$title Simple Nonlinear Variational Inequality (SIMPLEVI2,SEQ=47)

$onText
This model shows how to solve a simple nonlinear VI using EMP

VI is to find x in X:
F(x) (y - x) >= 0, for all y in X
X is a closed convex set, e.g. X = { x >= 0 | g(x) >= 0 }

Since the variational inequality is integrable w.r.t y this model can also be
expressed as NLP as done in the model 'nlpmod'.

Contributor: Michael Ferris, February 2010
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variables y;
equations F, g;

F..  sqr(y) - 1 =n= 0;
g..           y =g= 0;

model vimod /F, g/;

file fx / '%emp.info%' /;
putclose fx 'vi F y g';

solve vimod using emp;

* Now solve the equivalent NLP model and compare the results
scalar g_l, y_l;
g_l = g.l; y_l = y.l;

variables obj;
equations defobj;
defobj.. obj =e= power(y,3)/3 - y;

model nlpmod /defobj, g/;
solve nlpmod using nlp min obj;
abort$(abs(g_l-g.l) > 1e-9) 'bad g level', g_l, g.l;
abort$(abs(y_l-y.l) > 1e-9) 'bad y level', y_l, y.l;