Optimization toolbox presentation

MATLAB Optimization
Toolbox
Presented by
Chin Pei
February 28, 2003

Presentation Outline
 Introduction
 Function Optimization
 Optimization Toolbox
 Routines / Algorithms available
 Minimization Problems
 Unconstrained
 Constrained

Example

The Algorithm Description
 Multiobjective Optimization
 Optimal PID Control Example

Function Optimization
 Optimization concerns the minimization
or maximization of functions
 Standard Optimization Problem
IntroductionIntroduction Unconstrained MinimizationUnconstrained Minimization Constrained MinimizationConstrained Minimization
Multiobjective OptimizationMultiobjective Optimization ConclusionConclusion
( )~
~
min
x
f x
( )~
0jg x ≤
( )~
0ih x =
L U
k 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.
~
x is a column vector of design variables, which can
affect the performance of the system.

Constraints – Limitation to the design space.
Can be linear or nonlinear, explicit or implicit functions
( )~
0jg x ≤
( )~
0ih x =
L U
k k kx x x≤ ≤
Equality Constraints
Inequality Constraints
Side Constraints
Most algorithm require less than!!!

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

Minimization Algorithm

Minimization Algorithm (Cont.)

Equation Solving Algorithms

Least-Squares Algorithms

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.

Unconstrained Minimization
 Consider the problem of finding a set of
values [x1 x2]T
that solves
IntroductionIntroduction Unconstrained Minimization Constrained MinimizationConstrained Minimization
( ) ( )1
~
2 2
1 2 1 2 2
~
min 4 2 4 2 1x
x
f x e x x x x x= + + + +
[ ]1 2
~
T
x 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

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
~
T
x x x=
Objective function

Step 2 – Invoke Routine
x0 = [-1,1];
options = optimset(‘LargeScale’,’off’);
[xmin,feval,exitflag,output]=
fminunc(‘objfun’,x0,options);
Output arguments
Input arguments
Starting with a guess
Optimization parameters settings

Results
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

More on fminunc – Input
[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.
Ref. Manual: Pg. 5-5 to 5-9

More on fminunc – Output
[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

Options Setting – optimset
Options =
optimset(‘param1’,value1, ‘param2’,value2,…)
 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 Setting (Cont.)
Options =
 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 ]
The default is with { }
Parameter (param1)
Value (value1)

Options Setting (Cont.)
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.

Useful Option Settings
 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!!!

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.

fminunc and fminsearch
 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.

Constrained Minimization
IntroductionIntroduction Unconstrained MinimizationUnconstrained Minimization Constrained Minimization
[xmin,feval,exitflag,output,lambda,grad,hessian]
=
fmincon(fun,x0,A,B,Aeq,Beq,LB,UB,NONLCON,options,
P1,P2,…)
Vector of Lagrange
Multiplier at optimal
point

Example
( )~
1 2 3
~
min
x
f x x x x= −
2
1 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 72
A B
− − −   
= =   
   
0 30
0 , 30
0 30
LB UB
   
   = =   
      
function f = myfun(x)
f=-x(1)*x(2)*x(3);

Example (Cont.)
2
1 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 null
Matrix if the constraint does
not 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 72
A 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).
> In D:ProgramsMATLAB6p1toolboxoptimfmincon.m at line 213
In D:usrCHINTANGOptToolboxmin_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
2
1 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

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 by designing an
optimal PID controller for an plant.
Multiobjective Optimization ConclusionConclusion

Simulink Example
Goal: Optimize the control parameters in Simulink model optsim.mdl
in order to minimize the error between the output and input.
Plant description:
• Third order under-damped with actuator limits.
• Actuation limits are a saturation limit and a slew rate limit.
• Saturation limit cuts off input: +/- 2 units
• Slew rate limit: 0.8 unit/sec

Simulink Example (Cont.)
Initial PID Controller Design

Solving Methodology
 Design variables are the gains in PID
controller (KP, KI and KD) .
 Objective function is the error between
the output and input.

Solving Methodology (Cont.)
 Let pid = [Kp Ki Kd]T
 Let also the step input is unity.
 F = yout - 1
 Construct a function tracklsq for
objective function.

Objective Function
function F = tracklsq(pid,a1,a2)
Kp = pid(1);
Ki = pid(2);
Kd = pid(3);
% Compute function value
opt = simset('solver','ode5','SrcWorkspace','Current');
[tout,xout,yout] = sim('optsim',[0 100],opt);
F = yout-1;
Getting the simulation
data from Simulink
The idea is perform nonlinear least squares
minimization of the errors from time 0 to 100
at the time step of 1.
So, there are 101 objective functions to minimize.

The lsqnonlin
[X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA,JACOBIAN]
= LSQNONLIN(FUN,X0,LB,UB,OPTIONS,P1,P2,..)

Invoking the Routine
clear all
Optsim;
pid0 = [0.63 0.0504 1.9688];
a1 = 3; a2 = 43;
options =
optimset('LargeScale','off','Display','iter','TolX',0.001,'TolFun
',0.001);
pid = lsqnonlin(@tracklsq,pid0,[],[],options,a1,a2)
Kp = pid(1); Ki = pid(2); Kd = pid(3);

Results
Optimal gains

Results (Cont.)
Initial Design
Optimization Process
Optimal Controller Result

Conclusion
 Easy to use! But, we do not know what is happening
behind the routine. Therefore, it is still important to
understand the limitation of each routine.
 Basic steps:
 Recognize the class of optimization problem
 Define the design variables
 Create objective function
 Recognize the constraints
 Start an initial guess
 Invoke suitable routine
 Analyze the results (it might not make sense)
Multiobjective OptimizationMultiobjective Optimization Conclusion

Thank You!
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