MATH 5750/6880 OPTIMIZATION
HOMEWORK 4
Due: Mon Dec 1st 2008
1. Problem 16.21 (Nocedal and Wright p495). This asks you to redo
the derivation of the interior point method for convex quadratic
programming (§16.6 pp 480–482) so that equality constraints are
included as well.
2. Problem 18.3 (Nocedal and Wright p562). Here you need to imple-
ment sequential quadratic programming and use it to solve a small
non-linear problem. Here some implementation advice:
•The initial point x0and the solution x∗are swapped in the
textbook, and the solution x∗has been rounded in a strange
way.
•No globalization procedure is required for this problem, you
should get convergence with the full steps.
•An equality constrained quadratic programming (EQP) solver
is in the class website: eqp.m. It uses the nullspace method
described in §16.2.
•The objective function, gradient and Hessian computations for
the given function are tedious to do by hand, so they are given
in the file f 18 3.m. You are responsible for the gradients and
Hessians of the constraints.
•The initial guess λ0for the Lagrange multipliers is not very
relevant, as long as ∇2
xxL(xk, λk) is symmetric positive definite
in N(c0(xk)).
•Use kc(xk)k2< as the stopping criterion with say = 10−8.
This is not a general purpose stopping criterion and stopping
when ∇xL(xk, λk) is small does not work in general. A better
stopping criterion is: kc(xk)k2< and
ZT
k∇xL(xk, λk)
<
, but you are not required to implement this. Here Zkis a
representation of the nullspace of the linearized constraints at
the current iterate.
3. (extra credit) Problem 18.7 in (Nocedal and Wright p562) with
the second stopping criterion above.
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