Abstract: The sufficient conditions for unconditional extreme values are determined according to Hessian matrix, but the sufficient conditions for conditional extreme values are rather complex. Based on an analysis that the identification of the two kinds of sufficient conditions rely on different matrices, we derived a method of locating the sufficient conditions for conditional extreme values. By use of first order Taylor expansion in obtaining the relations among argument increments, a more precise matrix for determining the sufficient condition for conditional extreme values in high dimensions and with multiple constraints has been deduced. This helps a better understanding of the relationships and differences between the two kinds of sufficient conditions, and provides a means of finding a more precise sufficient condition for a conditional extreme value.