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Exercises II

4M020: Design Tools

Eindhoven University of Technology

Objective

Learn to use Matlabs constrained minimizer fmincon

Case Problem

Consider again the two-bar planar truss design problem with two design variables x1 = h and

x2 = d, see Exercises I. Aim is to design the truss such that the mass of the truss is minimizedwhile satisfying the constraints on the yield stress and the critical buckling force. There is also a

space limitation: height h has an upper limit Hmax (positive value).

Exercise 1: Getting started with fmincon

In Exercise 2 of Exercises I, the optimization problem was formulated in negative null form as:

minx

f(x) = C0 x2

2

S2 +x2

1

s.t. g1(x) = C1x11 0

g2(x) = C2

S2 +x2

1

x22x1

1 0

g3(x) = C3

S2 +x2

1

3/2x42x1

1 0

: x1,x2 > 0,

(1)

with C0 = 1.24, C1 = 0.25, C2 = 0.606, C3 = 10.32, = 100, and S= 1. The optimal solutionwas determined by visual inspection.

Now we find it numerically using fmincon.

a) Type help fmincon in Matlab. Also type doc fmincon. Read the documentation that

goes with the fmincon function in the Matlab optimization toolbox.

b) Type demo in Matlab. Unfold the Toolboxes directory (by clicking the +). Unfold the

Optimization directory. Select Tutorial. Read the sections entitled Constrained opti-

mization example: inequalities and Constrained optimization example: inequalities and

bounds.

c) Why is fmincon suitable to solve optimization problem (1)?

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d) Solve the two-bar truss design problem using fmincon. Make three small m-files, one con-

taining the call to fmincon (the main program), and the two others containing the objec-

tive function and the constraints, respectively.

Compare the outcome with the plot of the previous week. Which constraint(s) is (are) active

at the minimizer?

Exercise 2: Visualization of the search path offmincon

We change algorithm parameter settings in fmincon using optimset, and then visualize the

iteration path to investigate how fmincon searches for the constrained minimizer.

a) Matlab has various default parameter settings for the optimization algorithms in the MatlabOptimization Toolbox. Include the lines

options = optimset(fmincon);

options = optimset(options,LargeScale,off);

options = optimset(options,Display,Iter);

options = optimset(options,TolX,1e-4,TolCon,1e-4,...

TolFun,1e-4);

before calling fmincon, and include the options argument in your call of fmincon.

Solve problem (1) starting from initial design x0 = (3,4).

Explain what the above lines do.

b) The iteration points xk with the respective objective function values fk along the iteration path

can be obtained as follows: include

global HISTORY

HISTORY.x = [];

HISTORY.fval = [];

options = optimset(options,outputfcn,@outfun);

afteryour optimset options definition(s) and before the call to fmincon. Add to your

working directory the m-file outfun.m.

Run your program again. Investigate the contents ofHISTORY by typing HISTORY.x and

HISTORY.fval on the Matlab command line. Run the file plothistory.m to plot the

iteration path in the contour plot of the two-bar truss.

c) Change your initial design to the (infeasible) starting point x0 = (3,1) and see how the iterationpath offmincon changes.

d) fmincon is a Newton-type of local search algorithm. Gradients of the objective function and

constraints are an indispensible part of such an algorithm.

Did you provide gradient information? How does fmincon obtain the gradients if you did

not?

How can the user provide the gradients to fmincon? (you do not need to implement this

for this exercise)

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Exercise 3: FEM-model in the loop

Instead of the analytical expressions for objective function and constraints we now use the Matlab

FEM-2D model to calculate mass m, bar force P, and stress . The optimization problem is then

given by:

minx

f(x) = m(x)

s.t. g1(x) = C1x11 0

g2(x) =(x)

y1 0

g3(x) =P(x)

Pcrit(x)1 0

: x1,x2 > 0,

(2)

a) Edit the files of Exercise 1 d) such that the FEM-2D model is used for the calculation of the

objective function and constraints.

b) Run your program and compare with Exercise 1 and 2.

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