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# 【速搜问答】确定性算法是什么

9个月前 (04-13) 157次浏览

Chinese-English Translation:

Deterministic algorithm is to use the analytical properties of the problem to generate a certain sequence of finite or infinite points to make it converge to the global optimal solution. This kind of method searches the local minima according to a certain deterministic strategy, and tries to jump the local minima that have been obtained to achieve a global optimum. It can make full use of the analytical properties of the problem and thus has high computational efficiency.

Deterministic algorithm is to use the analytical properties of the problem to generate a certain sequence of finite or infinite points to make it converge to the global optimal solution. This kind of method searches the local minima according to a certain deterministic strategy, and tries to jump the local minima that have been obtained to achieve a global optimum. It can make full use of the analytical properties of the problem and thus has high computational efficiency.

Such as filling function method, hole function method, D.C. programming algorithm, interval method, monotone programming, branch and bound method and integral level set method, the construction of these algorithms all involve some local or global properties of the known objective function. Among them, the continuity and differentiability of functions are considered as local properties, while convexity, monotonicity, density, equicontinuity, Lipschitz continuity and level set are generally called global analytical properties.

Filled function method

Filled function is first proposed by Professor Ge Renpu of Xi’an Jiaotong University. Filled function method makes full use of the local properties of function in feasible region.

From the definition of filled function, we can see that if the current minimum is not the global minimum, by minimizing the filled function, we can jump out of the current local minimum of the original problem and reach a point where the function value of the original problem is smaller than the current local minimum. Therefore, minimizing the objective function of the original problem from this point will lead to a local minimum point with smaller value of the objective function of the original problem.

The filled function algorithm consists of two phases: minimization phase and filling phase. These two stages are used alternately until no better local minima can be found. In the first stage, the classical minimization algorithm can be used to find a local minimum of the objective function

The advantage of the filled function is that it makes more use of the properties of the function, so the convergence speed is faster, and the design and implementation of the algorithm is relatively easy; the disadvantage is that the filled function relies too much on some unknown parameters, which increases the workload before the algorithm design. A large number of experiments are needed to determine the range of parameters, so as to ensure that a satisfactory global optimal solution can be found, and the filled function is easy to implement The function uses line search, so it is very difficult to find the point of low basin valley.

Hole function method

The hole function method was first proposed by Levy and Montalvo in 1985. It consists of a series of cycles. Each cycle includes two stages: local minimization stage and hole drilling stage.

The first stage is the minimization stage. Starting from an initial point, a local minimization algorithm is applied to obtain a local minimization point of F (x)

here

A local minimum x ‘is found by properly minimizing the hole function

(1) 如果 x’=

(1) If x ‘=

(2) 如果 x’≠

(2) If x ‘≠

here,

(3) 如果 f(x’)≤f(

(3) If f (x ‘) ≤ f（

The hole function has the following defects

(1) 极的强度

(1) The strength of the pole

(2) 打洞函数可能找到另一局部极小点 x’，成立 f(x’)≥f(

(2) The hole function may find another local minimum x ‘, f (x’) ≥ F（

(3) 极的增加会引起打洞函数成为平坦，这时候极小点很难求。

(3) The increase of the minimum will cause the hole function to be flat, and the minimum is difficult to find.

D.C.规划算法

D. C. planning algorithm

By introducing the variable t, the following D.C. programming problem is solved

It can be transformed into an equivalent concave minimization problem (denoted as problem (4)):

Obviously, the objective function

Therefore, any D.C. programming problem can be solved by concave minimization algorithm. For concave minimization problems, some algorithms have been proposed, which are based on branch and bound techniques, cut plane methods, optimality conditions and integer programming, and their effectiveness depends on the structural characteristics of the problem to be solved.

Interval method

The basic idea of interval method is based on interval analysis, using interval variable instead of point variable to calculate interval according to interval arithmetic operation rules, and combining branch and bound method with Moore skelboe algorithm. For this kind of algorithm, Moore first proposed the concept of interval global optimization. In this research area, all algorithms include accurate interval calculation, and the efficiency of the algorithm depends on the construction methods of objective function, gradient and constrained interval expansion. This kind of method is generally divided into five basic steps: delimitation, branching, termination, deletion and splitting. It includes interval splitting rules, deletion rules and interval selection rules. Different interval algorithms are based on different processing methods of these rules.

Compared with other methods (i.e. generating approximate point sequence by point search), interval method has the outstanding advantage that for global optimization problems in low dimensional space, it can find all global minima within a given precision. In particular, for the univariate objective function in two-dimensional space, a highly efficient interval slope algorithm for global minimization of univariate function is established. The disadvantage is that when applied to high-dimensional global optimization problems, the algorithm has a large amount of computation, and it is very difficult to determine the interval splitting rules, deletion rules, interval selection rules and their test conditions.

branch-and-bound procedure

In the branch and bound algorithm, the feasible region is relaxed, and the original region is divided into more and more small regions step by step, which is called branch; in these small regions, the lower and upper bounds of the objective function are determined, which is called bound. At a certain stage of the algorithm, small regions whose inner and lower bounds are larger than the current minimum upper bounds are deleted. This process is called pruning, because these small regions obviously do not contain the optimal solution. When the difference between the maximum lower bound and the minimum upper bound tends to zero, and the subdivision region shrinks to a point, we can get the global minimum and the global minimum of the objective function.

All kinds of branch and bound algorithms have the following common characteristics in solving global optimization problems with continuous variables

(1) 对目标函数和可行域有较高的要求，以便于分支和定界。算法的效率与分支和定界方法的效率紧密相关。

(1) There are higher requirements for the objective function and feasible region to facilitate the branching and demarcation. The efficiency of the algorithm is closely related to the efficiency of branch and bound methods.

(2) 在算法实施时，需要储存越来越多的细分的小区域和目标函数在其上的下界，这使得在编程时，对数据结构的选择、计算机内存的使用提出了更高的要求。

(2) In the implementation of the algorithm, it is necessary to store more and more subdivided small areas and the upper and lower bounds of the objective function, which makes higher requirements for the selection of data structure and the use of computer memory in programming.

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