... λ,1)CONVEXITYConvex Sets Nonconvex Setsxyαx + (1 - α)y, 0 < α < 1xxyyxyConvex and nonconvex sets.αf(x) + (1 - α)f(y)xyCzf(z)A convex function. Linear interpolation underestimatesthe ... toaff(X).•Every nonempty convex set has a nonemptyrelative interior.DUALITY THEOREM FOR EQUALITIES•Assumptions:−The setXis convex and the functionsf,gjare convex overX.−The ... Danskin’s Theorem:Consider the functionf(x)=maxz∈Zφ(x, z), whereφ : n+m→is continuous,Zis compact, andφ(·,z)is convex for eachz ∈ Z.Thenfis convex and∂f(x)=Convex Hull∂xφ(x,...