HOM4PS-2.0para: Parallelization of HOM4PS-2.0 for solving polynomial systems

Solving polynomial least squares problems via semidefinite programming relaxations

“…One of the most successful methods is the polyhedral homotopy continuation method [23], which provides all isolated complex solutions of f (x) = 0. When one or some of isolated real solutions in a certain interval are to be found, it is more reasonable to formulate the problem as (24). We must say, however, that any comparison of the presented method with the polyhedral continuation method is not of our interest; the state-of-art software package [24] for the polyhedral homotopy continuation method computes all complex solutions of economic-n and cyclic-n polynomial systems much faster than the presented method that computes a single solution of (24).…”

“…When one or some of isolated real solutions in a certain interval are to be found, it is more reasonable to formulate the problem as (24). We must say, however, that any comparison of the presented method with the polyhedral continuation method is not of our interest; the state-of-art software package [24] for the polyhedral homotopy continuation method computes all complex solutions of economic-n and cyclic-n polynomial systems much faster than the presented method that computes a single solution of (24). The main concern here is comparing the sparse POP formulation (3) with the sparse PSDP formulation (7) through polynomial systems.…”

“…See Section 5.6 of [33]. When appropriate values for the bounds are not known in advance, we simply assign a very large number and a very smaller number, for instance, 1.0e+10 and −1.0e+10, to the bounds and solve (24). If an optimal value of desired accuracy is not obtained, then the attained optimal solution values are used for the lower and upper bounds after perturbing the values slightly.…”

“…We use f i (x) from the corresponding polynomial whose name is described in the first column [31]. The number in the column "iter" indicates the number of times that the problem is solved with updated lower and upper bounds; 1 means initial application of the sparse POP formulation (3) or the sparse PSDP formulation (7) for the problem (24). The initial bounds for the variables were given as [−5, 5] for the tested problems.…”

Polynomial homotopies on multicore workstations

“…All parallel homotopy algorithms described in [23,29,30,31,47,48,49,51] use the message passing library MPI [42] for interprocess communication on clusters of Linux nodes, developed with personal clusters and tested on the NCSA supercomputers. Also other continuation software for polynomial systems (Bertini [8], HOM4PS-2.0para [34], PHoMpara [24], and POLSYS GLP [45]) use MPI in their parallel implementations.…”