VL 04: Complementarity Modeling

Question 1

Define convexity for a function using a straight line

Answer 1

s. 7

Question 2

Define convexity for a function using the hessian matrix.

Answer 2

A function f is convex if and only if its Hessian matrix is positiv semi-definite.

A matrix Q is positiv semi-definite if all Eigenvalues are nonnegative.

det[Q-(lamba*E)] = 0
(gives you the Eigenvalues)

s. 9, 10

Question 3

How do you solve difficult KKT problems by hand?

Answer 3

case analysis for the lambdas

Question 4

What are the sufficient conditions for finding the optimum with the KKT method?

Answer 4

Constraint qualifications

• There are many different CQ
• Examples for CQ: Linearity or convexity of the optimization problem

Question 5

What is the first step when using the KKT method?

Answer 5

Check whether the objective function is convex or not!

Question 6

min -x^2

s.t. -1 <= x <= 1

Use the KKT Method.

Answer 6

-x^2 is not convex. Therefore you can’t use the KKT Method.

Question 7

True or false?

In order to use the KKT method you have to bring the optimizition problem into standardform.

Answer 7

True!

max objective function 
only g(x) <= 0 
and f(x) = 0 conditions

Question 8

True or false?

GAMS mcp solver can only solve greater or equal to condition.

Answer 8

True!