Hi, initially I cut the photo off the book called Nonlinear-and-Mixed-Integer-Optimization-Fundamentals-and-Applications-Topics-in-Chemical-Engineering and page 120.
My question is: in the GREEN box it says we have to use KKT gradient conditions with respect to a_i and as a result of that we obtain the set in RED box. You see the sum of u_i equals to 1 BUTTT when I take gradient conditions into account with respect to a_i, I find that all individual u_i must be 1. FOR INSTANCE say that Lagrangian function is sum of a_i + sum of u_i*[ g_i () - a_i ], and apply KKT gradient conditions to Lagrangian function with respect to a_1. Derivative of sum of a_i with respect to "a_1" is just " 1 " and
derivative of sum of u_i*[ g_i () - a_i ] with respect to "a_1" equals to just " -u_1", then summation of them must equal to "0", then all individual u_i must be 1 INSTEAD of summation of u_i equals 1??
Could you help me about that?