[Help-glpk] [Fwd: AW: [Fwd: Re: [Fwd: How to rewrite a nonlinear expression in a linear one]]]

[Andrew Makhorin]

-------- 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.

Regards,
Sina


-----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.


Jeff






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
        possible.
        
        I use glpk(gusek) and my model needs unfortunately two nonlinear
        expressions.
        
        Is there any possibility to rewrite these nonlinear expressions
        into
        linear ones to solve it with gusek?
        
        Or can I assign a solution value of var to an parameter or
        something
        like that to avoid the nonlinear type?
        
        
        
        Here’s an abstract of the model : (the “problems” are red
        labeled)
        
        
        
        
        
        ###Declarations####
        
        param T, integer; #horizont of time
        
        set D;     #  DemandPoints
        
        
        
        /*Periodenzeitraum*/
        
        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;                        #
        Population
        at DemandPoint j in periode t
        
        var I{j in D,t in 0..T}>=0;                        # Persons who
        are
        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;                               #
        add.
        infected persons in j in t
        
        var lambda{j in D,t in P}>=0;
        #Infectionrate
        
        
        
        /*Index of contagions*/
        
        param c, >=0,<=1;
        
        
        
        /*contactrate*/
        
        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]-
        I_neu[j,t];
        
        
        
        #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
        nonlinear
        and glpk can not solve NLP.
        
        
        
        #Calculate the new infected persons
        
        Neuinfiziert{j in D,t in P}: I_neu[j,t]=
        lambda[j,t]*susceptible[j,t-1];
        
        
        
        # example data
        
        
        
        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;
        
        
        
        end;
        
        
        
        
        
        
        
        I would be very happy about your help.
        
        Thanks in advance.
        
        
        
        Regards,
        
        Sina
        
        
        
        
        
        
        
        
        
        
        
        _______________________________________________
        Help-glpk mailing list
        address@hidden
        https://lists.gnu.org/mailman/listinfo/help-glpk