Page 377 - Applied Numerical Methods Using MATLAB

366 OPTIMIZATION
(cf) One might be disappointed with the reliability of the MATLAB optimization
routines to see that they may fail to find the optimal solution depending on the
initial guess. But, how can a human work be perfect in this world? It implies
the difficulty of nonlinear constrained optimization problems and can never
impair the celebrity and reliability of MATLAB. Actually, it demonstrates the
importance of studying some numerical stuff in addition to just getting used
to the various MATLAB commands and routines.
Here is a tip for the usage of “fmincon()”: it might be better to use with
an initial guess that is not at the origin, but in the admissible region satisfying
the constraints, even though it does not guarantee the success of the routine.
It might also be helpful to apply the routine with several values of the initial
guess and then choose the best result.

7.8 Constrained Optimization and Penalty Method
Consider again the constrained minimization problem having the objective
function (E7.3.1a) and the constraints (E7.3.1b).

2 2 2 2
Min f(x) ={(x 1 + 1.5) + 5(x 2 − 1.7) }{(x 1 − 1.4) + 0.6(x 2 − 0.5) } (P7.8.1a)
 
 
−x 1 0
−x 2
 
 0 
 
s.t. g(x) =  3x 1 − x 1 x 2 + 4x 2 − 7    (P7.8.1b)
  ≤  0 
2x 1 + x 2 − 3  
0
 
2
3x 1 − 4x − 4x 2 0
2
In Example 7.3, we made the MATLAB program “nm722.m” to solve the
problem and defined the objective function (E7.3.2a) having the penalized
constraint terms in the file named “f722p.m”.
2 2 2 2
Min l(x) ={(x 1 + 1.5) + 5(x 2 − 1.7) }{(x 1 − 1.4) + 0.6(x 2 − 0.5) }
5
+ v m ψ m (g m (x)) (P7.8.2a)
m=1
where

0 if g m (x) ≤ 0 (constraint satisfied)
ψ m (g m (x)) =
exp(e m g m (x)) if g m (x)> 0 (constraint viloated)
with e m = 1 ∀m = 1,. .., 5 (P7.8.2b)
(a) Whatistheweightingcoefficientvector vinthefilenamed“f722p.m”?Do
the points reached by the routines “fminsearch()”/“opt_
steep()”/“fminunc()” satisfy all the constraints so that they are in the
admissible region? If not, specify the constraint(s) violated by the points.
(b) Suppose the fourth constraint was violated by the point in (a). Then,
how would you modify the weighting coefficient vector v so that the
violated constraint can be paid more respect? Choose one of the fol-
lowing two weighting coefficient vectors: