Then, minimization problem (40) reduces to (43). The domain of the minimization problem is D as defined in (44). The Lagrangian associated with problem (43) is (45). λ$_{1}$ , λ$_{2}$ ,..., λ$_{N}$ , µ , ν are Lagrange multi- pliers. The Lagrange dual function associated with problem (43) is (46). L ( p , λ ,µ, ν ) is a convex function of each p$_{i}$ . The first order condition is 35 min p d $^{T}$p such that (A) p ≥ 0 (B) ∑ N i $_{=1}$p$_{i}$ = 1 (C) N i $_{=1}$p$_{i}$ ln ( p$_{i}$ q$_{i}$ ) ≤ Ω (43) D = { p ∈ R $^{N}$| p > 0 } (44) L ( p , λ ,µ, ν ) = d $^{T}$p + N ∑ i =1 λ$_{i}$ ( − p$_{i}$ ) + µ { N ∑ i =1 p$_{i}$ ln ( p$_{i}$ q$_{i}$ ) − ¯ Ω } + ν ( N ∑ i =1 p$_{i}$ − 1 ) (45) ∂L ( p , λ ,µ, ν ) ∂p$_{i}$ = ( d$_{i}$ − λ$_{i}$ + ν ) + µ { 1 + ln ( p$_{i}$ q$_{i}$ )} = 0 (47) 1 + ln ( p$_{i}$ q$_{i}$ ) = − d$_{i}$ − λ$_{i}$ + ν µ (48) p$_{i}$ = q$_{i}$ exp ( − 1 + λ$_{i}$ − d$_{i}$ − ν µ ) > 0 (49) g ( λ ,µ, ν ) = inf p > $_{0}$L ( p , λ ,µ, ν ) (46)