ECEN 434/754: Computer Project
Fall 2020
Project assignment: Consider the unconstrained problem:
minimize f(x) =
Xm
i=1
ln(1 ati
x)
Xn
i=1
ln(1 x2i
)
with variable x 2 Rn and dom(f) = fx : ati
x < 1; i = 1; 2; : : : ; m; jxij < 1; i = 1; 2; : : : ; ng.
Note that we can choose x(0) = 0 as our initial point. You can generate instances of this
problem by choosing ai from some distribution on Rn.
a) (10 Points) Derive an analytical expression for the gradient of the objective function.
Use the gradient method to solve the problem, using reasonable choices for the back-
tracking parameters, and a stopping criterion of the form krf(x)k2 . Plot in
separate gures the objective value (in log scale) and step length versus iteration num-
ber. Once you have determined p to high accuracy, you can also plot f p (in log
scale) versus iteration. Experiment with the backtracking parameters and to see
their eect on the total number of iterations required. Carry these experiments out for
several instances of the problem, of dierent sizes.
b) (10 Points) Derive an analytical expression for the Hessian of the objective function.
Redo the same problem instances from part a) using damped Newton’s method, with a
stopping criterion of the form (1=2)(x)2 , where (x)2 := rf(x)tr2f(x) 1rf(x).
Plot in separate gures f p (in log scale) and step length versus iteration number.
Describe the convergence behavior that you observe.
c) (10 Points) The cost of Newton’s methods is dominated by the cost of evaluating the
Hessian and nding its inverse. For large problems, it is sometimes useful to replace the
Hessian by a positive denite approximation that makes it easier to form and solve for
the search direction. For each of the approximate Newton methods described below,
test the method on the same instances from part a) and b) and compare the results to
those obtained using damped Newton’s method and gradient method.
i) (Re-using the Hessian)We evaluate and invert the Hessian only every N iterations,
where N > 1, and use the search direction x = H 1rf(x), where H is the
last Hessian evaluated
1
ii) (Diagonal approximation) We replace the Hessian by its diagonal, so we only
have to evaluate the n second derivatives @2f(x)=@x2i
, and computing the search
direction is very easy.
Report requirement:
1) Please include both gures and Matlab codes in your report. We will check both to
determine your project grade.
2) For each set of gures that you include in your report, please explicitly state the prob-
lem instance and algorithm parameters (backtracking parameters and , accuracy
parameters and ) of your choice.
3) Please submit your project report to Google Class by noon Monday, December 7th.
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