Language
Country/Area
No result found
The secret weapon of optimization: Do you know how one-dimensional line search finds the best solution?
In optimization problems, how to effectively find the local minimum of a function has always been a topic of great concern. As a basic iterative method to solve this problem, one-dimensional line search technology has undoubtedly become a secret weapon in the field of optimization. This method is not only applicable to simple single-variable situations, but can also be extended to complex multi-variable situations, helping researchers and engineers find more appropriate solutions.
A one-dimensional line search first finds a descent direction and then calculates a step size to determine how far to move in that direction.
First, let's understand the basic concept of 1D line search. Suppose we have a one-dimensional function f, and it is unimodal, which means that in some interval [a, z], it contains only one local minimum x*. In this case, the function f is strictly decreasing between [a, x*] and strictly increasing between [x*, z].
To find this minimum point, several different methods can be used, including zero-order and first-order methods. Zero-order methods do not make use of derivatives, but rely solely on evaluation of functions. Among them, the three-point search method is widely used. This method selects two points b and c, and gradually narrows the search range by comparing the size of f(b) and f(c). If f(b) ≤ f(c), then the minimum must be in [a, c]; otherwise, it must be in [b, z].
This gradual reduction method requires two function evaluations, although each reduction is about 1/2, so the convergence speed is linear and the convergence rate is about 0.71. If b and c are chosen so that the lengths of intervals a, b, c, and z are equal, the search interval will be reduced by 2/3 in each iteration, and the convergence rate will be improved to about 0.82.
Fibonacci search and golden section search are also variants of the zero-order search method, but both require only one function evaluation, so the convergence efficiency is higher, and the convergence rate is about 0.618, which is higher than the zero-order method. The best.
To further clarify, first-order methods assume that the function f is continuously differentiable, which means that we can not only evaluate the value of the function, but also calculate its derivatives. For example, binary search is a common search method. At each iteration, if we can find the midpoint c of the interval, by checking the value of the derivative f'(c), we can determine the location of the minimum.
However, if superlinear convergence is required, we need to use curve fitting methods. These methods fit the known function value with a polynomial and then find the minimum value of the fitted function as the new operating point. We have to mention Newton's method, which uses first and second order derivatives and converges quadratically when the initial point is close to a non-degenerate local minimum.
Curve fitting methods have superlinear convergence properties when the initial point is close to a local minimum, which makes them powerful in many application scenarios.
When multiple dimensions are involved, although the specific calculation process becomes more complicated, one-dimensional line search can still be performed in the presence of multiple dimensions. It first finds a descent direction and then determines the step size for efficient optimization. Often, such models can be combined with other methods such as simulated annealing to overcome the risk of getting stuck in local minima.
Through these methods, optimization can achieve higher performance and also help us better understand the mechanisms behind mathematical models. In the desire to find the best solution, whether in scientific research or commercial applications, one-dimensional line search has demonstrated its indispensable value.
Have you ever wondered what other innovative ways there will be to improve existing line search techniques in the future?
