Quaternion Polynomial Optimization

The TSSOS.QuaternionSOS submodule implements the quaternion moment-SOS relaxation, including correlative and term sparsity. Load its API explicitly:

using TSSOS.QuaternionSOS
using DynamicPolynomials
using COSMO

@ncpolyvar z[1:2]
pop = [-z[1]*z[2], 1-z[1]*z[2]]
bound = qtssos(pop, z, 1, 1; optimizer=COSMO.Optimizer, QUIET=true)
# bound ≈ -1

The variables are ordered as z[1:n] followed by their conjugates z[n+1:2n]. Use Hermitian (real-valued) objectives and scalar constraints. pop[1] is the objective, followed by nonnegative constraints and then numeq equalities. The return value is the relaxation bound; solve=false computes the supports without invoking an optimizer and returns nothing.

Common keyword arguments:

CS="MF": correlative sparsity using minimum fill. Also accepts "MD", "NC", or false. Supply cliques to specify variable cliques explicitly.
TS=false: term sparsity; also accepts "block", "MF", or "MD".
nb=0: constrain the first nb quaternion variables to unit norm.
conjubasis=true: include conjugate variables in the objective SOS basis.
ipart=true: use quaternion Gram matrices. Set false for the real coefficient relaxation used in the supplied examples.
addrcons=true: add real commutation identities; addicons=true additionally adds imaginary commutation identities to their multipliers.
rncnumeq=0, qncnumeq=0: counts of trailing noncommutative real and quaternion equalities, respectively, within the final numeq constraints. Quaternion equalities require ipart=true. Their multipliers have four independent real components, including singleton blocks, and enter the certificate together with their quaternion adjoints.
optimizer: a JuMP optimizer factory, or model: an empty JuMP.Model. Without either, TSSOS selects its default optimizer. QUIET=true suppresses normal progress output.

fsupp and fcoe may supply an explicit quaternion objective when ipart=true. Include the complete Hermitian representation, with adjoint terms and their coefficients. Each support term is a vector of three Vector{UInt16} parts: paired variable indices for scalar norm factors, their conjugate indices, and the remaining ordered word. Supply both arrays with matching lengths. This representation is useful when quaternion coefficients occur between noncommuting variables.

quaternion_to_real(pop, z) returns real polynomials and 4n commuting variables, in blocks containing the real, i, j, and k coordinates. It takes the real part after substitution. This permits comparison with tssos or cs_tssos; see example/quaternion.jl.

TSSOS.QuaternionSOS.qtssos — Function

qtssos(pop, z, n, d; numeq=0, CS="MF", TS=false, optimizer=nothing, kwargs...)

Compute a quaternion SOS lower bound at order d. The noncommuting variables z[1:n] are followed by their conjugates z[n+1:2n]. Put the Hermitian objective first and the numeq equalities last. Supply an SDP optimizer or an empty JuMP model; otherwise TSSOS selects its default solver. solve=false returns nothing after computing supports, without selecting a solver.

See docs/src/quaternion.md for sparsity, unit-norm, and coefficient options.

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TSSOS.QuaternionSOS.quaternion_to_real — Function

quaternion_to_real(pop, q)

Substitute real, i, j, and k coordinates for each quaternion and take the real part. Return (real_pop, x) with coordinates grouped in four blocks of length length(q) ÷ 2. The second half of q denotes conjugates of the first half.

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