Topic
  • 4 replies
  • Latest Post - ‏2013-02-27T03:47:57Z by SystemAdmin
SystemAdmin
SystemAdmin
754 Posts

Pinned topic Is it possible to reformulate these constraints to be solvable by CPLEX?

‏2013-02-24T06:30:43Z |
I have a model with constraints as shown in this link: http://i.imgur.com/PiLZLRB.gif?1

where z_{ijt} is a binary variable, and w_{kt} is a continuous variable. There are other constraints restricting the denominator of the fraction to be strictly positive. I recall reading that constraints with fractions could be written as second order cone constraints, but I'm not sure how/if that applies in this case. Any suggestions?
Updated on 2013-02-27T03:47:57Z at 2013-02-27T03:47:57Z by SystemAdmin
  • SystemAdmin
    SystemAdmin
    754 Posts

    Re: Is it possible to reformulate these constraints to be solvable by CPLEX?

    ‏2013-02-24T15:56:26Z  
    It's possible to linearize this. If you multiply both sides by the denominator (maintaining the direction of the inequality, since you know the denominator is positive), you end up with quadratic terms of the form z*w (binary times continuous). Assuming that w is bounded a priori, these products can be linearized.

    Paul

    Mathematicians are like Frenchmen: whenever you say something to them, they translate it into their own language, and at once it is something entirely different. (Goethe)
  • SystemAdmin
    SystemAdmin
    754 Posts

    Re: Is it possible to reformulate these constraints to be solvable by CPLEX?

    ‏2013-02-24T20:05:14Z  
    Thanks! I think this works, but I think I need an extra step and to do two types of linearizations; linearizing a product of binaries and a product of binary and continuous variables. Maybe I'm making it too complicated. Here are the steps as a link again as I'm not seeing how to do equations easily in here:

    http://i.imgur.com/4UoTzCU.png
  • SystemAdmin
    SystemAdmin
    754 Posts

    Re: Is it possible to reformulate these constraints to be solvable by CPLEX?

    ‏2013-02-25T23:16:04Z  
    Thanks! I think this works, but I think I need an extra step and to do two types of linearizations; linearizing a product of binaries and a product of binary and continuous variables. Maybe I'm making it too complicated. Here are the steps as a link again as I'm not seeing how to do equations easily in here:

    http://i.imgur.com/4UoTzCU.png
    Sorry, my initial response sounded simpler than it should have. What I had in mind introduces a pile of variables (which happily are continuous) and a bunch of constraints. You're correct that there's no way (at least none I know of) to insert nontrivial math here. (I'm pretty sure the forum software was designed for coders, not mathematicians.) I'm attaching a PDF of what I have in mind. I offer no warranty that it's the easiest or most efficient solution.

    Paul

    Mathematicians are like Frenchmen: whenever you say something to them, they translate it into their own language, and at once it is something entirely different. (Goethe)
  • SystemAdmin
    SystemAdmin
    754 Posts

    Re: Is it possible to reformulate these constraints to be solvable by CPLEX?

    ‏2013-02-27T03:47:57Z  
    Thank you for the very thorough response.