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Given a starting position and a search direction , the task of a line search is to determine a step size > that adequately reduces the objective function : (assumed i.e. continuously differentiable), i.e., to find a value of that reduces (+) relative to ().
If we pick b,c such that the partition a,b,c,z has three equal-length intervals, then the interval shrinks by 2/3 at each iteration, so the method has linear convergence with rate /. Fibonacci search: This is a variant of ternary search in which the points b , c are selected based on the Fibonacci sequence .
Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function.At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks.
The Barzilai-Borwein method [1] is an iterative gradient descent method for unconstrained optimization using either of two step sizes derived from the linear trend of the most recent two iterates. This method, and modifications, are globally convergent under mild conditions, [ 2 ] [ 3 ] and perform competitively with conjugate gradient methods ...
Plot of the Rosenbrock function of two variables. Here a = 1 , b = 100 {\displaystyle a=1,b=100} , and the minimum value of zero is at ( 1 , 1 ) {\displaystyle (1,1)} . In mathematical optimization , the Rosenbrock function is a non- convex function , introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for ...
Figure 1. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal.
In recent years, some interior-point methods have been suggested for convex minimization problems, but subgradient projection methods and related bundle methods of descent remain competitive. For convex minimization problems with very large number of dimensions, subgradient-projection methods are suitable, because they require little storage.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.