Explore a comprehensive lecture on nonnegative polynomials, nonconvex polynomial optimization, and their applications in learning. Delve into shape-constrained regression, difference of convex (DC) programming, and monotone regression, including problem definitions and NP-hardness. Examine SOS relaxation techniques, approximation theorems, and numerical experiments in low noise environments. Investigate DC decomposition, the Convex-Concave Procedure (CCP), and strategies for selecting optimal decompositions. Gain insights into undominated decompositions and learn to compare different approaches. Understand the wide-ranging applications of optimization over nonnegative polynomials and the power of SDP/SOS-based relaxations in this field.