initiation to matlab - lecture 5

Introduction Function Handle fsolve(.) fminsearch(.) fmincon(.) Complements Exercises Surface & Contour Plots

Initiation to Matlab - Lecture 5

Ludovic Calès1

1HEC - University of [email protected] - Room 253

M.Sc. Finance - Autumn 2012


This Course Outline

TodayFunction HandleSolve an equation with fsolve(.)Minimize an unconstrained problem with fminsearch(.)Minimize a constrained problem with fmincon(.)


Function Handle

ProblemThe optimization functions take the function to optimize asan input.However, a function is not a variable and therefore cannotbe passed in argument.

Solution: Function HandleThe function to optimize is transformed into a variable ofthe data type function handle.The handle, i.e. the variable representing a function, isobtained with the ’at’ sign (@) followed by the functionname.

myVariable = @FunctionName (1)


Example

ExampleA function handle for the cosine function – named cos in Matlab– is built as follows:

>> myCosineHandle = @cos;

Now, the variable myCosineHandle can be passed in argumentto another function. Remark that the function handle is notused to call the related function but to handle it as simply asany other variable.


fsolve(.)

Definition

The function fsolve(.) solve the equation f (X ) = ~0 where X and f (X )are vectors.

Syntax

[X , fval ,exitflag] = fsolve(myfcthandle,X0) (2)

where

myfcthandle function handle of fX0 initial valuesexitflag > 0 , succeeded,

< 0 , failed,0 , did not end (e.g. it reaches the max.

number of iterations),X solutionfval value of f in X .


Example

ProblemSolve the following problem

2x5 + 3x4 − 30x3 − 57x2 − 2x + 24 = 0

Solution1 Write the .m file of the function to solve:

function y = myFunction(x)y = 2*x.ˆ5 + 3*x.ˆ4 - 30*x.ˆ3 - 57*x.ˆ2 - 2*x + 24

end

2 use the fsolve(.) function with a guessed initial value.

>> [X , fval, exitflag] = fsolve(@myFunction,0)X =

0.56fval =

-3.324e-009exitflag =

1


fminsearch(.)

DefinitionThe function fminsearch(.) to find the minimum of a multivariatefunction f (x) returning a scalar, i.e. solves the problem min

xf (x)

Syntax

[X , fval ,exitflag] = fminsearch(myfcthandle,X0) (3)

where the arguments are those of the previous section.


Example: Bowl function

Problem

Find the minimum of f (x , y) = x2 + (y − 2)2, the bowl function.

Figure: Surface and contour plots of the bowl function.


Example: Bowl function

Solution1 Write the .m file for the bowl function:

function z = Bowl(x)z = x(1)ˆ2 + (x(2) -2)ˆ2;

end

2 Call fminsearch(.) as follows

>> [ X , fval , exitflag ] = fminsearch(@Bowl, [ 1 1 ]);X =

-0.0000 2fval =

2.7371e-009exitflag =

1


fmincon(.)

Definition

The function fmincon(.) solves the following minimization problem

minx

f (x) subject to

∣∣∣∣∣∣∣∣∣c(x) ≤ 0ceq(x) = 0A x ≤ bAeq x = beqlb ≤ x ≤ ub

(4)

where c(x) and ceq(x) are non-linear functions, A and Aeq are matrices, b, beq, lband ub are vectors.

Syntax

[X , fval, exitflag] = fmincon(myfcthandle,X0,A, b,Aeq, beq, lb, ub, nonlcon) (5)

where the arguments A, b, Aeq, beq, lb and ub are those given by the problem andnonlcon defines the functions c(x) and ceq(x). If some arguments do not apply, setthem to [ ].


Example: Himmelblau’s function

Problem

Find the minimum of f (x , y) = (x2 + y − 11)2 + (x + y2 − 7)2 such that x < 0and y > 0. Here, the system implied by the constraints is[

1 00 −1

]︸︷︷︸

A

[xy

]≤[

00

]︸︷︷︸

b

(6)

Figure: Surface and contour plots of Himmelblau’s function.


Example: Himmelblau’s function

Solution

1 Write the .m file for the Himmelblau function.

function z = Himmelblau(x)z = (x(1)ˆ 2+x(2)-11)ˆ 2 + (x(1)+x(2)ˆ 2-7)ˆ 2;

end

2 Define the parameters A and b as follows.

>> A = [ 1 0 ; 0 -1];>> b = [ 0 ; 0 ];

3 Call fmincon(.) as follows

>> [ X , fval , exitflag ] = fmincon(@Himmelblau,...[ -1 1 ], A, b, [], [], [], [], []);

X =-2.8050 3.1313

fval =4.0509e-007

exitflag =5


Modify Optimization Parameters with optimset(.)

DefinitionThe function optimset(.) modifies the optimization parameters.It takes as input a parameter name as a string followed by thevalue assigned to this parameter, and returns a structure whichis then provided as last argument to the functions fsolve(.),fminsearch(.) and fmincon(.).

Optimization Parameters’MaxIter’ : the number of iteration performed by thealgorithm (1000 by default)’TolFun’ : the precision desired for the result returned bythe objective function (10−6 by default)’TolX’ : the precision desired for the variables (10−6 bydefault).


Modify Optimization Parameters with optimset(.)

Syntax

options = optimset(′Param1′, value1,′ Param2′, value2, . . . )(7)

Example with fminsearch(.)>> options = optimset( ’TolFun’, 1e-10);>> [X, fval, exitflag] = fminsearch(@Bowl, [ 1 1 ], options);


Pass Extra Parameters to the Objective Functions

CaseSuppose that your objective function takes some parameters inaddition to its unknown variables. To pass these parameters asinput of the objective function while optimizing it, we use thefunction handle precising which input is the variable.

Syntax

OptimizationFct(@(x)myObjectiveFct(x ,a,b), . . . )

where x is the unknown variable, a and b are parameterspassed as input to my objective function. myObjectiveFct(.) isthe objective function and OptimizationFct(.) is theoptimization function, i.e. fsolve(.), fminsearch(.), fmincon(.).


Pass Extra Parameters to the Objective Functions

Example

Consider the function mypoly(x) = ax2 + bx + c where x is theunknown and, a, b and c are parameters. We want to find theminimum of mypoly(.) using fminsearch(.) and passing a, band c as input of mypoly(.).

1 The .m file of this function would befunction y = mypoly(x, a, b, c)

y = a*xˆ2 + b*x + cend

2 We set a, b and c and call fminsearch(.) as follows>> a=1; b=2; c=1;>> [X, fval, exitflag] = ...

fminsearch(@(x)mypoly(x, a, b, c), 1);


Exercise 1

Exercise 1

Find the minimum of f (x , y) = (1 − x)2 + 100 ∗ (y − x2)2

(Rosenbrock’s function).

Figure: Surface and contour plots of Rosenbrock’s function.


Solution

SolutionThe minimum of Rosenbrock’s function is (1,1).


Exercise 2

Exercise 2Find the minimum of f (x , y) = 20 + x2 + y2 − 10 (cos 2πx + cos 2πy)(Rastrigin’s function).

Figure: Surface and contour plots of Rastrigin’s function.


Solution

SolutionThe minimum of Rastrigin’s function is (0,0). However, thisfunction is known to be very difficult to solve and theminimization functions proposed by Matlab cannot find it easily.They require initial values near the minimum.


Surface & Contour Plots

Surface & Contour Plots

The function surf(X,Y,Z) plots a surface graph with Z = f(X,Y).

The function contour(X,Y,Z,n) plots a coutour graph with Z =f(X,Y) and n is the number of contour lines.

initiation to matlab - lecture 5

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