values.getsol Get the solution value of a nonlinear constraint.gettype Get the type of a nonlinear constraint.ishidden Test whether a constraint is hidden.sethidden Hide or unhide a nonlinear constraint.setinitval Set an initial value (start value) for a variable.settype Set the type of a nonlinear constraint.

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