Diet

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 14 in Conjugate Duality in Economic Analysis, 2026, pp 111-114 from Springer

Abstract: Abstract Stigler's diet problem is a classic primal/dual application. The problem for the consumer is to minimize the cost of buying food to achieve nutrition targets. Define the linear transformation $${\textit {{\sf { {A}}}}}:=\left [ \begin {array}[c]{ccc}{a}_{1} & \cdots & {a}_{m}\end {array} \right ] \!,$$ in which aj denotes the nutrition from one unit of food j. Diet x provides nutrition Ax. The consumer chooses x to minimize the food cost $$\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle$$ to achieve nutrition target y: $$\inf _{\boldsymbol {x}}\left ( \left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle + \delta _{ {\textit {{\sf { {A}}}}}\boldsymbol {x}\succeq \boldsymbol {y}}+ \delta _{\boldsymbol {x}\succeq \mathbf {0}}\right ) \!.$$ The reciprocal dual is $$\sup _{\boldsymbol {y}^{\ast }}\left ( \left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle -\delta _{ {\textit {{\sf { {A}}}}}^{\top }\boldsymbol {y}^{\ast }\preceq {x}^{\ast }}-\delta _{\boldsymbol {y}^{\ast }\succeq \mathbf {0}}\right ) \!.$$ Interpret the dual choice variable y* as the price (value) of nutrition. Choose y* to maximize the value $$\left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle$$ of nutrition, subject to food value A⊤y* less than or equal to food cost x*.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_14

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