阅读说明：本技术 一种新颖的动态约束优化测试函数集 (Novel dynamic constraint optimization test function set ) 是由 余俭 王勇 于 2019-03-06 设计创作，主要内容包括：本发明公开了一种新颖的动态约束优化测试函数集,属于进化计算领域。本发明设计的测试函数具有1)可扩展性、2)可调节性、3)多模态、4)可行区域变化的严重程度、5)全局和局部最优解已知等特点。本发明提出的测试函数的可扩展性、可调节性、多模性、可行域的变化程度可以模拟现实生活中复杂的问题,全局最优解和局部最优解已知能够很容易的评价动态约束优化算法的性能,在动态约束优化算法应用到实际问题中去之前可以充当一个有用的测试工具。(The invention discloses a novel dynamic constraint optimization test function set, and belongs to the field of evolutionary computation. The test function designed by the invention has the characteristics of 1) expandability, 2) adjustability, 3) multimodality, 4) severity of feasible region change, 5) known global and local optimal solution and the like. The expandability, the adjustability, the multimode and the change degree of the feasible region of the test function can simulate complex problems in real life, the performance of the dynamic constraint optimization algorithm can be easily evaluated by knowing the global optimal solution and the local optimal solution, and the test function can serve as a useful test tool before being applied to actual problems.)

1. A novel dynamic constraint optimization test function set is designed according to the following two main points: (1) simulating the condition of severe change of the feasible region by enabling the feasible region to track the change of the highest peak, and simulating the condition of slight change of the feasible region by enabling the feasible region to track the change of the peak with the fixed number; (2) simulating the test problem of different feasible domain numbers by designing the variable of a plurality of peaks with fixed feasible domain tracking; and finally, combining the two ideas pairwise to design test functions with different characteristics.

2. The method according to claim 1, wherein the feasible region is made to track the variation of the highest 1, 2, 3 peaks to simulate the condition of the feasible region changing drastically; enabling the feasible region to track the change of the fixed numbers 1, 6 and 10 to simulate the slightly changed condition of the feasible region;

3. the method according to claim 1, wherein the test function set of single, two and three feasible domains is designed.

4. The test function constructed according to the method of claim 1 is scalable, multi-modal, varying severity of feasible regions, known global and local optimal solutions, and the like.

Technical Field

The invention designs a novel dynamic constraint optimization test function set. The method can be applied to performance detection of various dynamic constraint optimization algorithms, and belongs to the field of evolutionary computation.

Background

In practical application, a large number of dynamic optimization problems exist. Unlike static optimization problems, the topology and the location of the optimal solution of a dynamic optimization problem may change accordingly as time (environment) changes. Therefore, when solving the dynamic optimization problem, the optimization algorithm should have the ability to continually adjust itself to identify and track the dynamic environment. As one of the important branches of the Dynamic optimization problem, Dynamic constraint optimization problems (DCOPs for short) are very common in the fields of scientific and engineering applications (such as on-line design and control of controllers, task allocation and scheduling, etc.).

In general, the dynamic constraint optimization problem can be roughly classified into the following three types: 1) the objective function is dynamically changing, the constraint conditions remain static; 2) the objective function is kept unchanged, and the constraint condition is dynamically changed; 3) both the objective function and the constraints are dynamically changing. In the first case, dynamic changes in the objective function may cause the position of the optimal solution to jump between two unconnected feasible domains; in the second case, due to the dynamic change of the constraint conditions, the position, size and shape of the feasible region may change, which directly results in that the position of the optimal solution may also change; in a third case, the constraint optimization problem before change and the constraint optimization problem after change may have completely different characteristics. At this time, the objective function and the constraint condition simultaneously affect the position of the optimal solution. And the first two variations also occur, so the third case is specifically studied herein. The dynamic constraint optimization problems are of various types and complex, and the reflected dynamic characteristics are different along with the difference of the problems. Generally, these dynamic characteristics may be caused by one or more combinations of objective functions, decision variables, and constraints, which results in a very large number of dynamic characteristics. The non-connectivity, multi-modal, periodic, predictable, and even non-detectability of changes to the problem of the feasible domain that may cause the problem. The test function of the dynamic constraint optimization problem designed by the researcher plays an important role in judging whether a dynamic constraint optimization algorithm is capable of processing the dynamic constraint optimization problem. Through a standard set of test functions, researchers can test the performance of one algorithm from multiple aspects, and can compare various algorithms on the same platform. However, one of the problems widely existing in the field of dynamic constraint optimization problems is the lack of standard test functions, and at present, a few standard test sets are designed on the dynamic constraint optimization problems.

Disclosure of Invention

Based on the extensive knowledge of some of the deficiencies of the existing test function sets and the characteristics of the dynamic constraint optimization problem, the present invention considers that the test function of the dynamic constraint optimization problem should have the following characteristics: 1) and (4) expandability. The number of decision variables and constraints should be extensible; 2) adjustability. The shape and the size of the feasible region can be freely adjusted; 3) the multi-mode can check the searching capability of a Dynamic Constrained Optimization algorithm (DCOEAs for short) on a complex objective function with a plurality of local optimal solutions; 4) the severity of the feasible region change, such as a slight change or a dramatic change; 5) the global and local optimal solutions are known. Under these conditions, the performance of DCOEAs can be conveniently evaluated. However, existing test functions do not satisfy all of these five characteristics. Therefore, the invention provides a universal test function set of the dynamic constraint optimization problem. A dynamic unconstrained multi-peak test function (Moving Peaks Benchmark) with good performance is selected as an objective function of the test function proposed by us, and an adjustable constraint condition is designed on the basis of the objective function, so that the size, the number and the change degree of a feasible domain can be flexibly controlled.

Drawings

FIG. 1 is a three-dimensional graph and a contour plot of MPB function at time t and time t +1

FIG. 2 is a test function constraint composition diagram

The specific implementation mode is as follows:

in view of the nature that the dynamic constraint optimization problem should have, the present invention employs the MPB function as the objective function of the test function we have designed. The reason for selecting the function is that the function is a multi-peak function, has good dynamic performance, the number, height, width and moving step length of peaks can be well controlled by adjusting some parameters, and the correlation between different peak movements can be controlled by setting the parameter of the correlation coefficient. This function has been widely used by researchers in recent years for the detection of dynamic unconstrained optimization algorithms. Its definition is as follows:

here H_i(t) and W_i(t) represents the height and width of the ith peak at time t, respectively; x_ij(t) represents the location of the ith peak in the jth dimension space at time t; p represents how many peaks there are in total; d represents the dimension of the decision space; each peak moving in a certain direction and speed, sRepresenting the step size of the shift, which represents the magnitude of the dynamically varying degree of change, the shift for a single peak can be defined as follows:

motion vector hereIs a random vector

And the motion vector of the previous time instant

The moving step s is normalized, and the correlation coefficient λ is set to 0 here, which indicates that the movement of each peak is not correlated with each other. For each peak, its equation of motion is as follows:

H_i(t)＝H_i(t-1)+Height_severity*

W_i(t)＝W_i(t-1)+Width_severity*

here is a value following a standard normal distribution with a mean of 0 and a variance of 1. Height _ visibility represents the degree of change in the Height of the peak, and Width _ visibility represents the degree of change in the Width of the peak.

In general, the parameters are set as shown in table one:

table 1: default parameter table

Table 2: test function examples

To better observe the movement of the individual peaks of the MPB function, FIG. 1 shows a three-dimensional map and a two-dimensional contour map of the MPB function. From fig. 1 we can see that the position, height and width of the peaks are changed at different times.

For understanding the characteristics of the dynamic constraint optimization problem, we add constraint conditions to the original MPB function to simulate the characteristics of the dynamic constraint optimization problem, and the framework of the proposed test function is as follows:

the moving rule of the peak is the same as that of the MPB function, and the difference from the MPB function is that the function frame designed by the user has one more parameter r_l，r_lThe radius of the ith feasible domain is shown, the difficulty of the test function can be well controlled by setting the parameter, and the performance of the dynamic constraint optimization algorithm can be better evaluated. Different test problems are designed according to different characteristics of the dynamic problems. For feasible domains, the invention designs a single feasible domain problem and a multiple feasible domain problem; in terms of the degree of change of the dynamic problem, the invention designs the problems that the feasible region changes greatly and slightly. The present invention has designed a total of 6 test functions based on the above design, as shown in table 2.

Compared with the test function of the existing dynamic constraint optimization problem, the test function provided by the invention has the following characteristics:

(1) and (4) expandability. The number of peaks of the MPB function is expandable, and the number of designed constraint conditions can also be expanded; in addition to this, the decision variables of the objective function and the constraints can also be extended

(2) Adjustability. The radius of each feasible field can pass through r_k(t) this parameter. The advantages of such an arrangement are the following two points: the position of each feasible field can be adjusted; the position of each locally optimal solution can also be adjusted,the global optimal solution can also be adjusted. The location of the global optimal solution may jump between two unconnected feasible domains. Since the heights of the peaks are random, it may also occur that a new, better optimal solution is generated without changing the original optimal solution.

(3) And (4) multi-modulus property. It is clear that the test function designed by the present invention is a multi-modal function.

(4) Degree of variation of the feasible region. In this article we use two different tracking methods to design different feasible domain variation degrees. The feasible region is made to track the peaks with fixed numbers to simulate the slight change of the feasible region, and the feasible regions are made to track the highest peaks to simulate the severe change of the feasible region.

(5) The global optimal solution and the local optimal solution are known. Since all the locally optimal solutions are the centers of the feasible domains, all the locally optimal solutions of the test functions designed by us are known, and naturally, the globally optimal solution is also known.

Based on the above discussion, the test function designed by the present invention satisfies five characteristics of the dynamic constraint optimization problem. As the idea proposed by Branke, the test function should be complex enough to simulate the actual problem in real life on one hand, and should be able to help people to simply analyze the working principle of the algorithm on the other hand. In fact, the expandability, adjustability, multimode and the change degree of a feasible domain of the test function provided by the invention can simulate complex problems in real life, and the performance of the DCOEA can be easily evaluated by known global optimal solution and local optimal solution. Therefore, in the field of evolutionary dynamic constraint optimization, the test function provided by the invention can serve as a good auxiliary research tool for people.