Karush–Kuhn–Tucker conditions
Value function

If we reconsider the optimization problem as a maximization problem with constant inequality constraints:

The value function is defined as

so the domain of V {\displaystyle V} is { a ∈ R m ∣ for some x ∈ X, g i (x) ≤ a i, i ∈ { 1, …, m } }. {\displaystyle \{a\in \mathbb {R} ^{m}\mid {\text{for some }}x\in X,g_{i}(x)\leq a_{i},i\in \{1,\ldots,m\}\}.}

Given this definition, each coefficient μ i {\displaystyle \mu _{i}} is the rate at which the value function increases as a i {\displaystyle a_{i}} increases. Thus if each a i {\displaystyle a_{i}} is interpreted as a resource constraint, the coefficients tell you how much increasing a resource will increase the optimum value of our function f {\displaystyle f}. This interpretation is especially important in economics and is used, for instance, in utility maximization problems.
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