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Introduction To Non–linear Optimization

Department of Mechanical EngineeringUniversiti Tenaga Nasional

PART I

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Optimization Tree

Figure 1: Optimization tree.

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What is Optimization?

Optimization is an iterative process by which a desired solution (max/min) of the problem can be found while satisfying all its constraint or bounded conditions.

Optimization problem could be linear or non-linear. Non –linear optimization is accomplished by numerical ‘Search Methods’. Search methods are used iteratively before a solution is achieved.

The search procedure is termed as algorithm.

Figure 2: Optimum solution is found while satisfying its constraint (derivative must be zero at optimum).

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Linear problem – solved by Simplex or Graphical methods.

The solution of the linear problem lies on boundaries of the feasible region.

Non-linear problem solution lies within and on the boundaries of the feasible region.

Figure 3: Solution of linear problem Figure 4: Three dimensional solution of non-linear problem

What is Optimization?(Cont.)

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Constraints• Inequality• Equality

Fundamentals of Non-Linear Optimization

Single Objective function f(x)• Maximization• Minimization

Design Variables, xi , i=0,1,2,3…..

Figure 5: Example of design variables and constraints used in non-linear optimization.

Maximize X1 + 1.5 X2 Subject to:X1 + X2 ≤ 1500.25 X1 + 0.5 X2 ≤ 50X1 ≥ 50X2 ≥ 25X1 ≥0, X2 ≥0

Optimal points• Local minima/maxima points: A point or Solution x* is at local point if there is no other x in its Neighborhood less than x* • Global minima/maxima points: A point or Solution x** is at global point if there is no other x in entire search space less than x**

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Figure 6: Global versus local optimization. Figure 7: Local point is equal to global point if the function is convex.

Fundamentals of Non-Linear Optimization (Cont.)

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Function f is convex if f(Xa) is less than value of the corresponding point joining f(X1) and f(X2). Convexity condition – Hessian 2nd order derivative) matrix of function f must be positive semi definite ( eigen values +ve or zero).

Fundamentals of Non-Linear Optimization (Cont.)

Figure 8: Convex and nonconvex set Figure 9: Convex function

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Mathematical Background

Slop or gradient of the objective function f – represent the direction in which the function will decrease/increase most rapidly

x

f

x

xfxxf

dx

dfxx

00

lim)()(

lim

.......)(!2

1)()( 2

2

2

xdx

fdx

dx

dfxxf

pp xxp

z

g

y

g

x

gz

f

y

f

x

f

J

Taylor series expansion

Jacobian – matrix of gradient of f with respect to several variables

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Hessian – Second derivative of f of several variables

Second order condition (SOC)• Eigen values of H(X*) are all positive• Determinants of all lower order of H(X*) are +ve

2

22

2

2

2

y

f

yx

fxy

f

x

f

H

First order Condition (FOC)

0*)( Xf

Mathematical Background (Cont.)

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Optimization Algorithm

Deterministic - specific rules to move from one iteration to next , gradient, Hessian

Stochastic – probalistic rules are used for subsequent iteration

Optimal Design – Engineering Design based on optimization algorithm

Lagrangian method – sum of objective function and linear combination of the constraints.

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Multivariable Techniques ( Make use of Single variable Techniques specially Golden Section)

Optimization Methods

Deterministic• Direct Search – Use Objective function values to locate minimum • Gradient Based – first or second order of objective function.• Minimization objective function f(x) is used with –ve sign – f(x) for maximization problem.

Single Variable• Newton – Raphson is Gradient based technique (FOC)• Golden Search – step size reducing iterative method

• Unconstrained Optimization a.) Powell Method – Quadratic (degree 2) objective function polynomial is non-gradient based. b.) Gradient Based – Steepest Descent (FOC) or Least Square minimum (LMS) c.) Hessian Based -Conjugate Gradient (FOC) and BFGS (SOC)

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• Constrained Optimization a.) Indirect approach – by transforming into unconstrained problem. b.) Exterior Penalty Function (EPF) and Augmented Lagrange Multiplier c.) Direct Method Sequential Linear Programming (SLP), SQP and Steepest Generalized Reduced Gradient Method (GRG)

Figure 10: Descent Gradient or LMS

Optimization Methods …Constrained

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Global Optimization – Stochastic techniques

• Simulated Annealing (SA) method – minimum energy principle of cooling metal crystalline structure

• Genetic Algorithm (GA) – Survival of the fittest principle based upon evolutionary theory

Optimization Methods (Cont.)

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Optimization Methods (Example)

Multivariable Gradient based optimizationJ is the cost function to be minimized in twodimension

The contours of the J paraboloid shrinks as it isdecrease

function retval = Example6_1(x)% example 6.1retval = 3 + (x(1) - 1.5*x(2))^2 + (x(2) - 2)^2;

>> SteepestDescent('Example6_1', [0.5 0.5], 20, 0.0001, 0, 1, 20)Where

[0.5 0.5] -initial guess value20 -No. of iteration0.001 -Golden search tol.0 -initial step size1 -step interval20 -scanning step

>> ans2.7585 1.8960

Figure 11: Multivariable Gradient based optimization

Figure 12: Steepest Descent

MATLAB Optimization Toolbox

PART II

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Presentation Outline

IntroductionFunction OptimizationOptimization ToolboxRoutines / Algorithms available

Minimization ProblemsUnconstrainedConstrained

ExampleThe Algorithm Description

Multiobjective OptimizationOptimal PID Control Example

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Function Optimization

Optimization concerns the minimization or maximization of functions

Standard Optimization Problem:

~

~min

xf 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.

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Function Optimization (Cont.)

~

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

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Optimization Toolbox

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

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Minimization Algorithm

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Minimization Algorithm (Cont.)

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Equation Solving Algorithms

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Least-Squares Algorithms

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Implementing Opt. Toolbox

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.

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Unconstrained Minimization

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

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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

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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

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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

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

fminunc(fun,x0,options,P1,P2,…)

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.

More on fminunc – Input

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[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

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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.

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Options =

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

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:

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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 =

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

and pass to the input of fminunc.

Options Setting (Cont.)

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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

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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.

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[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

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~

1 2 3~

minx

f 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

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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.)

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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.)

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

> In D:\Programs\MATLAB6p1\toolbox\optim\fmincon.m at line 213

In D:\usr\CHINTANG\OptToolbox\min_con.m at line 6

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.)

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Multiobjective Optimization

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.

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

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Simulink Example

x_dot_dot x_dot x

Program m-files reqd.1) BasketflyBall.m2) start_flyBasketBall.m3) Distflysim.m

3Out3

2

Out2

1Out1

f(u)

z_dot_dot

atan(u(1)/u(2))

theta

f(u)

h_dot_dot

gamefield.translation

blue.translation

object.translation

VR Sink

STOP

Stop Simulation

>=

1s

1sxo

1s

1sxo

1s

1s

if { }In1 Out1

u1if(u1 < x_dot_max)

If

1/cart_mass

Gain

AeroFac*(u(1)^2+u(2)^2)

Fd

(20 -0.6 0)

0

V0*cos(theta_0)

V0*sin(theta_0)

15

15

F0

Cart position1

Cart position

Ball position1

Ball position

Shooting a flying box Jeff_fly basket.mdl

Eq. of ball motion in z horz. direction

Eq. of ball motion in h vert. direction

Angle of ballAerodynamic drag force

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Simulink example – shooting ballfunction P = Distflysim(theta_0) F0=25.0; %Ncart_mass=2; %kgx_dot_max=50; %m/secro_air=1.224; %kg/m^3h0=0.5; %mz0=0;Cd=1;r_ball=0.05; %mA_ball=pi*r_ball^2;ball_mass=0.1; %kgg=-9.8; %m/sec^2theta_0; %radV0=50; %m/secF0=15.0; %NAeroFac=Cd*A_ball*ro_air/2;theta_0assignin('base','F0',F0);assignin('base','cart_mass',cart_mass);assignin('base','x_dot_max',x_dot_max);assignin('base','AeroFac',AeroFac);assignin('base','ball_mass',ball_mass);assignin('base','g',g);assignin('base','V0',V0);assignin('base','theta_0',theta_0);% Newrtp=rsimgetrtp('jeff_basket');% save ShotParams.mat Newrtp;% !jeff_basket -p ShotParams.mat% load jeff_basket;[t,x,y]=sim('jeff_flybasket',[0 10]);np=max(size(y));xf=y(np,1);zf=y(np,2);%hf=y(np,3);P=(xf-zf)^2;%+(hf-25)^2;

% BasketflyBallnit1.mF0=25.0; %Ncart_mass=1; %kgx_dot_max=50; %m/secro_air=1.224; %kg/m^3Cd=1;r_ball=0.05; %mA_ball=pi*r_ball^2;ball_mass=0.05; %kgg=-9.8; %m/sec^2theta_0=pi/2.5; %radV0=50; %m/secAeroFac=Cd*A_ball*ro_air/2;

%% Start_flyBasketBall.mInitialGuess= pi/2.5 ;X = fminsearch('Distflysim', InitialGuess)*180/pi;fprintf('\nShoot at %f deg \n', X);

1. Optimization toolbox for use with MATLAB, User Guide, The MathWorks Inc. 2006

2. Applied Optimization with MATLAB Programming, P. Venkataraman, Wiley InterScience, 2002

3. Optimization for Engieering Design, Kalyanmoy Deb, Prentice Hall, 1996.

4. http://mathdemos.gcsu.edu/mathdemos/maxmin/max_min.html

5. http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory /MaxMin.html

6. http://users.powernet.co.uk/kienzle/octave/optim.html

7. http://www.cse.uiuc.edu/eot/modules/optimization/SteepestDescent/

REFERENCES