# Algorithm for Optimization and Interpolation based on Hyponormality

## TL;DR

This paper introduces a unified linear algebra-based algorithm leveraging hyponormality to solve complex polynomial optimization and exponential interpolation problems, extending existing methods and avoiding algebraic geometry.

## Contribution

It presents the first algorithm capable of extracting global solutions from complex polynomial optimization problems and offers an alternative to Prony's method for interpolation.

## Key findings

- Successfully extracts global solutions for complex polynomial optimization.
- Provides an algebraic alternative to Prony's method for interpolation.
- Demonstrates enforcement of hyponormality to improve relaxation solutions.

## Abstract

On one hand, consider the problem of finding global solutions to a polynomial optimization problem and, on the other hand, consider the problem of interpolating a set of points with a complex exponential function. This paper proposes a single algorithm to address both problems. It draws on the notion of hyponormality in operator theory. Concerning optimization, it seems to be the first algorithm that is capable of extracting global solutions from a polynomial optimization problem where the variables and data are complex numbers. It also applies to real polynomial optimization, a special case of complex polynomial optimization, and thus extends the work of Henrion and Lasserre implemented in GloptiPoly. Concerning interpolation, the algorithm provides an alternative to Prony's method based on the Autonne-Takagi factorization and it avoids solving a Vandermonde system. The algorithm and its proof are based exclusively on linear algebra. They are devoid of notions from algebraic geometry, contrary to existing methods for interpolation. The algorithm is tested on a series of examples, each illustrating a different facet of the approach. One of the examples demonstrates that hyponormality can be enforced numerically to strenghten a convex relaxation and to force its solution to have rank one.