Page 374 - Applied Numerical Methods Using MATLAB

PROBLEMS 363
(c) Removing the third term from the objective function and splitting the
equality constraint into two reversed inequality constraints, we can
modify the problem as follows:

Min x f(x) =−3x 1 − 2x 2 (P7.5.3a)
subject to the constraints
 
3 4 ≤ 7

 −2 −1  x 1 ≥−3
  and (P7.5.3b)
 3 −2  x 2 ≤−2
3 −2 ≥−2

0 x 1 10
l = ≤ x = ≤ = u
0 x 2 10
Noting that this fits the linear programming, apply the routine “lin-
prog()” to solve this problem.
(d) Treating the equality constraint separately from the inequality con-
straints, we can modify the problem as follows:
Min x f(x) =−3x 1 − 2x 2 (P7.5.4a)
subject to the constraints
 
3 −2 = −2
 3 x 1 ≤ 7 0 x 1 10 = u
4  and l = ≤ x = ≤
x 2 0 x 2 10
−2 −1 ≥ −3
(P7.5.4b)
Apply the two routines “linprog()”and “fmincon()” to solve this
problem and see if the solutions agree with the solution obtained in (c).
(cf) Note that, in comparison with the routine “fmincon()”, which can solve a gen-
eral nonlinear optimization problem, the routine “linprog()” is made solely
for dealing with a class of optimization problems having a linear objective
function with linear constraints.
7.6 Nonnegative Constrained LS and Constrained Optimization
Consider the problem of minimizing a nonlinear objective function
2
T
Min x ||Cx − d|| = [Cx − d] [Cx − d] (P7.6.1a)
subject to the constraints

0
x 1
x = ≥ = l (P7.6.1b)
x 2 0
where
   
12 5.1
C =  34  , d =  10.8  (P7.6.1c)
51 6.8