Could you give more specific instances of your problem? Without what you hide behind the dots (...), the instance you give is trivial (b=0, m=c=1/2).

You can try the Lasserre moment relaxations of polynomial optimization problems, see the survey of Monique Laurent:

http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf

The code for your (unfortunately trivial) example in YALMIP (a MATLAB package) would be:

    clear b c m
    sdpvar b c m

    constraints = ...
    set(b>=0)+...
    set(2-3*m-c>=0) +...
    set((2-3*m-c)^2<=b^2*(1+m^2)) +...
    set(4-7*m-c>=0) +...
    set((4-7*m-c)^2<=b^2*(1+m^2))

    minobj=b

    relaxdeg=4

    [info,sol,mom,cert]=solvemoment(constraints,minobj,[],relaxdeg)

    sol{1}

    relaxdeg=5

    [info,sol,mom,cert]=solvemoment(constraints,minobj,[],relaxdeg)

    sol{1}

For this to work you need an SDP solver like SeDuMi. Implementations of the Lasserre moment approach other than YALMIP are SOSTools, GloptiPoly and SparsePOP. I have also some slides which might be helpful:

http://www.math.uni-konstanz.de/~schweigh/presentations/polopt-kirchberg.pdf