Have studied lagrangian and optimization primarily via Khan academy, as got bogged down with Simon and Blume Mathematics for Economists/Chiang and Wainwright Fundamental Methods of Mathematical Economics went through this but didn’t understand all). Have some specific questions:
in an equality constraint what is the intuition or economic meaning or geometric interpretation of a negative lamda multiplier (I understand how a positive one implies change in function for change in constraint)
I understand that there is need to allow for interior optimal solutions i.e. why KKT when dealing with inequality optimization, but why does restricting lamda to >= 0 and primal feasibility allow this vs. equality optimization (i understand intuition behind lagrangian stationary and complementary slackness but think this is same for both equality and inequality). A bit lost here
what is the intuition behind why the linear independence and rank of gradient of constraint relative to minimum of number of variables/constraints tell you if derivative of function and gradient are parallel and point in same direction?
Thank you