Optimizing a Round Down (or floor) Function; or How to tell Gurobi to always round down

[Danny]

Hello fellow Gurobi Optimizers,

I have the following Max problem:

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Max f = sum_i( a_i * y_i) problem           ''' a_i all Positive is the parameter and y_i is variable'''

s.t 

constraints on x

constraints on x and z

and the problem making constraints:

y <= (d - t(x,z)) / 4d) + 1     ''' d is a parameter,  and t(x,z) at most can be = 4d '''

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y is NOT a decision variable, z is the decison variable, and x is a function of z, and y is a function of x; y is an indicator variable, I want y to indicate: if t(x,z) > d then set y = 0 else if t(x,z) <= d set y = 1,  

and what y <= (d - t(x,z)) / 4d) + 1 does is :  

(1)  when  t(x,z) <= d  --->   0 < d - t(x,z)) / 4d  < 1  --->  1 <= (d - t(x,z)) / 4d) + 1  < 2

(2) when  t(x,z) > d  --->   d - t(x,z)) / 4d  < 0          --->  0 <  (d - t(x,z)) / 4d) + 1  < 1

The LP relaxation gives the optimal solutions with  y either >= 1  or   y < 1. The point is every feasible solution of the LP relaxation has ONLY ONE corresponding feasible integer solution and that is to always ROUND DOWN on y_values, so that in case (1) y = 1 and in case (2)  y = 0. I could actually represent this in the objective function as:

Max f = sum_i( a_i * ROUNDDOWN(y_i) ) 

So I figured I can get rid of y > 1 ( e.g. y = 1.46) by setting ub = 1 for y. But I don't know how to get rid of y < 1 (e.g. y = 0.35), what if I could tell Gurobi to always Rounddown when branching on a variable? How can I do that? Or should I reformulate or revise my model? Do you think this is the right for what I aim for? 

Thank you very much in advance

[Stuart Mitchell]

How bout making y a binary variable?

Stu

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Stuart Mitchell

PhD Engineering Science

Extraordinary Freelance Programmer and Optimisation Guru

www.stuartmitchell.com

[Danny]

Thanks Stuart,

y is a binary variable, it is an indicator (0 or 1), but the solver is oddly very slow in solving the problem, even for small problems. 

I am trying to see if I can relax it.