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## Re: [Help-glpk] Multiple LP problem solving

**From**: |
Andrew Makhorin |

**Subject**: |
Re: [Help-glpk] Multiple LP problem solving |

**Date**: |
Wed, 08 Jun 2011 15:42:56 +0400 |

>* I mean I have several objective functions within one LP. I'm trying to*
>* find the values of the variables x1, x2, ..., xN that minimize the*
>* following functions:*
>* z1 = a1.x1 + b1.x2 + ...*
>* z2 = a2.x1 + b2.x2 + ...*
>* z3 = a3.x1 + b3.x2 + ...*
>* *
>* *
>* The variables are common to each function, but the coefficient are*
>* not. I'm basically trying to minimize "Cx - d", given a coefficient*
>* matrix C and a constant vector d.*
>* *
>* *
>* Btw, I'm used to working with MATLAB and the lsqlin() function to*
>* solve that kind of LP.*
>* *
>* *
>* Am I clear enough?*
Now you are. Thanks.
There exist two main ways to solve multicriteria problems. The first one
is using a convolution of the objectives, and the second one is a
sequential optimization. I guess that Matlab's lsqlin implements the
first approach minimizing the sum of squares of the objectives. This
leads to a quadratic programming problem that is not supported by glpk,
however, instead you might minimize the maximum of the objectives, i.e.
minimize obj: z;
s.t. z >= z1;
z >= z2;
z >= z3;
. . .
The latter differs from the least squares approach only in the metrics.