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Taming non convexity?

Periodic Reporting for period 3 - TAMING (Taming non convexity?)

Reporting period: 2018-09-01 to 2019-08-31

"The Moment-SOS approach (or the Moment-SOS hierarchy) has been already considered a breakthrough in the field of optimization and more recently in Theoretical Computer Science (TCS). Indeed the TCS community calls the SOS-hierarchy (aka the ""Lasserre hierarchy"") a META algorithm as it applies ""canonically"" to most combinatorial optimization problems in graphs. In particular is is now recognized as a fundamental tool to prove/disprove Khot's celebrated ""Unique Games Conjecture"" (UGC) in TCS. One challenge is to make the moment-SOS approach scalable so as to potentially solve some large scale problems, and in particular the Optimum Power Flow problem (OPF) of strategic importance in the Management of Energy networks. Another challenge is to show that the moment-SOS approach can be applied in various other applications to provide better solutions than those provided
by other methods, e.g. in Control, Optimal Control and Robotics, in some inverse problems, in Statistics, Machine Learning. Therefore, and according to the accepted proposal TAMING, the research plan consists of mainly two types of research investigations: (I) Theoretical and algorithmic research on the moment–SOS approach, and (II) versatility and power of the moment-SOS approach."
I. THEORETICAL & ALGORITHMIC RESEARCH: [1], [2], [3], [4], [36], [38]

II. APPLICATIONS OF THE MOMENT-SOS HIERARCHY: [5], [6], [7], [34], [39]. OPTIMIZATION: [8], [9], [10], [11], [12]. INVERSE PROBLEMS: [13], [20], [14], [15], [25], [21], [16], [37]. PROBABILITY: [22], [23], [40]. STATISTICS, MACHINE LEARNING, DATA ANALYSIS: [26], [18], [24], [35]. APPROXIMATION THEORY: [26], [32], [33]. NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS: [30], [31]

[1] J.B. Lasserre, Kim-Chuan Toh, S. Yang. Eur. J. Comput. Optim. 5, pp. 87–117, 2017
[2] T. Weisser, J.B. Lasserre, K. Toh. Math. Program. Comput. 9, pp. 1--32, 2017.
[3] C. Josz. On the relationship between real and complex linear systems. 2016. Submitted
[4] C. Josz. Algorithms for optimization and interpolation based on hyponormaliy. 2017. Submitted
[5] M. Korda, D. Henrion. Optim. Letters, pp. 1--8, 2017.
[6] M. Korda, D. Henrion, C. N. Jones. Systems & Control Letters 100, pp. 1--5, 2017.
[7] M. Korda, D. Henrion, C. N. Jones. Automatica 67, pp. 54--66, 2016
[8] J.B. Lasserre. SIAM J. Optimization 26, pp. 247—273, 2016.
[9] V. Jeyakumar, J.B. Lasserre, G. Li, Tien Son Pham. SIAM J. Optim., pp. 753—780, 2016.
[10] R. Hess R., D. Henrion, J.B. Lasserre, Tien Son Pham. SIAM J. Control & Optim. 54, pp. 1633—1656, 2016.
[11] F. Bugarin, D. Henrion, J.B. Lasserre. Math. Program. Comput. 8, pp. 83—111, 2016.
[12] J.B. Lasserre. Oper. Res. Letters. 44, pp. 158—164, 2016
[13] E. Pauwels, D. Henrion, J.B. Lasserre. SIAM J. Control & Optim. 54, pp. 1798—1825, 2016
[14] J.-P. Laumond, N. Mansard, J.B. Lasserre. Geometric and Numerical Foundations of Movements. Springer Tracts in Advanced Robotics, New York, 2017.
[15] Y. De Castro, F. Gamboa, D. Henrion, J.B. Lasserre. IEEE Trans. Information. Theory 63, pp. 621—630, 2017
[16] J.B. Lasserre. Adv. Comput. Math. 42, pp. 1129—1148, 2016.
[17] J.B. Lasserre. Adv. Applied Math. 91, pp. 137--163, 2017.
[18] J.B. Lasserre, E. Pauwels. Proceeding 2016 NIPS Conference, Barcelona, 2016.
[19] C. Josz, D. Molzahn. SIAM J. Optim. 28, pp. 1017--1048, 2018
[20] E. Pauwels, D. Henrion, J.B. Lasserre. J.-P. Laumond, N. Mansard and J.B. Lasserre (Eds.), Springer Tracts in Advanced Robotics 117, Springer, Switzerland, 2017, pp. 113—132.
[21] J.B. Lasserre. SIGMA 11 (077), 2015.
[22] J.B. Lasserre, Y. Emin. Math. Operations Research, doi/10.1287/moor.2018.0980
[23] J.B. Lasserre. IEEE Control Systems Letters 1, pp. 50-55, 2017.
[24] J.B. Lasserre, E. Pauwels. Adv. Comp. Math. 45, pp. 1439--1468, 2019
[25] C. Josz, J.B. Lasserre, B. Mourrain. Adv. Comp. Math. 45, pp. 1401--1437, 2019
[26] Y. De Castro, F. Gamboa, D. Henrion, R. Hess, J.B. Lasserre. The Annals of Statistics 47, pp. 127--155, 2019
[27] J. Rouot, J.B. Lasserre. Proc. 56th IEEE CDC Conference, Melbourne, December 2017.
[28] S. Foucart, J.B. Lasserre. J. Approximation Theory 235, pp. 74--91, 2018
[29] J.B. Lasserre. Proceedings ICM 2018, Rio de Janeiro.
[30] M. Korda, D. Henrion, J.B. Lasserre. arXiv:1804.07565
[31] S. Marx, T. Weisser, D. Henrion, J.B. Lasserre. Math. Control & Related Fields, 2019
[32] S. Foucart, J.B. Lasserre. Computational Methods and Function Theory. 2019
[33] S. Marx, E. Pauwels, T. Weisser, D. Henrion, J.B. Lasserre. Tractable semi-algebraic approximation using Christoffel-Darboux kernels. submitted
[34] J.B. Lasserre. SIAM J. Appl. Algebra & Geometry 3, pp. 372--389, 2019
[35] J.B. Lasserre & V. Magron. SIAM J. Optim. 28, pp. 3127--3144, 2018
[36] J.B. Lasserre & V. Magron. SIAM J. Optim. 29, pp. 2128--2145, 2019
[37] F. Bréhard, M. Joldes and J.B. Lasserre. Proceedings ISSAC'2019, Beijing, 2019, pp. 66--73.
[38] J.B. Lasserre. Submitted. arXiv:1907.09784
[39] J.B. Lasserre. European Control Conference (ECC 2019), Naples, Italy, July 2019
[40] J.B. Lasserre & T. Weisser. Math. Program Séries A. 2019
I. The Lasserre-hierarchy adapted by Cedric Josz and D. Molzahn for solving some Optimum Power Flow problems (OPFs) with thousands variables is already considered a breakthrough in the management of Power Systems. It is not yet clear whether the alternative sparse version [2] of [1] will bring significant improvements. However for other large scale applications it could make a difference. It should be noted that the SOS-algorithm is also at the core of the 2018 ERC consolidator grant UTOPEST (of David Steurer, ETH, Zurich) and is called a meta-algorithm by the Theoretical Computer Science research community.

II. All contributions in II.1 -> II.5 describe a completely new methodology for difficult problems: (i) Optimal design for polynomial regression in Statistics, (ii) nonlinear hyperbolic PDE's, and (iii) the Christoffel
function for Statistics & Machine Learning should have a noticeable impact very soon. In recognition of the impact of this methodology:

•The paper [37] got the ISSAC 2019 Best Paper Award at the ISSAC 2019 conference in Beijing, July 2019.
•J.B. Lasserre was an invited speakers (with David Vogan from MIT) at the 90th Anniversary of the Department of Mathematics, National University of Singapore (NUS), September 2019.
• was invited as Simons CRM Professor at CRM in Montreal (Canada)
•was an invited speaker at the International Congress of Mathematics (ICM 2018, Rio de Janeiro)
•was invited to speak at several specialized workshop at prestigious Institutes of Mathematics
•was Keynote or plenary speaker at various conferences and workshops.
•was speaker at the prestigious Mathematical Colloquia (Charles University in Praha, Tcheky) in 2017.
•was awarded the 2015 John von Neumann Theory Prize of the INFORMS Society as well as the L. Khachiyan Prize of the Optimization Society of INFORMS (for Lifetime achievement in Optimization)
•will hold an “optimization chair” in the newly created ANITI Institute (Toulouse, France) for Artificial Intelligence.

•Didier Henrion has been invited to speak at several workshops and conferences

•Cédric Josz and Didier Henrion got (jointly) in 2016 the Optimization Letters Best Paper Award.

•Swann Marx spent the whole month of January 2019 at the Deusto University (Bilbao, Basque country) at the invitation of Enrique Zuazua