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

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.

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

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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