Title: LP and SDP-relaxations for Polynomial Programming with applications in mathematical finance Author: Jean B. Lasserre Abstract: We consider the general global optimization problem where both criteria and constraints are polynomials functions (polynomial programming). Recently, results in real algebraic geometry on the representation of polynomials, positive on a compact set, have permitted to define a hierarchy of SDP-relaxations whose associated sequence of optimal values converges to the global optimum, under relatively weak assumptions. On the other hand, LP-relaxations had been already defined with finite convergence for 0-1 programs. In fact, using an old representation result of Krivine, we show that LP-relaxations also converge for general polynomial programming problems. We then compare the relative advantages of both LP and SDP-relaxations. We also illustrate the power of SDP-relaxations for certain option pricing problems in mathematical finance.