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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.

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