I have many polynomial equations in many variables which I want to jointly minimize (in a mean square sense, but you could pick a different reasonable measure which favors anything where all quantities go to zero). For example, I am looking to make the 6 equations below as "small" as possible (a-j are unknown real numbers).
This example probably actually has a solution where all equations are zero, but I also have cases which have no zero solution, so I'd rather not do the "repeatedly eliminate variables and solve for the quadratic root" approach (also, this approach takes too long; is there even any machine which could find a full zero for these equations within 10 minutes?). I'm thinking there might be some software tool that considers the "terrain" smartly and is locally minimizing on many global fronts...or maybe that is impractical. So, is there a free math tool (like Sage) which can minimize things for me (and be certain that no other point is better within some tolerance)? I'm open to theoretical advice, but feel like the options will all look like brute force.
Should I give up if I need to minimize a similar set of equations within 10 minutes on one machine?
a^2 + b^2 + c^2 - 4.52 0.136*a^2 - 0.15*a + d^2 + e^2 + f^2 - 3.84 1.12*a^2 + 0.415*a + b^2 - 2.0*b*d + c^2 - 2.0*c*e + d^2 + e^2 + f^2 - 0.593 0.602*a^2 - 0.0411*a*b - 0.851*a*d - 0.94*a + 0.634*b^2 - 0.489*b*d + 0.219*b + 0.407*d^2 + 0.588*d + h^2 + i^2 + j^2 - 0.612 0.0676*a^2 - 0.0495*a*b + 0.258*a*d + 0.155*a + 0.095*b^2 + 0.14*b*d + 0.157*b + c^2 - 2.0*c*h + 0.407*d^2 + 0.588*d + h^2 + i^2 + j^2 - 2.0 0.813*a^2 - 0.192*a*b - 0.843*a*d - 1.01*a + 0.634*b^2 - 2.03*b*d + 0.302*b + 2.04*d^2 + 0.212*d + e^2 - 2.0*e*h + f^2 - 2.0*f*i + h^2 + i^2 + j^2 - 2.76