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Polynomial Optimization, Efficiency through Moments and Algebra

Periodic Reporting for period 1 - POEMA (Polynomial Optimization, Efficiency through Moments and Algebra)

Période du rapport: 2019-01-01 au 2020-12-31

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 is targeting 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 is training 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 is developed inside the network. After the ESRs started investigating their subjects during the second year of the project, they are now able to 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.

Despite the difficult situation that we are all facing, 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. Some changes in the secondment plans have been implemented. The collaboration between the different sites is very effective, with new joint works between partners or joint organization and contribution to scientific events.
The research investigations in the second period of the project shall gain momentum and lead to effective visible outputs. The development of new algebraic techniques for polynomial optimization will be 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 is expected.

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

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 will be 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.

In the second period of the project, a special emphasis will be given on academic and real-life applications, leading to or solvable by techniques of polynomial optimization. Specific optimization problems such as power network management, water consumption, mechanism optimization, and portfolio optimization will be studied to demonstrate the potential of polynomial optimization to find better solutions to industrial decision optimization problems and to solve novel practical decision optimization problems. The methods and tools developed in other work-packages will be confronted to practical problems, in order to validate and improve the research results.

The training and networking activity will be further sustained to ensure 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 shall also increase its international visibility via scientific meetings and dissemination actions, in order to increase the R\&D career perspectives of the young researchers, the resonance of research results in the academic area and their application impact for solving real-life problems.
POEMA Workshop 1 in Florence