Sparse Polynomial Optimization Research Articles

CS-TSSOS: Correlative and Term Sparsity for Large-Scale Polynomial Optimization

This work proposes a new moment-SOS hierarchy, called CS-TSSOS , for solving large-scale sparse polynomial optimization problems. Its novelty is to exploit simultaneously correlative sparsity and term sparsity by combining advantages of two existing frameworks for sparse polynomial optimization. The former is due to Waki et al. [ 40 ] while the latter was initially proposed by Wang et al. [ 42 ] and later exploited in the TSSOS hierarchy [ 46 , 47 ]. In doing so we obtain CS-TSSOS—a two-level hierarchy of semidefinite programming relaxations with (i) the crucial property to involve blocks of SDP matrices and (ii) the guarantee of convergence to the global optimum under certain conditions. We demonstrate its efficiency and scalability on several large-scale instances of the celebrated Max-Cut problem and the important industrial optimal power flow problem, involving up to six thousand variables and tens of thousands of constraints.

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds

Circuit polynomials are polynomials with properties that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds. Recent work has shown that sums of nonnegative circuit polynomials (or SONC polynomials for short) can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner very efficiently both in theory and in practice, if the polynomial is bounded from below and its support satisfies a certain nondegeneracy assumption. The quality of the SONC bound depends on the circuits used in the computation but finding the set of circuits that yield the best attainable SONC bound among the astronomical number of candidate circuits is a non-trivial task that has not been addressed so far. We propose an efficient method to compute the optimal SONC lower bound by iteratively identifying the optimal circuits to use in the SONC bounding process. The method is derived from a new proof of the result that every SONC polynomial decomposes into SONC polynomials on the same support. This proof is based on convex programming duality and motivates a column generation approach that is particularly attractive for sparse polynomials of high degree and with many unknowns. The method is implemented and tested on a large set of sparse polynomial optimization problems with up to 40 unknowns, of degree up to 60, and up to 3000 monomials in the support. The results indicate that the method is efficient in practice and requires only a small number of iterations to identify the optimal circuits.

Nonnegative Polynomials and Circuit Polynomials

The concept of sums of nonnegative circuit (SONC) polynomials was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONCs. As a first result, we provide sufficient conditions for nonnegative polynomials with general Newton polytopes to be a SONC, which generalizes the previous result on nonnegative polynomials with simplex Newton polytopes. Second, we prove that every SONC admits a SONC decomposition without cancellation. In other words, SONC decompositions preserve sparsity of nonnegative polynomials, which is dramatically different from the classical sum of squares decompositions and is a key property to design efficient algorithms for sparse polynomial optimization based on SONC decompositions.

Solving sparse polynomial optimization problems with chordal structure using the sparse bounded-degree sum-of-squares hierarchy

The sparse bounded degree sum-of-squares (sparse-BSOS) hierarchy of Weisser et al. (2017) constructs a sequence of lower bounds for a sparse polynomial optimization problem. Under some assumptions, it is proved by the authors that the sequence converges to the optimal value. In this paper, we modify the hierarchy to deal with problems containing equality constraints directly, without eliminating or replacing them by two inequalities. We also evaluate the sparse-BSOS hierarchy on a well-known bilinear programming problem, called the pooling problem, as well as a discrete-time optimal control problem.

A Multigrid Approach to SDP Relaxations of Sparse Polynomial Optimization Problems

We propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support. The problems we consider arise from the discretization of infinite dimensional optimization problems, such as PDE optimization problems, boundary value problems, and some global optimization applications. In many of these applications, the level of discretization can be used to obtain a hierarchy of optimization models that capture the underlying infinite dimensional problem at different degrees of fidelity. This approach, inspired by multigrid methods, has been successfully used for decades to solve large systems of linear equations. However, multigrid methods are difficult to apply to semidefinite programming (SDP) relaxations of polynomial optimization problems. The main difficulty is that the information between grids is lost when the original problem is approximated via an SDP relaxation. Despite the loss of information, we develop a multigrid approach and propose prolongation operator...

Bounded error identification of Hammerstein systems through sparse polynomial optimization

In this paper we present a procedure for the evaluation of bounds on the parameters of Hammerstein systems, from output measurements affected by bounded errors. The identification problem is formulated in terms of polynomial optimization, and relaxation techniques, based on linear matrix inequalities, are proposed to evaluate parameter bounds by means of convex optimization. The structured sparsity of the formulated identification problem is exploited to reduce the computational complexity of the convex relaxed problem. Analysis of convergence properties and computational complexity is reported.

Sums of Squares and Semidefinite Program Relaxations for Polynomial Optimization Problems with Structured Sparsity

Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of the supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite program (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations.

Generalized Lagrangian Duals and Sums of Squares Relaxations of Sparse Polynomial Optimization Problems

Sequences of generalized Lagrangian duals and their sums of squares (SOS) of polynomials relaxations for a polynomial optimization problem (POP) are introduced. The sparsity of polynomials in the POP is used to reduce the sizes of the Lagrangian duals and their SOS relaxations. It is proved that the optimal values of the Lagrangian duals in the sequence converge to the optimal value of the POP using a method from the penalty function approach. The sequence of SOS relaxations is transformed into a sequence of semidefinite programing (SDP) relaxations of the POP, which correspond to duals of modification and generalization of SDP relaxations given by Lasserre for the POP.

Path-following energy optimization in unilateral contact problems

Path-following (load incrementation) methods are studied in this paper for elastostatic analysis problems with unilateral contact relations in the framework of a large displacement theory by means of the parametric optimization techniques. Finite element discretization yields sparse polynomial optimization problems with equality and inequality constraints. For such sparse problems generically appearing singularities along the path of solutions are completely classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee these generic situations. This clarifies stability and uniqueness questions for the solution along the examined path.