Hand in hard copies of your code and results, and answers to the written question, through Canvas.
Exercise 4.7 from the text.
Write a Matlab routine called DoglegTR.m to implement the dogleg trust-region algorithm with a modi ed Hessian. Use the trust-region framework of Algorithm 4.1, with the dogleg procedure of p. 73-75 to nd the approximate solution of the trust-region subproblem. For the subproblem Hessian Bk use the true Hessian r2f(xk) modi ed in that the eigenvalues i are replaced by max( i; ), for i = 1; 2; : : : ; n, for some positive parameter .
^
Use parameters = 10, = :01, 0 = 1, = :01.
The header line of your routine should be
function [inform,x] = DoglegTR(fun, x, trparams)
where the input parameters are:
fun - a pointer to a function (such as obja, objb, objc)
x - a strucure with elds x.p, x.f, x.g, and x.h, in which x.p contains the point x, while x.f, x.g, and x.h contain the function, gradient, and Hessian values corresponding to x. On input, x.p is set to the starting point values x = struct(’p’, [-1.2, 1.0]); while x.f, x.g, and x.h are left blank. (The latter values can be used by DoglegTR to store the function information at iterates of the algorithm.)
trparams - a structure containing parameter values for the test:
trparams = struct(’maxit’,100,’delta’,.01,’hatDelta’,10,...
’eta’,.01,’Delta0’,1,’toler’,1.0e-6);
(The parameter names and values are discussed above and below.)
Your routine should call on fun to evaluate the objective function, gradient, and Hessian at computed points, using
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x.f = feval(fun,x.p,1);
and
x.g = feval(fun,x.p,2);
and
x.h = feval(fun,x.p,4);
You should terminated when either krf(xk)k2 10 6 or 100 function evaluations have been taken, whichever comes rst.
The output inform is a structure containing two elds: inform.status is 1 if the gradient tolerance is achieved and 0 if not, while inform.iter is the number of steps taken. The output x is the solution structure, with point, function, and gradient values at the nal value of xk.
Matlab functions that implement the three functions below can be found on Canvas, under the names obja.m, objb.m, and objc.m. Your program will be tested using the code DoglegMain.m, also posted. For each of the three functions above, the code will be run with initial point x0 = ( 1:2; 1)T .
(a) f(x) = x12
+ 5x22 + x1
5x2
(b) f(x) = x12
+ 5x1x2 + 100x22
x1 + 4x2
(c) f(x) = 100(x2 x12)2
+ (1
x1)2
Note that the global variables numf, numg, and numH are incremented by the function evaluation routines. Be sure to set these to zero at the start of DoglegTR.m.
Use the following line in your code, to print out information at each iteration:
fprintf(1,’ iter %3d: f=%12.5e, ||Df||=%12.5e, Delta=%7.2e\n’,...
inform.iter, x.f, norm(x.g), Delta);
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