© 2008 solutions 4u sdn bhd. all rights reserved introduction to optimization methods matlab...

© 2008 Solutions 4U Sdn Bhd. All Rights Reserved

Introduction to Optimization Methods

MATLAB Optimization Toolbox

Optimization Toolbox


Presentation Outline

• Introduction

• Function Optimization

• Optimization Toolbox

• Routines / Algorithms available

• Minimization Problems

• Unconstrained

• Constrained• Example

• The Algorithm Description

• Multiobjective Optimization

• Optimal PID Control Example



Optimization concerns the minimization or maximization of functions

Standard Optimization Problem:

~

~minxf x

~

0jg x

~

0ih x

L Uk k kx x x

Equality ConstraintsSubject to:

Inequality Constraints

Side Constraints

~

f x is the objective function, which measure and evaluate the performance of a system. In a standard problem, we are minimizing the function. For maximization, it is equivalent to minimization of the –ve of the objective function.

Where:

~x is a column vector of design variables, which can

affect the performance of the system.

Function Optimization



~

0ih x

L Uk k kx x x

Equality Constraints

Inequality Constraints

Side Constraints

~

0jg x Most algorithm require less than!!!

Constraints – Limitation to the design space. Can be linear or nonlinear, explicit or implicit functions

Function Optimization (Cont.)



Is a collection of functions that extend the capability of MATLAB. The toolbox includes routines for:

• Unconstrained optimization

• Constrained nonlinear optimization, including goal attainment problems, minimax problems, and semi-infinite minimization problems

• Quadratic and linear programming• Nonlinear least squares and curve fitting

• Nonlinear systems of equations solving• Constrained linear least squares

• Specialized algorithms for large scale problems




Minimization Algorithm



Minimization Algorithm (Cont.)



Equation Solving Algorithms



Least-Squares Algorithms



Most of these optimization routines require the definition of an M- file containing the function, f, to be minimized. Maximization is achieved by supplying the routines with –f. Optimization options passed to the routines change optimization parameters.

Default optimization parameters can be changed through an options structure.

Implementing Opt. Toolbox



Consider the problem of finding a set of values [x1 x2]T that solves

1

~

2 21 2 1 2 2

~min 4 2 4 2 1x

xf x e x x x x x

1 2~

Tx x x

Steps:

• Create an M-file that returns the function value (Objective Function). Call it objfun.m

• Then, invoke the unconstrained minimization routine. Use fminunc

Unconstrained Minimization



Step 1 – Obj. Function

function f = objfun(x)

f=exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1);

1 2~

Tx x x

Objective function



Step 2 – Invoke Routine

x0 = [-1,1];

options = optimset(‘LargeScale’,’off’);

[xmin,feval,exitflag,output]=

fminunc(‘objfun’,x0,options);

Output argumentsInput arguments

Starting with a guess

Optimization parameters settings



xmin =

0.5000 -1.0000

feval =

1.3028e-010

exitflag =

1

output =

iterations: 7

funcCount: 40

stepsize: 1

firstorderopt: 8.1998e-004

algorithm: 'medium-scale: Quasi-Newton line search'

Minimum point of design variables

Objective function value

Exitflag tells if the algorithm is converged.If exitflag > 0, then local minimum is found

Some other information

Results



[xmin,feval,exitflag,output,grad,hessian]=

fminunc(fun,x0,options,P1,P2,…)

More on fminunc – Input

fun : Return a function of objective function.

x0 : Starts with an initial guess. The guess must be a vector

of size of number of design variables.

Option : To set some of the optimization parameters. (More after few slides)

P1,P2,… : To pass additional parameters.



[xmin,feval,exitflag,output,grad,hessian]=

fminunc(fun,x0,options,P1,P2,…)

xmin : Vector of the minimum point (optimal point). The size is the number of design variables.

feval : The objective function value of at the optimal point.

exitflag : A value shows whether the optimization routine is terminated successfully. (converged if >0)

Output : This structure gives more details about the optimization

grad : The gradient value at the optimal point.

hessian : The hessian value of at the optimal point

More on fminunc – Output



Options =

optimset(‘param1’,value1, ‘param2’,value2,…)

Options Setting – optimset

The routines in Optimization Toolbox has a set of default optimization parameters. However, the toolbox allows you to alter some of those parameters, for example: the tolerance, the step size, the gradient or hessian values, the max. number of iterations etc. There are also a list of features available, for example: displaying the values at each iterations, compare the user supply gradient or hessian, etc. You can also choose the algorithm you wish to use.



Options =


LargeScale - Use large-scale algorithm if possible [ {on} | off ]

The default is with { }

Parameter (param1)Value (value1)

Options Setting (Cont.)

Type help optimset in command window, a list of options setting available will be displayed. How to read? For example:



LargeScale - Use large-scale algorithm if possible [ {on} | off ]• Since the default is on, if we would like to turn off, we just type:

Options = optimset(‘LargeScale’, ‘off’)

Options =


and pass to the input of fminunc.

Options Setting (Cont.)



Display - Level of display [ off | iter | notify | final ]

MaxIter - Maximum number of iterations allowed [ positive integer ]

TolCon - Termination tolerance on the constraint violation [ positive scalar ]

TolFun - Termination tolerance on the function value [ positive scalar ]

TolX - Termination tolerance on X [ positive scalar ]

Highly recommended to use!!!

Useful Option Settings



fminunc and fminsearch

fminunc uses algorithm with gradient and hessian information. Two modes:

• Large-Scale: interior-reflective Newton• Medium-Scale: quasi-Newton (BFGS)

Not preferred in solving highly discontinuous functions.

This function may only give local solutions.. fminsearch is generally less efficient than fminunc for problems of order greater than two. However, when the problem is highly discontinuous, fminsearch may be more robust.

This is a direct search method that does not use numerical or analytic gradients as in fminunc. This function may only give local solutions.



[xmin,feval,exitflag,output,lambda,grad,hessian] =

fmincon(fun,x0,A,B,Aeq,Beq,LB,UB,NONLCON,options,P1,P2,…)

Vector of LagrangeMultiplier at optimal point

Constrained Minimization



~

1 2 3~

minxf x x x x

21 22 0x x

1 2 3

1 2 3

2 2 0

2 2 72

x x x

x x x

1 2 30 , , 30x x x

Subject to:

1 2 2 0,

1 2 2 72A B

0 30

0 , 30

0 30

LB UB

function f = myfun(x)

f=-x(1)*x(2)*x(3);

Example



21 22 0x x For

Create a function call nonlcon which returns 2 constraint vectors [C,Ceq]

function [C,Ceq]=nonlcon(x)

C=2*x(1)^2+x(2);Ceq=[];

Remember to return a nullMatrix if the constraint doesnot apply

Example (Cont.)



x0=[10;10;10];

A=[-1 -2 -2;1 2 2];

B=[0 72]';

LB = [0 0 0]';

UB = [30 30 30]';

[x,feval]=fmincon(@myfun,x0,A,B,[],[],LB,UB,@nonlcon)

1 2 2 0,

1 2 2 72A B

Initial guess (3 design variables)

CAREFUL!!!

fmincon(fun,x0,A,B,Aeq,Beq,LB,UB,NONLCON,options,P1,P2,…)

0 30

0 , 30

0 30

LB UB

Example (Cont.)



Warning: Large-scale (trust region) method does not currently solve this type of problem, switching to medium-scale (line search).

>

Optimization terminated successfully:

Magnitude of directional derivative in search direction less than 2*options.TolFun and maximum constraint violation is less than options.TolCon

Active Constraints:

2

9

x =

0.00050378663220

0.00000000000000

30.00000000000000

feval =

-4.657237250542452e-035

21 22 0x x

1 2 3

1 2 3

2 2 0

2 2 72

x x x

x x x

1

2

3

0 30

0 30

0 30

x

x

x

Const. 1

Const. 2

Const. 3

Const. 4

Const. 5

Const. 6

Const. 7

Sequence: A,B,Aeq,Beq,LB,UB,C,Ceq

Const. 8

Const. 9

Example (Cont.)



Previous examples involved problems with a single objective function.

Now let us look at solving problem with multiobjective function by lsqnonlin.

Example is taken for data curve fitting

In curve fitting problem the the error is reduced at each time step producing multiobjective function.

Multiobjective Optimization

lsqnonlin in Matlab – Curve fitting

)2(1

0986.1

1

1bx

beR

clc; %recfit.mclear; global data; data= [ 0.6000 0.999

0.6500 0.998 0.7000 0.997 0.7500 0.995 0.8000 0.982 0.8500 0.975 0.9000 0.932 0.9500 0.862 1.0000 0.714 1.0500 0.520 1.1000 0.287 1.1500 0.134 1.2000 0.0623 1.2500 0.0245 1.3000 0.0100 1.3500 0.0040 1.4000 0.0015 1.4500 0.0007 1.5000 0.0003 ]; % experimental data,`1st coloum x, 2nd coloum R

x=data(:,1); Rexp=data(:,2); plot(x,Rexp,'ro'); % plot the experimental data hold on b0=[1.0 1.0]; % start values for the parameters b=lsqnonlin('recfun',b0) % run the lsqnonlin with start value b0, returned parameter values stored in b Rcal=1./(1+exp(1.0986/b(1)*(x-b(2)))); % calculate the fitted value with parameter b plot(x,Rcal,'b'); % plot the fitted value on the same graph

Find b1 and b2

>>recfit

>>b =

0.0603 1.0513

%recfun.mfunction y=recfun(b) global data; x=data(:,1); Rexp=data(:,2); Rcal=1./(1+exp(1.0986/b(1)*(x-b(2)))); % the calculated value from the model %y=sum((Rcal-Rexp).^2); y=Rcal-Rexp; % the sum of the square of the difference %between calculated value and experimental value



Numerical Optimization

Newton –Raphson Method1.Root Solver – System of nonlinear equations 2.(MATLAB – Optimization – fsolve)

)('

)(

11

i

iii xf

xfxx

2. One dimensional Solver – (MATLAB - OPTIMIZATION Method )

)(''

)('

11

i

iii xf

xfxx

Steepest Gradient Ascent/Descent Methods

iii

iii

dxfxf

dxfx

).(.)(

).(1

ii dxf ).( Is the magnitude of descent direction.

)(2 ixfd achieve the steepest descent )(2 ixfd achieve steepest

ascent



Steepest Gradient

75.0)1(25.2)1(325.23

yxx

f

075.1)1(4)1(25.275.1425.2

yxy

f

284375.05625.05.0)1 ,75.01( hhhf

0)1(25.2)75.0(3 x

f

5625.075.1)1(4)75.0(25.2 y

f

263281.0316406.059375.0)5625.01 ,75.0( hhhf

The partial derivatives can be evaluated at the initial guesses, x = 1 and y = 1,

Therefore, the search direction is –0.75i.

This can be differentiated and set equal to zero and solved for h* = 0.33333. Therefore, the result for the first iteration is x = 1 – 0.75(0.3333) = 0.75 and y = 1 + 0(0.3333) = 1. For the second iteration, the partial derivatives can be evaluated as,

Therefore, the search direction is –0.5625j.

This can be differentiated and set equal to zero and solved for h* = 0.25. Therefore, the result for the second iteration is x = 0.75 + 0(0.25) = 0.75 and y = 1 + (–0.5625)0.25 = 0.859375.

Solve the following for two step steepest ascent

yxyxyyxf 25.175.125.2),( 2

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

0

2

1

max



%chapra14.5 … ContdclearclcClf x2

ww1=0:0.01:1.2; ww2=ww1; [w1,w2]=meshgrid(ww1,ww2); J=-1.5*w1.^2+2.25*w2.*w1-2*w2.^2+1.75*w2;cs=contour(w1,w2,J,70);%clabel(cs);holdgridw1=1; w2=1; h=0;for i=1:10 syms h dfw1=-3*w1(i)+2.25*w2(i); dfw2=2.25*w1(i)-4*w2(i)+1.75; fw1=-1.5*(w1(i)+dfw1*h).^2 + 2.25*(w2(i)+dfw2*h).*(w1(i)+… dfw1*h)-2*(w2(i)+dfw2*h).^2+1.75*(w2(i)+dfw2*h); J=-1.5*w1(i)^2+2.25*w2(i)*w1(i)-2*w2(i)^2+1.75*w2(i) g=solve(fw1); h=sum(g)/2; w1(i+1)=w1(i)+dfw1*h; w2(i+1)=w2(i)+dfw2*h; plot(w1,w2) xi

pause(0.05) End

MATLAB OPTIMIZATION TOOLBOX

w1(i), w2(i) function J=chaprafun(x)w1=x(1);w2=x(2)J=-(-1.5*w1^2+2.25*w2*w1-2*w2^2+1.75*w2);

%startchapra.mclcclearx0=[1 1];options=optimset('LargeScale','off','Display','iter','Maxiter',… 20,'MaxFunEvals',100,'TolX',1e-3,'TolFun',1e-3);[x,fval]=fminunc(@chaprafun,x0,options)



Newton –Raphson – Four Bar MechanismSine and Cosine angle components - All angles referenced from global x - axis

In above equations θ1 = 0 as it is along the x-axis and other three angles are time varying.

1st angular velocity derivative



• If input is applied to link2 - DC motor then ω2 would be the input to the syste

In Matrix for

2. Numerical Solution for Non Algebraic Equations

)('

)(

1

_

1

i

iii xf

xfxx



)('

)(_

ii qf

qfq



)('

)(_

_

i

iii

f

f

Newton-Raphson

%fouropt.mfunction f=fouropt(x)the = 0; r1=12; r2=4; r3=10; r4=7;f=-[r2*cos(the)+r3*cos(x(1))-r1*cos(0)-r4*cos(x(2)); r2*sin(the)+r3*sin(x(1))-r1*sin(0)-r4*sin(x(2))];

%startfouropt.mclcclearx0=[0.1 0.1];options=optimset('LargeScale','off','Display','iter','Maxiter',… 200,'MaxFunEvals',100,'TolX',1e-8,'TolFun',1e-8);[x,fval]=fsolve(@fouropt,x0,options);theta3=x(1)*57.3theta4=x(2)*57.3

Foursimmechm.m, foursimmech.mdl and possol4.m



Simulink PID Controller Optimization

Kp=72, Ki=36, Kd=160.2

------------------------- s2 - 0.2 s - 0.1

Y(s) ------- =

U(s)

Step

Signal Constraint

Scope

u y

Plant

e u

PID Controller1

1

uSum

Kp

Proportional

1s

Integrator

Ki

Integral

Kd

Derivative

s

0.1s+1

Band-limitedDerivative

1

e

UNIM513tune1.mdl

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