LINE SEARCH Meaning and
Definition
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Line search is a numerical optimization method used to find the minimum or maximum of a function by searching along a given direction, typically in search of a local minimum. It is commonly employed in gradient-based optimization algorithms, such as Newton's method, conjugate gradient, and quasi-Newton methods.
The line search process involves determining an appropriate step size along a specific search direction, that is, scanning along a line in the multidimensional parameter space. The goal is to find the step size that minimizes or maximizes the objective function according to certain criteria. The step size is often defined as a scalar scaling factor that scales the search direction vector.
Line search can be performed using various techniques. One common approach is the Armijo-Goldstein condition, which checks whether the selected step size satisfies certain conditions, such as sufficient decrease in the objective function value, to ensure convergence. Other methods include the Wolfe conditions and the strong Wolfe conditions, which impose additional constraints such as curvature conditions to guarantee convergence and prevent overshooting.
By iteratively updating the step size and search direction, line search algorithms effectively navigate through the parameter space, approaching the desired optimum value. However, it is important to note that line search methods are sensitive to the choice of search direction, and may converge to a local minimum rather than a global minimum if the search direction is not carefully chosen.
Etymology of LINE SEARCH
The term "line search" can be broken down into its constituent parts for better understanding:
1. Line: The word "line" originates from the Latin word "linea" meaning "line, string, or thread". It has been used to refer to a straight or curved continuous extent since the 14th century. In mathematics and geometry, a line represents a straight or curved path connecting two points.
2. Search: The word "search" comes from the Old French word "cerchier" meaning "to search, seek out, or look for". It can be traced back to the late 13th century and has remained relatively unchanged in meaning throughout its history, referring to the act of seeking or looking for something.
Combining these two terms, "line search" refers to the process of systematically looking for or seeking out something along a straight or curved path.
