this lecture will give a brief introduction of hilbert's 17th problem: is every nonnegative polynomial a summation of squares of rational functions? after artin's affirmative solution, there is a generalization to positivstellensatz on the solvability of semialgebraic systems. the positivstellensatz can be applied to solve global optimization problems of polynomial functions. the basic tool is semidefinite programming (sdp), a very nice convex optimization model.