A Unified Approach to Optimization via Generalized Moment Problems (25rit035)

The Banff International Research Station will host the "A Unified Approach to Optimization via Generalized Moment Problems" workshop in Banff from September 21 to October 4, 2025.

\noindent
Optimization problems, seeking to minimize (or maximize) a function
subject to constraints, are fundamental to numerous fields. While
Lasserre's SDP hierarchy {[}JBL01{]} has revolutionized polynomial optimization, its
application to broader classes of functions remains limited. This
proposal outlines a novel approach to extend Lasserre's methodology by
unifying recent advances in generalized moment problems, \((p,q)\)-mean
polynomials, and techniques from differential algebra. This unified
framework will enable the optimization of functions with nice algebraic
representations, including those arising from algebraic differential
equations. This research is expected to:
\begin{itemize}
\item
\textbf{Significantly extend the scope of Lasserre's SDP hierarchy} to
encompass a much wider class of functions.
\item
\textbf{Develop new theoretical and computational tools} for
optimization in semigroup algebras, unifying moment problems,
\((p,q)\)-polynomials, and differential algebra.
\item
\textbf{Advance understanding of Positivstellensatz} and its
applications in optimization.
\item
\textbf{Provide practical solutions to challenging optimization
problems} in diverse fields, including control theory, robotics,
engineering, and machine learning.
\end{itemize}
By developing this unified framework, this research aims to make a
significant contribution to both the theory and practice of
optimization, enabling the solution of problems previously considered
intractable.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.