Page 375 - Applied Numerical Methods Using MATLAB

364 OPTIMIZATION
(a) Noting that this problem has no other constraints than the lower bound,
apply the constrained linear least-squares routine “lsqlin()”tofind
the solution.
(b) Noting that the lower bounds for all the variables are zeros, apply the
MATLAB built-in routine “lsqnonneg()” to find the solution.
(c) Applythegeneral-purposeconstrainedoptimizationroutine“fmincon()”
to find the solution.
7.7 Constrained Optimization Problems
Solve the following constrained optimization problems by using the MAT-
LAB built-in routine “fmincon()”.

(a) 3 2 2 (P7.7.1a)
Min x x − 5x + 6x 1 + x − 2x 2 + x 3
1 1 2
subject to the constraints
2
2
x + x − x 3 ≤ 0    
0
1 2 x 1
2
2
2
x + x + x ≥ 6 and x =  x 2  ≥  0  = l (P7.7.1b)
1 2 3
x 3 ≤ 5 x 3 0
Try the routine “fmincon()” with the initial guesses listed in
Table P7.7.
Table P7.7 The Results of Applying ‘‘fmincon()’’ with Different Initial Guess
Initial Guess Lower
o
x 0 Bound x o f(x ) Remark (warning ?)
[0 0 0] 0 No feasible solution (w)
[1 1 5] 0 Not a minimum
(a)
[0 0 5] 0 Minimum
[1 0 2] 0 [1.29 0.57 2] 2.74
[0 0 0] 0 [0 0 0] 0
(b1)
[10 10 10] 0 Maximum (good)
[0 0 0] 0 No feasible solution (w)
(b2) [10 10 10] 0 Not a minimum, but the max
[0.1 0.1 3] 0 One of many minima (w)
[0 0 0] 0
(c1) [0.1 0.1 0.1] 0 [1 1 1] 3 Maximum (good)
[0 1 2] 0
[0 0 0] 0 [1 1 1] 3 Not a minimum, but the max
(c2) [0.1 0.1 0.1] 0
[0 1 2] 0 One of many minima
[1.0 0.5] 0 Weird (warning)
[0.2 0.3] 0 [10.25 0] ∞
(d)
[2 5] 0 [5.77 8.17] 25.98
[100 10] 0 Minimum