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Then, minimization problem (40) reduces to (43). 

The domain of the minimization problem is D as defined in (44). 

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The Lagrangian associated with problem (43) is (45). λ$_{1}$ , λ$_{2}$ ,..., λ$_{N}$ , µ , ν are Lagrange multi- pliers. 

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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 

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