linesearch

Linesearch is a technique used in optimization, root-finding, and numerical analysis to efficiently search for a local minimum of a function in multidimensional space. The method is based on the concept of iteratively exploring the function's neighborhood to find a lower value. At each iteration, the method moves along the negative gradient of the function, stopping when it finds a sufficient reduction in the function value.

The linesearch process typically involves the following steps: initialization, objective function evaluation, direction selection, and line search. The goal is to find a step length along the direction vector, which ensures sufficient progress towards the local minimum while satisfying the conditions of slope, optimality, and compatibility. Conventional line search algorithms are usually implemented based on one-dimensional minimization techniques, such as backtracking, Armijo rule, or strong Wolfe conditions. These algorithms exploit the one-dimensional structure of the line search problem to select an optimal step length efficiently.

Several variations and improvements of the basic linesearch algorithm have been developed, including the Wolfe conditions, gradient tracking, and derivative-free methods. These adaptive techniques can handle complex constraints, provide faster convergence rates, and adjust the algorithm's settings according to the specific optimization problem at hand. Line search plays a crucial role in many numerical optimization methods, such as quasi-Newton, truncated Newton, and interior-point algorithms, as well as in direct search and derivative-free optimization methods.

Although linesearch algorithms have been extensively studied and applied in various optimization contexts, their choice and implementation can significantly impact the performance of numerical methods in solving complex optimization problems.