Question: A scalar multi-objective least squares problem. We consider the special case of the multi-objective least squares problem in which the variable \(x\) is a scalar, and the \(k\) matrices \(A_{i}\) are all \(1 \times 1\) matrices with value \(A_{i}=1\), so \(J_{i}=\left(x-b_{i}\right)^{2}\). In this case our goal is to choose a number \(x\) that is simultaneously close to all the numbers \(b_{1}, \ldots, b_{k}\). Let \(\lambda_{1}, \ldots, \lambda_{k}\) be positive weights, and \(\hat{x}\) the minimizer of the weighted objective. Show that \(\hat{x}\) is a weighted average of the numbers \(b_{1}, \ldots, b_{k}\), i.e., it has the form

where \(w_{i}\) are nonnegative and sum to one. Give an explicit formula for the combination weights \(w_{i}\) in terms of the multi-objective least squares weights \(\lambda_{i}\).

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