[Matlab algorithm] Basic concepts of solving multidimensional functions

multidimensional function

Multidimensional functions are defined in ${\mathbb{R}}^{}$function on${}^n$$n$ is the dimension of the function. For example,$f(x,y)=x^2+y^2$ is a two-dimensional function,$f(x,y,z)=x^2+y^2+z^2$ is a three-dimensional function.

Optimization problem

The optimization problem refers to finding the function f ( x ) f(x) under given constraints$The\; maximum\; or\; minimum\; value\; of\; f\left(x\right)$ .

Optimization algorithm

Optimization algorithms refer to algorithms used to solve optimization problems.

Types of optimization problems

According to the function $Due\; to\; the\; properties\; of\; f\left(x\right)$ , optimization problems can be divided into the following types:

• Unconstrained optimization problem

Unconstrained optimization problems refer to optimization problems without any constraints. For example, find the function $f(x)=x^{Minimum\; value\; of\; 2}$ .

• Constrained optimization problem

Constrained optimization problems refer to optimization problems with constraint conditions. For example, find the function $f\left(x\right)=x^{Minimum\; value\; of\; 2}$ , where$x\ge 0$。

Classification of optimization algorithms

According to the strategy for solving optimization problems, optimization algorithms can be divided into the following types:

• direct method

The direct method refers to directly solving the optimal solution of the optimization problem. For example, Newton's method is a direct method.

• Iterative method

The iterative method refers to gradually approaching the optimal solution through iteration. For example, gradient descent is an iterative method.

According to the properties of function f(x) , optimization algorithms can be divided into the following types:

• Convex optimization problem

The convex optimization problem means that the function f(x) is a convex function and the constraints are convex sets. For convex optimization problems, there is a unique global optimal solution.

• Non-convex optimization problem

Non-convex optimization problem means that the function f(x) is a non-convex function, or the constraints are non-convex sets. For non-convex optimization problems, there may be multiple local optimal solutions or even no global optimal solution.

Commonly used methods for solving multidimensional functions

Commonly used optimization algorithms include the following:

• gradient descent method

Gradient descent is a simple and easy-to-use iterative method. The basic idea of ​​this method is that at the current point $(x_k,y_k)$ , along the function$The\; gradient\; direction\; of\; f\left(x\right)$ is searched until it converges to the optimal solution.

• conjugate gradient method

The conjugate gradient method is an improved gradient descent method. The basic idea of ​​this method is that at the current point $(x_k,y_k)$ , along the functionf ( x ) f(x)Search in the gradient direction of$f$$($$x$$)\; ,\; but\; in\; each\; search,\; the\; direction\; of\; the\; previous\; search\; must\; be\; considered.$

• Newton's method

Newton's method is an iterative method based on the second derivative of a function. The basic idea of ​​this method is that at the current point $(x_k,y_k)$ , along the functionf ( x ) f(x)The direction of the inverse matrix of the second derivative matrix of$f$$($$x$$)\; is\; searched.$

• simulated annealing

Simulated annealing is an iterative method based on simulating physical phenomena. The basic idea of ​​this method is to search for the optimal solution by gradually lowering the temperature starting from the initial point.

Conclusion

Optimization problem is an important issue in the field of mathematical optimization. There are many kinds of optimization algorithms, and each algorithm has its advantages and disadvantages. In practical applications, it is necessary to choose an appropriate algorithm based on specific problems.

Additional information

• The solution of the optimization problem can be divided into the following steps:
  1. Determine the objective function.
  2. Determine constraints.
  3. Choose an appropriate algorithm.
  4. Algorithm.
  5. Evaluate algorithm performance.

For example, the following code block represents a gradient descent script for a two-dimensional function:

def gradient_descent(f, x0, eps):
  """
  梯度下降法求解多维函数的最优解。

  Args:
    f: 目标函数。
    x0: 初始点。
    eps: 精度。

  Returns:
    最优解。
  """

  x = x0
  while True:
    dx = -grad(f, x)
    x = x + dx
    if np.linalg.norm(dx) < eps:
      break
  return x

The following table represents the advantages and disadvantages of commonly used methods for solving multidimensional functions:

method	advantage	shortcoming
gradient descent method	Simple and easy to use	Easy to fall into local optimal solution
conjugate gradient method	Fast convergence speed	Sensitive to initial values
Newton's method	The fastest convergence speed	Large amount of calculation
Quasi-Newton method	Fast convergence and insensitive to initial values	Sensitive to second derivatives of functions
simulated annealing	Suitable for multimodal functions and not easy to fall into local optimal solutions	Slow convergence
genetic algorithm	Suitable for complex search spaces and not easy to fall into local optimal solutions	Slow convergence and sensitive to initial values
particle swarm algorithm	The convergence speed is fast and it is not easy to fall into the local optimal solution.	Sensitive to initial values
bat algorithm	The convergence speed is fast and it is not easy to fall into the local optimal solution.	Sensitive to initial values
Ant Colony Algorithm	It is suitable for search spaces with structure and is not easy to fall into local optimal solutions.	Slow convergence and sensitive to initial values
bee colony algorithm	It is suitable for search spaces with structure and is not easy to fall into local optimal solutions.	Slow convergence and sensitive to initial values