Constrained Optimization
5
Most problems in structural optimization must be formulated as constrained minimization problems. In a typical structural design problem the objective function
is a fairly simple function of the design variables (e.g., weight),

EGM6365 Homework#4
1. Make a Matlab script to solve the following unconstrained optimization problem using
a Newton method.
Minimize
f ( x) = 1 + 2( x + 1)2 x 3 + e x
Use the initial point x = 6 and -6. The iteration should stop when the function gradient

EGM6365 Homework #3
1. A linear programming problem is given as
Maximize
f = 2 x1 + 5 x2
Subject to
3 x1 + 2 x2 12
2 x1 + 3x2 6
(1)
x1 0, x2 unrestricted
(i)
(ii)
Solve Eq. (1) using MATLAB.
Convert Eq. (1) into the standard form and solve the standards p

EGM6365 Homework #2
1. Consider the following design optimization problem:
Minimize
2
f (x) = x12 + x2 4 x1 + 4
Subject to
g1 (x) = x1 0
g 2 ( x) = x2 0
g3 (x) = x2 (1 x1 )3 0
(i)
(ii)
Find the optimum point graphically
Show that the optimum point does no

EGM6365 Homework #1
1. A beer can is supposed to hold at least specific amount of beer and meet other design
requirements. The can will be produced in billions, so it is desirable to minimize the cost
of manufacturing them. Since the cost can be related d

EGM6365 Homework#4
1. Make a Matlab script to solve the following unconstrained optimization problem using
a Newton method.
Minimize
f ( x) = 1 + 2( x + 1)2 x 3 + e x
Use the initial point x = 6 and -6. The iteration should stop when the function gradient

EGM6365 Homework #3
1. A linear programming problem is given as
Maximize
f = 2 x1 + 5 x2
Subject to
3 x1 + 2 x2 12
2 x1 + 3x2 6
(1)
x1 0, x2 unrestricted
(i)
(ii)
Solve Eq. (1) using MATLAB.
Convert Eq. (1) into the standard form and solve the standards p

EGM6365 Homework #2
1. Consider the following design optimization problem:
Minimize
2
f (x) = x12 + x2 4 x1 + 4
Subject to
g1 (x) = x1 0
g 2 ( x) = x2 0
g3 (x) = x2 (1 x1 )3 0
(i)
(ii)
Find the optimum point graphically
Show that the optimum point does no

Educational article
Struct Multidisc Optim 21, 120127 Springer-Verlag 2001
A 99 line topology optimization code written in Matlab
O. Sigmund
Abstract The paper presents a compact Matlab implementation of a topology optimization code for compliance minimiz

Chapter 5: Constrained Optimization
5.5 Gradient Projection and Reduced Gradient Methods
Rosens gradient projection method is based on projecting the search direction into
the subspace tangent to the active constraints. Let us rst examine the method for
t

EGM6365 Project #1 Optimization of Space Structure
A space frame structure as shown in the figure consists of 25 truss members. Initially, all
members have the same circular cross-sections with diameter d = 2.0 in. At nodes 1 and 2,
a constant force F = 6