# Variable metric method

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Originally referred to the Davidon-Fletcher-Powell (DFP) method, this is a family of methods that choose the direction vector in unconstrained optimization by the subproblem: $LaTeX: \textstyle d^* \in \arg\max\{\nabla f(x)*d: ||d|| = 1\},$ where $LaTeX: \textstyle ||d||$ is the vector norm (or metric) defined by the quadratic form, $LaTeX: \textstyle d'Hd.$ With $LaTeX: H$ symmetric and positive definite, the constraint $LaTeX: \textstyle d'Hd = 1$ restricts d by being on a "circle" -- points that are "equidistant" from a stationary point, called the "center" (the origin in this case). By varying $LaTeX: H,$ as in the DFP update, to capture the curvature of the objective function, $LaTeX: f,$ we have a family of ascent algorithms. Besides DFP, if one chooses $LaTeX: H=I,$ we have Cauchy's steepest ascent. If $LaTeX: f$ is concave and one chooses $LaTeX: H$ equal to the negative of the inverse hessian, we have the modified Newton's method.