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The magic from zero to one: How to find the minimum value of a function using the zero-order method?
In the field of mathematical optimization, finding the minimum value of a function is an important task. Whether in machine learning, economic models, or engineering design, being able to find minima accurately and efficiently can bring considerable benefits. In this process, the zero-order method has become a popular choice due to its unique advantages.
Basic concepts of zero-order method
Zero-order methods do not rely on information about the derivative of the function, but only use the function value for optimization. This gives them great flexibility in dealing with certain minima problems where derivatives are not available.
In many practical applications, functions may be messy, piecewise discontinuous, or hidden in a black box model. Here, zero-order methods can provide valuable solutions.
Main types of zero-order methods
There are several main zero-order methods for finding minima of one-dimensional functions, such as the ternary search, the Fibonacci search, and the golden section search.
Ternary Search Method
The basic idea of this method is to determine the possible location of the minimum value by comparing the function values at three points. Its main advantage is that it can quickly narrow the search range and gradually find a more precise minimum location.
Fibonacci search
Compared to the ternary search method, the Fibonacci search method uses the Fibonacci sequence in mathematics to make each step of the search more efficient. Each step only requires one function evaluation, which greatly reduces the time cost in the calculation process.
Golden Section Search Method
This method is similar to the Fibonacci method, but each step is divided based on the golden ratio, which can ensure the best search efficiency.
What these methods have in common is that they neither rely on the derivative of the function nor require the continuity of the function, thus expanding their application scope.
Comparison with first-order methods
Although the zero-order method has many advantages, in some cases, the first-order methods such as the improved bisection method and Newton's method also show their excellent performance.Improved dichotomy
This method requires the function to be differentiable and guides the direction of finding the minimum value by calculating the derivative of the function at a certain point. It generally converges faster than the zero-order method, but has difficulty with non-smooth or discontinuous functions.
Newton's method
Newton's method, which expands the function into a quadratic polynomial, can reach quadratic convergence when approaching the minimum point, which makes it possible to converge quickly in the early stages of optimization.
Zero-Order Methods in Multidimensional Search
When dealing with multidimensional functions, the zero-order method is also indispensable. By determining a direction of descent, these methods can continually search for lower function values. This process demonstrates a high degree of flexibility and scalability.
ConclusionIn many practical applications, the zero-order method is used in combination with other optimization strategies, such as simulated annealing, to overcome the limitations of the current local minimum, which can effectively expand the solution space.
In summary, the zero-order method is a powerful and flexible optimization tool that can not only deal with discontinuities and non-smoothness of functions, but also find the optimal solution in high-dimensional space. As the research on function minima deepens, these methods will play an increasingly important role in future scientific and technological development. In this context, what method do you think should be used to find the minimum value in your application scenario?
