J.Pelfort Test Problem Full Integer Convex.
For those bold fellas there is yet another solution, please try to find it
by Surrogate Dual Method starting with multipliers 1/3.
Solving it by Outer Lagrangian Linearization for D gives you the point [0,1,3,2] D -52.42
this is due because lambda (1)=2.125 is a linear underestimator of its true value 2.6944.
So you have to look for the nearest value in the Lagrangian Relaxation which is [0,0,2,2].
the problem ends because [2.125, .9167, 0 ] is optimal in Linearized Dual.
Suppose x1 and x4 are meant to be Integer and x2 and x3 Continuous
One could be tempted to fix x1=0 and x4=2 and then solve the continuos problem for x2,x3.
That gives x2 =.475 and x3=2.05 f=-50.762 but I regret to say that this
does not work. It is time to revisit Bender's decomposition or Outer Aproximation methods or B&B.
The Optimal solution is in reality x1=0, x4 =1 , and then x2=0.8 , x3=2.4 , F = -53.36
Do enjoy fellas...
