An Introduction to Computational Algebraic Geometry and Polynomial Optimization

A polynomial optimization problem (POP) is the task to minimize a multivari-
ate real polynomial given finitely polynomial inequalities as constraints. Both

systems of polynomial equations and POPs appear in countless applications in
various areas of science and engineering.
Traditionally, computational algebraic geometry deals with solving systems
of polynomial equations over the complex numbers. In the first part of my talk,
I will highlight two of the canonical approaches – Gr ̈obner bases and homotopy
continuation methods – to tackle these systems. I will also point out some of
the problems that one faces when considering real instead of complex numbers.
In the second part of the talk, I will explain how certificates of nonnegativity
can be used to attack POPs in practice. Exemplary, I will compare semidefinite
programming using the classical sums of squares (SOS) certificates with relative
entropy programming using sums of nonnegative circuit polynomials (SONC)
certificates, which were recently developed by Iliman and myself.

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