Line search methods

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Author names: Elizabeth Conger
Steward: Dajun Yue and Fengqi You



An algorithm is a line search method if it seeks the minimum of a defined nonlinear function by selecting a reasonable direction vector that, when computed iteratively with a reasonable step size, will provide a function value closer to the absolute minimum of the function. Varying these will change the "tightness" of the optimization. For example, given the function f(x), an initial x_k is chosen. To find a lower value of f(x), the value of x_{k+1} is increased by the following iteration scheme


Step Length

Step Direction

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Steepest Descent Method

Newton Method

Quasi-Newton Method


Solution to 48 States Traveling Salesman Problem


\begin{bmatrix} G(x,y) & 0 & -A(x)^T \\ 0 & Y & W \\ A(x) & -I & 0 \end{bmatrix} \begin{bmatrix} \Delta x \\ \Delta s \\ \Delta y \end{bmatrix} = \begin{bmatrix} -\nabla f(x) + A(x)^T y \\ \mu e - W Y e \\ -g(x) + s \end{bmatrix}


1. Sun, W. & Yuan, Y-X. (2006) Optimization Theory and Methods: Nonlinear Programming (Springer US) p 688.

2. Anonymous (2014) Line Search. (Wikipedia).

3. Nocedal, J. & Wright, S. (2006) Numerical Optimization (Springer-Verlag New York, New York) 2 Ed p 664.