OR 6320: Nonlinear Programming. Spring 2010.
Assignment 1 solutions.
1. a) We nd f (u; v ) = (sin u; 3v 2 cos(v 3 ) and 2 f (u; v ) = Diag(cos u; 6v cos(v 3 )9v 4 sin(v 3 ).
At x = 0, the gradient vanishes and the second derivative is Diag(1; 0), which is

OR 6320: Nonlinear Programming. Spring 2010.
Assignment 2 solutions.
1. a) We have to show that epi v is convex. Let (wi , i ) epi v for i = 1, 2, so that i v (wi ).
Choose > 0. Then, since
i inf cfw_f (x) + g (Ax wi ,
x
we can nd xi such that
i f (xi ) +

OR 6320: Nonlinear Programming. Spring 2010.
Assignment 4 solutions.
1. Let the columns of Z form a basis for the null space of A, so that x satises Ax = b i
x = x + Zdz for some dz . Then an unconstrained problem equivalent to minx cfw_f (x) : Ax = b
is

OR 6320: Nonlinear Programming. Spring 2010.
Comments on the nal.
1. You all did pretty well on this. Its useful to list a few key properties before
starting (a) or (b) this is similar to proving some lemmas to prepare for a theorem.
First, we have yz = Z

OR 6320: Nonlinear Programming. Spring 2010.
Monday May 17, 9 am - Tuesday May 18, 12 m (thats noon!).
This is to be all your own work. You may use any result from class or homeworks,
and any standard real analysis or linear algebra result. Do not consult