We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex to numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: 1) to use a different relaxation order for each constraint, and 2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries.Key words. Multi-ordered Lasserre hierarchy, Hermitian sum-of-squares, chordal sparsity, semidefinite programming, optimal power flow.AMS subject classifications. 90C22, 90C06, 90C26, 28A99, 14Q99, 47N10.1. Introduction. Polynomial optimization encompasses NP-hard non-convex problems that arise in various applications and it includes, as special cases, integer programming and quadratically-constrained quadratic programming. The Lasserre hierarchy [59,76,77], which draws on algebraic geometry [80], enables one to solve such problems to global optimality using semidefinite programming. A big challenge today is to make it applicable to large scale real world problems. Recent approaches in this direction include the use of chordal sparsity [95], the BSOS hierarchy [57] and Sparse-BSOS hierarchy [96], the DSOS and SDSOS hierarchies [3,49,55], and ADMM for sum-of-squares [97]. The Lasserre hierarchy has two dual facets, moments and sumsof-squares, and most approaches to reduce the computational burden can be viewed as a restriction on the sum-of-squares: [95] restricts the number of variables, [57] restricts the degree, and [3] restricts the number of terms inside the square. Following this line of research, we propose to restrict sum-of-squares to Hermitian sum-of-squares [34] for optimization problems with oscillatory phenomena (e.g. power systems [11,17,65], imaging science [13,16,89], signal processing [4,21,69,70], automatic control [93], and quantum mechanics [45]). In addition, we propose to restrain the use of high degree sum-of-squares to only some constraints by using a different degree for each constraint. Finally, we show that if the polynomials defining the objective and the constraints are even (i.e. all the monomials have ...

Finding a global solution to the optimal power flow (OPF) problem is difficult due to its nonconvexity. A convex relaxation in the form of semidefinite programming (SDP) has attracted much attention lately as it yields a global solution in several practical cases. However, it does not in all cases, and such cases have been documented in recent publications. This paper presents another SDP method known as the momentsos (sum of squares) approach, which generates a sequence that converges towards a global solution to the OPF problem at the cost of higher runtime. Our finding is that in the small examples where the previously studied SDP method fails, this approach finds the global solution. The higher cost in runtime is due to an increase in the matrix size of the SDP problem, which can vary from one instance to another. Numerical experiment shows that the size is very often a quadratic function of the number of buses in the network, whereas it is a linear function of the number of buses in the case of the previously studied SDP method.

In this paper, we publish nine new test cases in MATPOWER format. Four test cases are French very high-voltage grid generated by the offline plateform of iTesla: part of the data was sampled. Four test cases are RTE snapshots of the full French very high-voltage and high-voltage grid that come from French SCADAs via the Convergence software. The ninth and largest test case is a pan-European ficticious data set that stems from the PEGASE project. It complements the four PEGASE test cases that we previously published in MATPOWER version 5.1 in March 2015. We also provide a MATLAB code to transform the data into standard mathematical optimization format. Computational results confirming the validity of the data are presented in this paper.

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set K described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, when the POP feasibility set K is compact, we show that there is no duality gap between each primal and dual SDP problem in Lasserre's hierarchy, provided a redundant ball constraint is added to the description of set K. Our proof uses elementary results on SDP duality, and it does not assume that K has an interior point.

Abstract-The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate lowimpedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems.

Abstract-A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.

We show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the n-dimensional torus. Therefore the semidefinite programming approach initiated by in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of ℓ 1 -norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive sparse recovery LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).

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