Bolzano-Weierstrass theorem

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A fundamental existence theorem guaranteeing a maximum and a minimum.

A continuous function on a non-empty compact set achieves a minimum and a maximum over the set.

A useful corollary to the Bolzano-Weierstrass theorem is:

Suppose LaTeX: f is continuous on its non-empty feasible region, LaTeX: F, and that LaTeX: F is a closed set. Then, LaTeX: f achieves a maximum on LaTeX: F if there exists LaTeX: x^0 such that LaTeX:  \{ x \in F : f(x) \ge f(x^0) \} is bounded.

In particular, if we have a quadratic objective of the form LaTeX: f(x) = x^T Q x + c^T x , it is sufficient for LaTeX: Q to be negative definite. Although often credited to Weierstrass alone, it was proven by Bolzano in 1817, and it was known to Cauchy near the end of the 19th century.


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