Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables

Abstract

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 ...

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

“…Global convergence is a consequence of the aforementioned sparse Positivstellensatz. table below and [131]). The relaxation order is typically augmented at a hundred or so constraints before reaching global optimality.…”

“…Searching for even sums-of-squares reduces to block diagonal SDP's (see [131,Section 7] for explanations).…”

Grid-Aware versus Grid-Agnostic Distribution System Control: A Method for Certifying Engineering Constraint Satisfaction

“…This can be attributed to the fact that the convex relaxation exploits inaccuracies in the outer approximation of the AC power flow constraints (3e), (3f) to artificially achieve lower voltage magnitudes. Tightening the relaxation via improvements such as those in [22,[29][30][31]35] may lead to less conservative results. We leave implementations of such improvements to future work.…”

A new bounded degree hierarchy with SOCP relaxations for global polynomial optimization and conic convex semi-algebraic programs

“…This is often done using one or more of the following approaches, where the scheme: (i) exploits special structures to restrict sum-of-squares in the hierarchy, such as sparsity, symmetry, of the underlying polynomial optimization problem to improve performance [14,24,25]. The Lasserre hierarchy has recently been shown to solve industrial-scale optimal power flow problems in electrical engineering with several thousands of variables and constraints [15] using restricted sum-of squares in the hierarchy; or (ii) employs alternative conic programming hierarchies with less computational cost such as the scaled diagonally dominant sum of squares (SDSOS) hierarchies [1,2,17] or (iii) restricts the degree of the sum-of-squares of the hierarchy by using a different representations of positivity to the Putinar representation, such as the Krivine-Stengle's certificate of positivity in real algebraic geometry [21,25]. This approach proposed a bounded degree hierarchy of semidefinite programming (SDP) relaxations where the size of the semidefinite matrix involved in the hierarchy, in contrast to the standard Lasserre hierarchy, is fixed.…”