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Polynomial Optimization, Efficiency through Moments and Algebra

Periodic Reporting for period 2 - POEMA (Polynomial Optimization, Efficiency through Moments and Algebra)

Okres sprawozdawczy: 2021-01-01 do 2023-06-30

Non-linear optimization problems are present in many real-life applications and in scientific areas such as operations research, control engineering, physics, information processing, economy, biology, etc. However, efficient computational procedures, that can provide the guaranteed global optimum, are lacking for them. The project aims to develop new polynomial optimization methods, combining moment relaxation procedures with computational algebraic tools to address this type of problems. Recent advances in mathematical programming and computer algebra have shown that the polynomial optimization problems can be approximated by sequences of Semi-Definite Programming problems, which provides a powerful way to compute global solutions of non-linear optimization problems and to guarantee the quality of computational results. Advanced algebraic algorithms allow to compute all the solutions of polynomial systems, with efficient implementations for exact and approximate solutions, were developed in the past twenty years.

Combining the expertise of active European teams working in these domains, the project has targeted important challenges in polynomial optimization and demonstrating the impact of this research on practical applications such as water distribution network management, energy flow in power systems, urban traffic management, oceanography and environmental monitoring and finance. The network has trained successfully 15 young researchers to master high-level mathematics, algorithm design, scientific computation and software development, and to solve optimization problems for real-world applications.
The successful campaign of recruitment of 15 ESRs leads to the constitution of a task force of young and talented researchers during the first year of the project. Thanks to a large communication on the PhD positions available in POEMA through various channels including academic networks, social networks, mailing lists in our community, etc, we received an impressive number of applications (more than 230). A strong review and selection process was organized by the position committee, with interviews taking place online and at the kick-off meeting.

New research in polynomial optimization has been developed inside the network. After the ESRs started investigating their subjects during the second year of the project, they all have produce new results in the area of algebraic computation for moments, mix-integer nonlinear programming, structure and robustness and applications as assessed by the publication accepted or submitted for publication. The important number of publications acknowledging POEMA support also shows the extreme vitality of the research activities of the network members.

During the project, significant advances have been made in particular by the ESRs and the other network members on exact representations of positive polynomials by Sum of Square, on the power of approximations of the moment and SOS hierachies, on the exploitation of symmetries and sparsity in polynomial optimization, on the extension of polynomial optimization in graph problems and copositive matrices, on the use of moment-sum-of-squares hierarchies for analysing solutions of dynamical systems and partial differential equations. Applications of these works have been developed in Optimal Power Flow problems, in finance, in geometric modeling, structural mechanics and truss topology optimization, VSLI design, the analysis of biological data and of protein interactions. Thanks to theses methodological progresses, problems which were previously unattainable fall now within the realm of feasible polynomial optimization problems.

The network POEMA also contributed to software developments, with the production of dedicated packages such as MomentTools, TSSOS, Loraine (in Julia) for polynomial optimization or msolve (in C) for finding the roots of polynomial equations.

The training and networking events have been organized as closely as possible to the initial plan. The kick-off meeting and first workshop took place on site and the first learning week, second and third workshops happened online. The international visibility of POEMA network in the community is very high. The ESRs are strongly involved and contributing to this success. We organized a very successful final conference, with excellent invited talks, very good presentations of the ESRs and an important attendance of and contributions from the members of the network and experts outside the network.

At this time, most of the PhD have defended successfully and the young PhDs are now either working in the academic sector as post-doctorate in order to pursue a career in research or working for government bodies or in industrial sectors such as financial technology.
The research investigations in the second period of the project gained momentum and lead to more effective visible outputs. The development of new algebraic techniques for polynomial optimization has been pursued, exploiting algebra, semidefinite programming, moment and/or sums of squares decomposition techniques. Its impact on the construction of more reliable numerical solvers for polynomial optimization and/or efficient algorithms based on symbolic computation can be observed.

General solution methods based on the dual theories of moments and sums of squares of polynomials and the design of efficient algorithms have been further developed. Applications within combinatorial optimization, finance and operations research, quantum information and energy networks have been investigated.

Exploiting structures which appear in polynomial optimization problems in order to improve the performance of the Semidefinite Programming solvers and the numerical quality of their solution has been been considered with a focus on the improvement of the performance of SDP relaxation methods by exploiting the underlying algebraic structure, of the numerical solution of SDP relaxation technique, the development of new relaxation methods, which exploit the structure of the nonlinear optimization problem.

Software developments have helped validating the methods and solving polynomial optimisation problems, which were not unttainable before.

The network has been very active in terms of training and collaboration, ensuring the individual research of the ESRs, their solid knowledge in technical aspects of polynomial optimization, the acquisitions of complementary skills, the intersectorial knowledge exchange with algebraists, geometers, computer scientists and industrial actors facing real-life optimization problems. The network has also considerably increased its international visibility through scientific meetings and dissemination actions, in order to improve the career prospects of young researchers, the resonance of research results in the academic field and their impact on the resolution of concrete problems.

The network has also made an important contribution to structuring a new community on polynomial optimization at European level, training new experts in a multidisciplinary field and developing new synergies between public and private stakeholders.
In particular, the ESR secondments provided excellent examples of cross-sector mobility and training. This has helped to create a network of interactions between research and industry players, which continues after the end of the project.
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