Engineering Optimization: Theory and Practice, Fourth Edition

490 Nonlinear Programming III: Constrained Optimization Techniques

7.45 Find the extrapolated solution of Problem 7.44 by using quadratic relations forX(r)and

f (r).

7.46 Give a proof for the convergence of exterior penalty function method.

7.47 Write a computer program to implement the interior penalty function method with

the DFP method of unconstrained minimization and the cubic interpolation method of

one-dimensional search.

7.48 Write a computer program to implement the exterior penalty function method with

the BFGS method of unconstrained minimization and the direct root method of

one-dimensional search.

7.49 Write a computer program to implement the augmented Lagrange multiplier method with

a suitable method of unconstrained minimization.

7.50 Write a computer program to implement the sequential linear programming method.

7.51 Find the solution of the welded beam design problem formulated in Section 7.22.3 using

the MATLAB functionfminconwith the starting pointX 1 = { 0. 4 , 6. 0 , 9. 0 , 0. 5 }T

7.52 Find the solution of the following problem (known as Rosen–Suzuki problem) using the

MATLAB functionfminconwith the starting pointX 1 = { 0 , 0 , 0 , 0 }T:

Minimize

f (X)=x 12 +x^22 + 2 x 32 −x^24 − 5 x 1 − 5 x 2 − 21 x 3 + 7 x 4 + 100

subject to

x^21 +x^22 +x 32 +x 42 +x 1 −x 2 +x 3 −x 4 − 100 ≤ 0

x^21 + 2 x 22 +x^23 + 2 x^24 −x 1 −x 4 − 10 ≤ 0

2 x 12 +x 22 +x^23 + 2 x 1 −x 2 −x 4 − 5 ≤ 0

− 100 ≤xi≤ 100 , i= 1 , 2 , 3 , 4

7.53 Find the solution of the following problem using the MATLAB functionfminconwith

the starting pointX 1 = { 0. 5 , 1. 0 }T:

Minimize

f (X)=x 12 +x 22 − 4 x 1 − 6 x 2

subject to

x 1 +x 2 ≤ 2

2 x 1 + 3 x 2 ≤ 12

xi≥ 0 , i= 1 , 2

7.54 Find the solution of the following problem using the MATLAB functionfminconwith

the starting point:X 1 = { 0. 5 , 1. 0 , 1. 0 }:

Minimizef (X)=x^21 + 3 x^22 +x 3

subject to

x 12 +x^22 +x^23 = 16