NCeigMin

description:

[opt,decom_sohs,decom_S,base,SDP_data,Z,Zg,H,Hg,decom_err,Y] = NCeigMin(f,S,d,params) computes the lower bound for the minimum eigenvalue of f on all matrices A = D_S, where S contains polynomials, i.e. lower bound for inf lambda_min (f(A)) s.t. g(A)\succeq 0 for all g in S

It solves:

     min <L_f,H>

     H ... hankel (moment) matrix

     H(1) = 1

     H >= 0

     H^g >= 0

     (H_g)_p,q = L(p^*gq), for all g in S

where H is of order |W_{d/2}| and H_g is of order |W_{d/2-deg(g)/2}| and d is input

This is dual to:

     sup eps

     s.t. f-eps = SOHS + \sum_{g\in S}\sum_i h_j^*gh_j

arguments:

f is an NCpoly representing a polynomial

S is a set of nc polynomails defining D_S

d is a starting degree for the hierarchy (even number)

With params.precision we can set the smallest value that is considered to be nonzero in numerical calculations; if the command is called without it, we assume the precision set with the command NCsetPrecision or the value set in NCparam.m.

params.messages is used to optionally turn on (1) and off (0) verbose output; default value is 1 (on).

params.solver sets the solver to be used for SDP and overrides the value set in the global option file NCparam.m. (currently SeDuMi, SDPA-M or SDPT3 are supported) 

params.eps sets the desired accuracy iy you are using SeDuMi as SDP solver. Setting params.eps=0 lets SeDuMi run as long as it can make progress.

params.justSDP_data == 1 means that the program ends when the SDP_data is prepared and nothing else is computed. It is optional; the default value is 0.

params.decomposition == 0 means that no SOHS decomposition over the module will actually be computed. It is optional; the default value is 1.

output:

opt ... optimal value of the SDP

decom_sohs ... sohs part of the sohs decomposition over the module M_{S,d}

decom_S ... weights of polynomials gin S

base ... is a list of monomials which appear in the SOHS decomposition over S

SDP_data ... is a structure holding all the data used in SDP solver

Z ... dual solution, that represent sohs

Zg.... dual solution that contains wights

H ... Hankel matrix

Hg ... shifted Hankel matrices for g in S

decom_err ... how much sohs decomposition differs from f

possible usage:

NCeigMin(f,S,d),NCeigMin(f,S,d,params)