matlab optimization

Matlab Optimization

Matlab Optimization

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1.

Optimization toolbox

Optimization toolbox

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2.

Solution of linear programs

Solution of linear programs

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3.

Metabolic flux balance analysis example

Metabolic flux balance analysis example

4.

4.

Solution of nonlinear programs

Solution of nonlinear programs

5.

Matlab Optimization Toolbox

Minimization

bintprog Solve binary integer programming problems

fgoalattain Solve multiobjective goal attainment problems

fminbnd Find minimum of single-variable function on fixed interval

fmincon Find minimum of constrained nonlinear multivariable function

fminimax Solve minimax constraint problem

fminsearch Find minimum of unconstrained multivariable function using derivative-free method

fminunc Find minimum of unconstrained multivariable function

fseminf Find minimum of semi-infinitely constrained multivariable nonlinear function

linprog Solve linear programming problems

quadprog _{Solve quadratic programming problems}

Least Squares

lsqcurvefit Solve nonlinear curve-fitting (data-fitting) problems in least-squares sense

lsqlin Solve constrained linear least-squares problems

lsqnonlin Solve nonlinear least-squares (nonlinear data-fitting) problems

Linear Programming (LP)

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Optimization of a linear objective function with linear 

equality and/or inequality constraints

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Standard LP form:

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Matrix

A

must have more columns than rows

(under-determined problem)

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Common solvers: CPLEX, MOSEK, GLPK 

Further information

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x – vector of variables to be determined (decision variables) A – matrix of known coefficients

b – vector of known coefficients

c – vector of weights

Matlab LP Solver:

linprog

 Solves linear programming (LP) problems of the form:

 Syntax:

x = linprog(f,A,b,Aeq,beq,lb,ub)

Set A=[]and b=[]if no inequality constraints exist Set Aeq=[]and beq=[]if no equality constraints exist Replace f with -f to find the maximum

 Defaults to a large-scale interior point method with options for a

medium-scale simplex method variation or the simplex method

 See

help linprog

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Metabolic Network Model

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Intracellular reaction pathways describing carbon metabolism

» Consumption of carbon energy sources (e.g. glucose)

» Conversion of carbon sources to biomass precursors (cell growth) » Secretion of byproducts (e.g. ethanol)

» Each node corresponds to a metabolite

» Each path (line) corresponds to a reaction

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Stoichiometric matrix,

A

» Row for each intracellular species (m rows) » Column for each reaction (n columns)

» The entry at the ith _{row and j}th _{column (a}

i,j) corresponds to the

stoichiometric coefficient of species ‘i’ participating in reaction ‘j’ » Av = 0, stoichiometric balance on the metabolites where v is the

vector of reaction fluxes

 –  More reactions (unknowns) than species (equations)

 –  Solution requires either enough measurements for the system to

Flux Balance Analysis (FBA)

 Linear programming (optimization) approach for resolving an

under-determined metabolic network model

 Objective function based on an assumed cellular objective such as

maximization of growth

 LP formulation:

 Growth rate, m, represented as a linear combination of intracellular 

fluxes of the biomass precursors

 Flux bounds represent physiochemical or thermodynamic constraints

on the reaction fluxes

» Extracellular conditions place limits on fluxes (e.g. oxygen availability) » Thermodynamics constrain the direction a reaction may proceed:

reversible or irreversible

 The solution is the set of fluxes that maximizes cellular growth while

satisfying the bounds and stoichiometric constraints

U   L T  v v v v  Av v w

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 Download the stoichiometric matrix to the

Matlab working directory and load into Matlab

>> load A.txt

 Specify the indices of key fluxes: glucose,

ethanol, oxygen, and biomass

>> ig = 22; ie = 20; >> io = 19; imu = 17;  Av = 0 >> [m n] = size(A); >> b = zeros(m,1);  Objective function >> w = zeros(n,1); >> w(imu) = 1;

 Specify flux bounds (all fluxes irreversible,

glucose uptake fixed)

>> vb = [zeros(n,1) Inf*ones(n,1)];

Flux Balance Analysis Example

 Yeast metabolic network model from HW #2

 Slightly modified to improve suitability for Flux Balance Analysis

(FBA)

 19x22 stoichiometric matrix

 Under-determined with 3 degrees of freedom

 Use FBA to determine solution corresponding to optimal cell growth  Solve the LP

>> v = linprog(-w,[],[],A,b,vb(:,1),vb(:,2)); Optimization terminated.

 View predictions for growth, oxygen uptake, and ethanol

secretion

>> mu = w'*v, vo2 = v(io), ve = v(ie) mu = 101.9302 vo2 = 108.3712 ve = 2.4147e-014

 All calculated values relative to a fixed glucose uptake rate

FBA Example cont.

 Determine sensitivity of model predictions to the oxygen uptake

rate to assess the tradeoff between achievable ethanol yields and cellular growth

 Create a vector of oxygen uptake rates to be considered

>> vo = 1:1:125;

 Implement a forloop to iterate over each entry in the oxygen

uptake vector (vo). For each iteration (inside the loop), update the upper bound* on oxygen uptake, solve the LP, and store the solution (mu, ve)

>> for i=1:length(vo) vb(io,2) = vo(i); v = linprog(-w,[],[],A,b,vb(:,1),vb(:,2)); mu(i) = w'*v; ve(i) = v(ie); end

 Plot the results

>> plot(vo,mu,vo,ve); >> xlabel('Oxygen Flux')

>> legend('Growth Rate','Ethanol Flux)

  Notice the tradeoff between cell growth and ethanol production.

Highest ethanol productivity is achieved in batch fermentation  by initially operating aerobically to rapidly increase cell density

then switching to anaerobic conditions to produce ethanol.

Nonlinear Programming (NLP)

Optimization of a nonlinear objective function with

nonlinear equality and/or inequality constraints

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Standard NLP form:

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System must have more variables than equality constraints

(under-determined problem)

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Common solvers: CONOPT, NPSOL

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 Non-convex problems can converge to a local optimum

x – vector of variables to be determined (decision variables) h(x) – vector function of equality constraints

g(x) – vector function of inequality constraints

 f (x) – scalar objective function

0

x

g

0

x

h

x

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)

(

)

(

:

subject to

)

(

min f  

 Nonlinear least-squares: lsqnonlin

x = lsqnonlin(@fun,x0,lb,ub)

where fun is a user-defined function that returns the vector value F ( x) ,

x0 is the initial guess (starting point), and lb and ub are the bounds on x

 Constrained nonlinear multivariable optimization : fmincon

where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c( x) and ceq( x) are functions that return vectors, and f ( x) is a function that returns a scalar 

x = fmincon(@fun,x0,A,b,Aeq,beq,lb,ub,@cfun)

where funis the function for  f ( x) and cfunis a function that returns c( x) and ceq( x)

f = fun(x) [c,ceq] = cfun(x)

Matlab NLP Solvers:

lsqnonlin

and

fmincon

ub  x lb beq  x  Aeq b  x  A  x ceq  x c  x  f    x         0 ) ( 0 ) ( : s.t. ) ( min 2 2 3 2 2 1 2 1 2 ) ( ) ( ) ( ) ( ) ( min  f    x  f    x  f    x  f    x  f  _n x n i i  x     

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Batch Fermentation Example

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Parameter estimation problem for penicillin fermentation

Model equations

» Batch cell growth is modeled by the logistic law

where y₁ is the cell concentration, k ₁ is the growth constant & k ₂ is the cessation (limiting nutrient) constant

» Penicillin production is modeled as

where y₂ is the penicillin concentration, k ₃ is the production constant & k ₄ is the degradation (hydrolysis) constant

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Dynamic parameter estimation

» Use experimental data from two batch penicillin fermentations

» Find values for the unknown parameters (k ₁, k ₂, k ₃, k ₄) that minimize the

sum of squared errors between the data & model predictions

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1

k 

 y

 y

k 

dt 

dy

 y

k 

 y

k 

dt 

dy

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Matlab Exercise: Batch Data Sets

Cell Penicillin Cell Penicillin Time concentration concentration concentration concentration (hours) (% dry weight) (units/mL) (% dry weight) (units/mL)

0 0.4 0 0.18 0 10 0 0.12 0 22 0.99 0.0089 0.48 0.0089 34 0.0732 1.46 0.0642 46 1.95 0.1446 1.56 0.2266 58 0.523 1.73 0.4373 70 2.52 0.6854 1.99 0.6943 82 1.2566 2.62 1.2459 94 3.09 1.6118 2.88 1.4315 106 1.8243 3.43 2.0402 118 4.06 2.217 3.37 1.9278 130 2.2758 3.92 2.1848 142 4.48 2.8096 3.96 2.4204 154 2.6846 3.58 2.4615 166 4.25 2.8738 3.58 2.283 178 2.8345 3.34 2.7078 190 4.36 2.8828 3.47 2.6542 Batch 1 Batch 2

 Load & plot the experimental data:

 Choose an initial guess, integrate the model, & plot the simulated profiles:

 Estimate parameter values that minimize the sum of squared errors between

the experimental measurements & model predictions:

Matlab Exercise: Solution

>> pendat = xlsread('penicillin.xls'); >> tdat = pendat(:,1); >> ydat = pendat(:,2:end); >> plot(tdat,ydat,'o'); >> xlabel('Time [h]'); >> ylabel('Concentration'); >> k0 = [0.1 4 0.01 0.01]; >> y0 = [0.29 0]; >> ts = [min(tdat) max(tdat)];

>> dy = @(t,y,k) [k(1)*y(1)*(1-y(1)/k(2)); k(3)*y(1)-k(4)*y(2)]; >> [tsim,ysim] = ode45(dy,ts,y0,[],k0);

>> hold on, plot(tsim,ysim,':');

>> options = optimset('Display','iter');

>> k = lsqnonlin(@simerr,k0,[],[],options,dy,ts,y0,tdat,ydat); >> [tsim,ysim] = ode45(dy,ts,y0,[],k);

Matlab Exercise: simerr.m

function e = simerr(k0,dy,ts,y0,tdat,ydat)

% Integrate the model

sol = ode45(dy,ts,y0,[],k0);

% Evaluate solution at the data points y = deval(sol,tdat)';

% Error between data and model e = ydat - y;

% Find missing measurements n = find(isnan(ydat));

% Zero error for missing measurements if ~isempty(n)

e(n) = zeros(size(n)); end