Page 368 - Applied Numerical Methods Using MATLAB

PROBLEMS 357
Table 7.3 The Names of MATLAB Built-In Minimization Routines in MATLAB 5.x/6.x
Unconstrained Minimization Constrained Minimization
Minimization Non-Gradient- Gradient- Linear Nonlinear
Methods Bracketing Based Based Linear Nonlinear LS LS Minimax
MATLAB 5.x fmin fmins fminu lp constr leastsq conls minimax
MATLAB 6.x fminbnd fminsearch fminunc linprog fmincon lsqnonlin lsqlin fminimax

The program also applies the general constrained minimization routine “fmin-
con()” to solve the same problem for cross-check. Readers are welcome to run
the program and see the results.

>> nm733
xo_lp = [0.3333 1.5000], fo_lp = -4.0000
cons_satisfied = -0.0000 % <= 0(inequality)
-0.8333 % <= 0(inequality)
-0.0000 % = 0(equality)
xo_con = [0.3333 1.5000], fo_con = -4.0000
In this result, the solutions obtained by using the two routines “linprog()”and
“fmincon()” agree with each other, satisfying the inequality/equality constraints
and it can be assured by Fig. 7.15.
In Table 7.3, the names of MATLAB built-in minimization routines in MAT-
LAB version 5.x and 6.x are listed.

PROBLEMS

7.1 Modification of Golden Search Method
In fact, the golden search method explained in Section 7.1 requires only
one function evaluation per iteration, since one point of a new interval
coincides with a point of the previous interval so that only one trial point
is updated. In spite of this fact, the MATLAB routine “opt_gs()”imple-
menting the method performs the function evaluations twice per iteration.
An improvement may be initiated by modifying the declaration type as

[xo,fo] = opt_gs1(f,a,e,fe,r1,b,r,TolX,TolFun,k)

so that anyone could use the new routine as in the following program,
where its input argument list contains another point (e)aswellasthe new
end point (b) of the next interval, its function value (fe), and a parameter
(r1) specifying if the point is the left one or the right one. Based on this
idea, how do you revise the routine “opt_gs()” to cut down the number
of function evaluations?