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Examination: Continuous Optimization 3TU-and LNMB-course, Utrecht December 22, 2009, 13.00-16.00

Summary: Examination: Continuous Optimization
3TU- and LNMB-course, Utrecht December 22, 2009, 13.00-16.00
Ex. 1
(a) Given a Rn
, show that the matrix aaT
is positive semidefinite.
(b) Show that a symmetric matrix A Rnn
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
, R). Suppose a feasible point x satisfies the KKT-conditions
(Karush-Kuhn-Tucker) 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 Wolfe-Dual (WD) of (CO).
Ex. 3 Let fi : C R, i I := {1, . . . , m} be convex functions on the convex compact set
C Rn


Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente


Collections: Engineering