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Re: [Help-glpk] Can't use variable within abs()?

From: glpkuser
Subject: Re: [Help-glpk] Can't use variable within abs()?
Date: Mon, 27 Aug 2007 17:56:29 -0700 (PDT)

Andrew Makhorin wrote:
>> I'm getting an error saying that "argument for abs has invalid type".
>> I'm trying to model a load-balancing scenario where 10 weighted
>> items are distributed across 3 buckets. I want to minimise the
>> difference between each bucket's total weight and the average weight
>> of all buckets, so I'm trying to do the following, which is not
>> working:
>> minimize z:  sum{b in BUCKETS} abs(BucketWeight[b] - AveBucketWeight);
> You cannot use variables in abs(), because this leads to non-linear
> objective not allowed in glpk.
> However, in your case the objective is piecewise linear and convex,
> so it can be easily reformulated as follows:
> var u{b in BUCKETS}; /* auxiliary variable */
> s.t. aaa{b in BUCKETS}: u[b] >= BucketWeight[b] - AveBucketWeight;
> s.t. bbb{b in BUCKETS}: u[b] >= AveBucketWeight - BucketWeight[b];
> minimize z: sum{b in BUCKETS} u[b];

Thank you for your help. I'm a beginner with linear programming and O.R.;
could you point me to a reference where I can find more info on piecewise
linearity and convex-ness with respect to re-formulating or fixing
non-linear models? For example, would the Winston book on Operations
Research have it?
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