The Minimize command in Maple's Optimization package hits a complex number in its iteration, and throws an error message. I couldn't get this package to produce anything worth discussing.
But, the given equation, of the form f(x,y,z)=0, can be solved explicitly for x = x(y,z) by Maple. There are four branches to the solution, only two of which are real.
Hence, each of the two real branches can be plotted with plot3d on the square 0<=y,z<=1. From these two plots, it is easy to see that one branch has a minimum of approximately 0.123 at (y,z) = (0,1).
The other branch has a minimum on the face y=0. Substituting y=0 into this branch gives a function x=x(z) which can be graphed to observe that there is a minimum of approximately -0.236 at (y,z) = (0,0.4). This function can be differentiated, etc., and fsolve will produce the critical value z = 0.4567706470, from which the actual constrained minimum of x = -0.2363733816 can be found.