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I'm looking to solve problems of the form in R:

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where

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and

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is an indicator function that equals 1 if the argument (.) is true and zero otherwise.

I've looked into packages lpSolve, Rcplex, and crs but I couldn't quite grasp how I would place my problem into the functions offered by those packages. I don't really know how I would incorporate my indicator functions in the objective. I thought about making my controls the binary W's themselves but then I would have to provide the lambdas (weights), however, my interest is finding the optimal combination of lambdas, and not the W's.

obsidian64
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1 Answers1

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You basically want

z<=r => w=1

This can be written as 

w=0 => z>r

or

z >= r + 0.001 - M*w

where M is a large enough constant (but preferably not too large). Some solvers like Cplex and Gurobi have indicator constraints: this way the last implication can be expressed directly without resorting to a big-M formulation.

Erwin Kalvelagen
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  • Thank you kindly for your response. Most of the MIP solvers require the problem be placed in the following way: min c'x s.t. A'x <= b where x are the control variables. In my case, I'm trying to determine the optimal combination of lambda's but they do not appear in my objective function, only the W's do. So I was wondering how to place my problem into this framework. – obsidian64 Nov 06 '17 at 21:45
  • Not all variables need to go in the objective. This is the same as saying that often many variables have an objective coefficient of zero. – Erwin Kalvelagen Nov 06 '17 at 21:56
  • Thanks again. I'm now able to formulate this problem in order to "fit" it into the Rcplex function of the 'Rcplex' package (which was a nightmare to install but that's another story). – obsidian64 Nov 09 '17 at 22:13