Path following

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In this context a path is a piecewise differentiable curve in space. The idea is to follow such a path (if one exists) from some initial point to a solution. One example of a path following algorithm is the barrier penalty function, with the path created by the parameter LaTeX: u in the solution, LaTeX: x^*(u) in LaTeX: \textstyle \argmax \left \{f(x) + uP(x): x \in X^0\right\}, where LaTeX: X^0 is the strict interior of the feasible region. (This is called the central path, or the path of the analytic center.)

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