However, the last column of ∇ $^{2}$G is The combination of (83) and (84) yields (85). ∇ $^{2}$G$_{·}$$_{N}$$_{+1}$ =     ∂ $^{2}$G ∂ω$_{1}$∂µ ∂ $^{2}$G ∂ω$_{2}$∂µ ... ∂ $^{2}$G ∂ω$_{N}$ ∂µ ∂ $^{2}$G ∂µ 2     = 1 µ 2 [ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ] 2     θ$_{1}$e$^{ω$_{1}$}$ µ ∑ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{1}$ ) e$^{ω}$ i µ θ$_{2}$e$^{ω$_{2}$}$ µ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{2}$ ) e$^{ω}$ i µ ... θ$_{N}$e$^{ω}$ N µ ∑ N i $_{=1}$θ$_{i}$ ( ω$_{i}$ − ω$_{N}$ ) e$^{ω}$ i µ 1 µ ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ∑ N i $_{=1}$θ$_{i}$ ω 2 $_{i}$e$^{ω}$ i µ − 1 µ [ ∑ N i $_{=1}$θ$_{i}$ ω$_{i}$e$^{ω}$ i µ ] 2     (84) N ∑ k =1 ω$_{k}$ µ ∇ $^{2}$G$_{·}$$_{,k}$ + ∇ $^{2}$G$_{·}$$_{N}$$_{+1}$ =     0 0 ... 0 0     (85) det ∇ $^{2}$G = det [ ∇ $^{2}$G$_{·}$$_{,}$$_{1}$ ∇ $^{2}$G$_{·}$$_{,}$$_{2}$ ... ∇ $^{2}$G$_{·}$$_{,N}$ ∇ $^{2}$G$_{·}$$_{,N}$$_{+1}$ ] = det ∇ $^{2}$G$_{·}$$_{,}$$_{1}$ ∇ $^{2}$G$_{·}$$_{,}$$_{2}$ ... ∇ $^{2}$G$_{·}$$_{,N}$ ∑ N k =1 ω$_{k}$ µ ∇ $^{2}$G$_{·}$$_{,k}$ + ∇ $^{2}$G$_{·}$$_{N}$$_{+1}$ ] = det [ ∇ $^{2}$G$_{·}$$_{,}$$_{1}$ ∇ $^{2}$G$_{·}$$_{,}$$_{2}$ ... ∇ $^{2}$G$_{·}$$_{,N}$ 0 ] = 0 (86) 49 Because both ∂ $^{2}$G ∂µ 2 and det ∇ $^{2}$G are nonnegative, G is a convex function. (9) is a convex optimization problem because both the objective function and the feasible set are convex. Therefore,