Linearity interval

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When a univariate function is piece-wise linear, it has the form LaTeX: a_i x + b_i for LaTeX: x in the interval LaTeX: [c_i, c_{i+1}], where LaTeX: a_i is not equal to LaTeX: a_{i+1}. (The phrase usually means the function is continuous.) This arises in linear programming when considering the optimal value as a function of varying right-hand sides (or cost coefficients) in fixed proportions: LaTeX: b+td (or LaTeX: c+td), where LaTeX: d is an admissible change vector and LaTeX: t is the (scalar) variable. Then, for the range of LaTeX: t where the LP has a solution, the optimal value function has this piece-wise linear (continuous) form, and the intervals that comprise the domain are called linearity intervals.

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