# Simple example application of Karush-Kuhn-Tucker conditions to minimization problem

I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a good reference textbook or monograph with an example?

Moreover, I am wondering if there is a computer algebra system (CAS) program or function that can be used to experiment with multivariate equations. The input to the program would be a multivariate equation, whereas the output would be some sort of analytic check for convergence (i.e. that a minimum exists). Perhaps such a program would quickly help to check equations for convergence.

• The KKT conditions are necessary conditions, just like Lagrange multipliers. Generally they cannot be used to show that a minimum exists unless you know something more about the problem (convexity, for example). – copper.hat Aug 1 '12 at 18:55
• @copper.hat: Thanks, copper.hat. Is there a good expository reference that deals with the KKT conditions, and perhaps demonstrates how to use them? How do I show convexity? – Nicholas Kinar Aug 1 '12 at 19:15
• This is a tough one for me. Frankly, it took me a long time to grasp the KKT conditions (and similar), and it wasn't until I acquired some familiarity with nonsmooth analysis (Frank Clarke, "Optimization and Nonsmooth Analysis") in the context of minimax problems ($\min_x \max_i f_i(x)$) that I began to get a grip on understanding. However, that is a tough route to follow, if understanding KKT is your goal. I have only glanced at the book, but Boyd & Vandenberghe's "Convex Optimization" has a section on KKT for convex problems, which might be good to develop intuition. – copper.hat Aug 1 '12 at 20:23
• Oh, these references are just lovely, and the Boyd & Vandenberghe book is available online as well: stanford.edu/~boyd/cvxbook, so I can read it immediately. Please write this up as an answer. – Nicholas Kinar Aug 1 '12 at 20:34
• You are welcome, hope it helps you. – copper.hat Aug 1 '12 at 20:54