| # Localization via Range Measurement: proof of intermediate result Localization > Outer Sphere Approximations > Proof In the localization via range measurement problem, the application of Lagrange duality leads to the function of , We can express the function in closed-form, as follows. The objective function of the maximization problem defining expresses as where for notational convenience we use the matrix and vector with components , . The above immediately implies that the maximum in the definition of is when the coefficient of is positive. If that coefficient is zero, then we must have for to be finite. Thus, we impose the constraint Assume that the first constraint holds. When this constraint is in place, computing is a concave quadratic maximization problem without any constraints, and can be solved in closed form by taking derivatives. The corresponding maximizer is Plugging that value of in the objective leads to Assume now that the second constraint holds: , and . Then reduces to the constant part in (), that is, . To summarize: