 
Summary: Examination: Continuous Optimization
3TU and LNMBcourse, Utrecht December 22, 2009, 13.0016.00
Ex. 1
(a) Given a Rn
, show that the matrix aaT
is positive semidefinite.
(b) Show that a symmetric matrix A Rn×n
is positive semidefinite if and only A · C 0
holds for all positive semidefinite matrices C.
(Here, for symmetric matrices, A · C denotes the "inner product", A · C = i,j aijcij)
Ex. 2 Consider the convex problem
(CO) min f(x) s.t. gj(x) 0, j = 1, . . . , m, x Rn
with convex functions f, gj C1
(Rn
, R). Suppose a feasible point x satisfies the KKTconditions
(KarushKuhnTucker) with a multiplier vector y 0.
(a) Show that (x, y) is a saddle point for the Lagrangian function L(x, y) of (CO).
(b) Show also that (x, y) is a solution of the WolfeDual (WD) of (CO).
Ex. 3 Let fi : C R, i I := {1, . . . , m} be convex functions on the convex compact set
C Rn
