Question: Modifying a diet to meet nutrient requirements. We consider a set of \(n\) basic foods (such as rice, beans, apples) and a set of \(m\) nutrients or components (such as protein, fat, sugar, vitamin \(\mathrm{C})\). Food \(j\) has a cost given by \(c_{j}\) (say, in dollars per gram), and contains an amount \(N_{i j}\) of nutrient \(i\) (per gram). (The nutrients are given in some appropriate units, which can depend on the particular nutrient.) A daily diet is represented by an \(n\) -vector \(d\), with \(d_{i}\) the daily intake (in grams) of food \(i\).

The current daily diet is specified by the \(n\) -vector \(d^{\text {curr }}\). Explain how to find the closest diet \(d^{\text {mod }}\) to \(d^{\text {curr }}\) that satisfies the nutrient requirements given by the \(m\) -vector \(n^{\text {des }}\), and has the same cost as the current diet \(d^{\text {curr }}\).

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