| The dilemma of mathematical optimization today is that many, often simple-sounding problems cannot be solved efficiently with current methods. In this project the researchers analyze the symmetry and other structural properties of these problems. This will open new computational ways to find solutions. The main objective of the proposed research is the combination of semidefinite programming and harmonic analysis. The goal is to use this combination to solve computational difficult problems which cannot be attacked by current techniques. Over the last two decades semidefinite programming became one of the strongest general purpose tools for the design and analysis of efficient algorithms in optimization. Over the last two centuries harmonic analysis (Fourier analysis) became the strongest general purpose tool to exploit qualitative and quantitative structure of mathematical objects, like functions and operators. Now Fourier analysis is omnipresent in our modern technological life. The foundations of this combination will be established. The aim is to gain insight into the mathematical structure of the considered computational problems and to establish a convenient theoretical framework for computer-assisted proofs. Harmonic analysis predominantely deals with transcendental objects, i.e.\ objects in infinite-dimensional spaces. The corresponding theory of infinite-dimensional semidefinite programming still has to be developed, as well as the design, the analysis and the implementation of numerical approximation algorithms for infinite-dimensional semidefinite programs. To demonstrate that this combination is extremely fruitful many applications and difficult computational problems will be considered. These applications come from different areas: continuous combinatorial optimization (approximations to continuous versions of NP-hard optimization problems), geometry (packing problems), mathematical physics (energy minimization and minimal surfaces), statistics (correlation theory of stochastic processes), and engineering (signal processing). The techniques used are concrete and computational. With a symbiosis of human and computer reasoning substantial advances in difficult computational problems are to be expected. Intensive use of computer mathematics will be made, applying both symbolic algorithms from pure mathematics and numerical algorithms from applied mathematics. |
