2Optimization conditions•Definitions of Global and local minima–We want to find a global but only afford to have local•Unconstrained optimization problem–KT condition (f’ = 0)–2ndorder necessary condition (f’’ PSD)–Sufficient condition (f’’ PD)•Condition for global minimum–Convex objective on convex constraint set–When the obj and constraint set become convex?•Equality constrained problem–Introduce Lagrangian–KT condition0,0LLx( , )( )( )i iLfhxxx
3Optimization conditions•Inequality constrained problem–Introduce slack variables:–KT condition–Complementary slackness (igi= 0)•2nd-order necessary condition–is P.S.D. for all feasible directions•Sufficient condition–is P.D. for all feasible directions2( , , )( )()iiiLsfgsxx0,0,0LLLsx2xL2xL
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