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Re: [Help-glpk] [Fwd: AW: [Fwd: Re: [Fwd: How to rewrite a nonlinear exp

From: Jeffrey Kantor
Subject: Re: [Help-glpk] [Fwd: AW: [Fwd: Re: [Fwd: How to rewrite a nonlinear expression in a linear one]]]
Date: Tue, 2 Apr 2013 10:34:02 -0400

Hi Sina,

There are tools appropriate for this type of problem, but that discussion is probably outside the scope of the glpk-help list.  So I'll email you privately.


On Tue, Apr 2, 2013 at 9:52 AM, Andrew Makhorin <address@hidden> wrote:
-------- Forwarded Message --------
From: Sina Burkhardt <address@hidden>
To: 'Andrew Makhorin' <address@hidden>
Subject: AW: [Fwd: Re: [Help-glpk] [Fwd: How to rewrite a nonlinear
_expression_ in a linear one]]
Date: Tue, 2 Apr 2013 10:51:13 +0200

Hi Jeff,

thanks for your answer.

You're right. The idea is to use this SIR type of model as a small part in another model which is a linear one.
Is there another tool like MathProg that I can use for?

The problem is that I must get a solution of my model until Friday, so in the next 3 days, to do some analysis on it and so on.


-----Ursprüngliche Nachricht-----
Von: Andrew Makhorin [mailto:address@hidden]
Gesendet: Dienstag, 2. April 2013 01:34
An: Sina Burkhardt
Betreff: [Fwd: Re: [Help-glpk] [Fwd: How to rewrite a nonlinear _expression_ in a linear one]]

-------- Forwarded Message --------
From: Jeffrey Kantor <address@hidden>
To: Andrew Makhorin <address@hidden>
Cc: GLPK <address@hidden>
Subject: Re: [Help-glpk] [Fwd: How to rewrite a nonlinear _expression_ in a linear one]
Date: Mon, 1 Apr 2013 18:47:28 -0400

Hi Sina,

This looks like a compartmental SIR type of model with multiple contagions.  These models are inherently nonlinear because of the denominator in the term you indicated was a problem, and additional terms for infection rate, etc.  I'm afraid there's not much you can do if you're looking for a global linearization without imposing some pretty rigid controllers in place which force linearization.

You could obtain a local linearization valid in the neighborhood of a steady state.  That might be useful if you're eventually looking for
optimal control policies.   You could also create a convex outer
approximation for the dynamics, but I'm not sure how helpful that would be for this problem.

So if what you're looking for is a global linearization of this inherently nonlinear model, I'm afraid MathProg may not be the right tool for the job.


On Mon, Apr 1, 2013 at 6:04 PM, Andrew Makhorin <address@hidden> wrote:
        -------- Forwarded Message --------
        From: Sina Burkhardt <address@hidden>
        To: address@hidden
        Subject: How to rewrite a nonlinear _expression_ in a linear one
        Date: Mon, 1 Apr 2013 23:31:16 +0200

        Hi all,

        I’m currently writing my Master thesis and I hope someone can
        help me to
        solve the following problem(s) with my model as quickly as

        I use glpk(gusek) and my model needs unfortunately two nonlinear

        Is there any possibility to rewrite these nonlinear expressions
        linear ones to solve it with gusek?

        Or can I assign a solution value of var to an parameter or
        like that to avoid the nonlinear type?

        Here’s an abstract of the model : (the “problems” are red


        param T, integer; #horizont of time

        set D;     #  DemandPoints


        set P, default{1..T};             # Planungshorizont T

        var susceptible{j in D,t in 0..T}>=0,integer;

        var N{j in D,t in 0..T}>=0,integer;                        #
        at DemandPoint j in periode t

        var I{j in D,t in 0..T}>=0;                        # Persons who
        infected at DemandPoint j in periode t

        var I_nB{j in D,t in 0..T}>=0,integer #Infected without
        treatment at
        DemandPoint j in periode t

        var I_neu{j in D,t in P}>=0;                               #
        infected persons in j in t

        var lambda{j in D,t in P}>=0;

        /*Index of contagions*/

        param c, >=0,<=1;


        param kappa{D};

        param beta{j in D}:=kappa[j]*c;

        /*Init. in  t=0*/

        init_Population{j in D}: N[j,0]=init_N0[j];

        init_Infiziert{j in D}: I[j,0]=init_I0[j];

        init_Gesund{j in D}: susceptible[j,0]=N[j,0]-I[j,0];

        population{j in D,t in P}: N[j,t]=susceptible[j,t]+I[j,t];
        #with var
        I{j in D,t in P}

        Gesunde{j in D,t in P}: susceptible[j,t]= susceptible[j,t-1]-

        #Calculate infectionrate

        s.t. infekt_rate{j in D,t in P}: lambda[j,t]=beta[j]*
        (I_nB[j,t-1] /N[j,t-1]); #Here’s the first Problem because its
        and glpk can not solve NLP.

        #Calculate the new infected persons

        Neuinfiziert{j in D,t in P}: I_neu[j,t]=

        # example data


        set D:= D1;

        param T:=3;

        param c:=0.2;

        param kappa:= D1 10;

        param init_N0:= D1            100000;

        param init_I0:= D1              1000;


        I would be very happy about your help.

        Thanks in advance.



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