let me see if i understand your problem coorectly you want to minimize the function J which is constrained by the inequality A*x < 0, here im assuming you want A*x < b where b is a mineq x 1 zero vector. please correct me if i'm wrong.
i think you can use fmincon function of matlab for this problem very efficiently.
use different random values of x0 because it might give local minima (fmincon is generally used for convex functions because we can not be sure if the minima given is local or global) and compare J(x) for all these x obtained and compare J(x) for these x's and the minimum of these will give you your answer. now its not 100% certain because there are chances that it will always get stuck on the local minima and never reach global minima but i think if you do it for 100 or 1000 random x0 it should give u the correct anwer.