matlab optimization

Matlab Optimization 1. Optimization toolbox 2. Solution of linear programs 3. Metabolic flux balance analysis example 4. Solution of nonlinear programs 5. Batch fermentation example

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

lsqnonneg

Solve nonnegative least-squares constraint problem

Linear Programming (LP) 

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Optimization of a linear objective function with linear  equality and/or inequality constraints Standard LP form: min x

cT x

subject to: Ax  b x0 x – vector of variables to be determined (decision variables) A – matrix of known coefficients b – vector of known coefficients c – vector of weights cT x – scalar objective function; linear combination of the decision variables 

Matrix A must have more columns than rows (underdetermined problem)

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Common solvers: CPLEX, MOSEK, GLPK 

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Further information http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html

Matlab LP Solver: linprog 

Solves linear programming (LP) problems of the form: min  x

 f  T  x

subject to:  A  x  b  Aeq  x  beq lb   x  ub where f , x, b, beq, lb, and ub are vectors and A and Aeq are matrices 

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

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Defaults to a large-scale interior point method with options for a medium-scale simplex method variation or the simplex method See help linprog for additional details and options

Metabolic Network Model 

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 

Stoichiometric matrix, A » Row for each intracellular species (m rows) » Column for each reaction (n columns)

» The entry at the ith row and jth column (ai,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  become square (n-m measurements) or optimization

Flux Balance Analysis (FBA) 

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Linear programming (optimization) approach for resolving an underdetermined metabolic network model Objective function based on an assumed cellular objective such as maximization of growth LP formulation:

max v

m  

wT v

subject to:  Av  0  L U  v vv

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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

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The solution is the set of fluxes that maximizes cellular growth while satisfying the bounds and stoichiometric constraints

Flux Balance Analysis Example 

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Yeast metabolic network model from HW #2 Slightly modified to improve suitability for Flux Balance Analysis (FBA)

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19x22 stoichiometric matrix

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Under-determined with 3 degrees of freedom

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Use FBA to determine solution corresponding to optimal cell growth

Download the stoichiometric matrix to the Matlab working directory and load into Matlab

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

>> load A.txt 

Specify the indices of key fluxes: glucose, ethanol, oxygen, and biomass

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Av = 0

101.9302

>> [m n] = size(A); >> b = zeros(m,1); 

vo2 =

Objective function

108.3712

>> w = zeros(n,1); >> w(imu) = 1; 

Specify flux bounds (all fluxes irreversible, glucose uptake fixed) >> vb = [zeros(n,1) Inf*ones(n,1)];

View predictions for growth, oxygen uptake, and ethanol secretion >> mu = w'*v, vo2 = v(io), ve = v(ie) mu =

>> ig = 22; ie = 20; >> io = 19; imu = 17; 

Solve the LP

ve = 2.4147e-014 

All calculated values relative to a fixed glucose uptake rate normalized to 100%

FBA Example cont. 

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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;

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Implement a for loop 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

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Plot the results >> plot(vo,mu,vo,ve); >> xlabel('Oxygen Flux') >> legend('Growth Rate','Ethanol Flux)

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 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. *A fixed oxygen uptake rate (lower bound equal to upper bound) was not specified to avoid forcing the cell to

Nonlinear Programming (NLP) 

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Optimization of a nonlinear objective function with nonlinear equality and/or inequality constraints Standard NLP form: min f  (x) x

subject to:

h ( x)  0 g ( x)  0

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 

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System must have more variables than equality constraints (under-determined problem) Common solvers: CONOPT, NPSOL  Non-convex problems can converge to a local optimum

Matlab NLP Solvers: lsqnonlin and fmincon 

Nonlinear least-squares: lsqnonlin n

min  x

  f  ( x)

2

i

  f  1 ( x) 2   f  2 ( x) 2   f  3 ( x) 2     f  n ( x) 2

i 1

  f  1 ( x )   f  2 ( x )  where fun is a user-defined function that returns the vector value F ( x) ,  F ( x )    f  3 ( x )     x0 is the initial guess (starting point), and lb and ub are the bounds on x    f   (  x )  n  x = lsqnonlin(@fun,x0,lb,ub)

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Constrained nonlinear multivariable optimization : fmincon

min  x

s.t.:

 f  ( x) c( x)  0

where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c( x)

ceq( x)  0

and ceq( x) are functions that return vectors, and f ( x) is a function that

 A  x  b

returns a scalar 

 Aeq  x  beq lb   x  ub x = fmincon(@fun,x0,A,b,Aeq,beq,lb,ub,@cfun) where fun is the function for  f ( x) and cfun is a function that returns c( x) and ceq( x)

f = fun(x)

[c,ceq] = cfun(x)

Batch Fermentation Example 

Parameter estimation problem for penicillin fermentation

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Model equations » Batch cell growth is modeled by the logistic law

   y    k 1 y1 1  1  dt    k 2  

dy1

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 dy2  k 3 y1  k 4 y2 dt  where y2 is the penicillin concentration, k 3 is the production constant & k 4 is the degradation (hydrolysis) constant 

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

Matlab Exercise: Batch Data Sets Batch 1 Time (hours) 0 10 22 34 46 58 70 82 94 106 118 130 142 154 166 178 190

Cell concentration (% dry weight) 0.4 0.99 1.95 2.52 3.09 4.06 4.48 4.25 4.36

Penicillin concentration (units/mL) 0 0 0.0089 0.0732 0.1446 0.523 0.6854 1.2566 1.6118 1.8243 2.217 2.2758 2.8096 2.6846 2.8738 2.8345 2.8828

Batch 2 Cell concentration (% dry weight) 0.18 0.12 0.48 1.46 1.56 1.73 1.99 2.62 2.88 3.43 3.37 3.92 3.96 3.58 3.58 3.34 3.47

Penicillin concentration (units/mL) 0 0 0.0089 0.0642 0.2266 0.4373 0.6943 1.2459 1.4315 2.0402 1.9278 2.1848 2.4204 2.4615 2.283 2.7078 2.6542

Matlab Exercise: Solution 

Load & plot the experimental data: >> >> >> >> >> >>

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

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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,':');

Estimate parameter values that minimize the sum of squared errors between the experimental measurements & model predictions: >> >> >> >>

options = optimset('Display','iter'); k = lsqnonlin(@simerr,k0,[],[],options,dy,ts,y0,tdat,ydat); [tsim,ysim] = ode45(dy,ts,y0,[],k); plot(tsim,ysim);

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