Nonnegative Polynomials, Nonconvex Polynomial Optimization, and Applications to Learning

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.

Intro
Optimizing over nonnegative polynomials
1. Shape-constrained regression
2. Difference of Convex (DC) programming Problems of the form min fo (x)
Monotone regression: problem definition
NP-hardness and SOS relaxation
Approximation theorem
Numerical experiments (1/2) • Low noise environment
Difference of Convex (dc) decomposition
Existence of dc decomposition (2/3)
Convex-Concave Procedure (CCP)
Picking the "best" decomposition for CCP
Undominated decompositions (1/2)
Comparing different decompositions (1/2)
Main messages • Optimization over nonnegative polynomials has many applications Powerful SDP/SOS-based relaxations available.
Uniqueness of dc decomposition