Convex Optimization and Lagrangian Duality

Finally the Lagranage dual function is given by g(~λ, ~ν) = inf~x L(~x,~λ, ~ν) We now make a couple of simple observations. Observation. When L(·, ~λ, ~ν) is unbounded from below then the dual takes the value −∞. Observation. g(~λ, ~ν) is concave1 as it is the infimum of a set of affine2 functions. If x is feasible solution of program (10.2)(10.4), then we have the following L(x,~λ, ~ν) = f0(x) + ∑m i=1 λifi(x) + ∑p j=1 νjhj(x) ≤ f0(x) for ~λ ≥ 0 A function g(x) is concave is for any 0 ≤ α ≤ 1, αg(x) + (1− α)g(y) ≤ g(αx + (1− α)y). That is, linear in {λi} and {νj}.