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Global Optima

Finding global maxima and minima is harder. There is one case that is of interest.

We say that a domain is convex if every line drawn between two points in the domain lies within the domain.

We say that a function f is convex if the line connecting any two points lies above the function. That is, for all x,y in the domain and tex2html_wrap_inline567 , we have tex2html_wrap_inline569 , as before (see Chapter 2).

To check that a function is convex on a domain, check that its Hessian matrix H(x) is positive semidefinite for every point x in the domain. To check that a function is concave, check that its Hessian is negative semidefinite for every point in the domain.






The determinants of the principal minors are det tex2html_wrap_inline587 , det tex2html_wrap_inline589 and det tex2html_wrap_inline591 . So tex2html_wrap_inline593 is positive semidefinite for all tex2html_wrap_inline595 in tex2html_wrap_inline597 . This implies that f is convex over tex2html_wrap_inline597 .



Michael A. Trick
Mon Aug 24 14:09:40 EDT 1998