**Non-Euclidean Geometry Data and Characteristics**

数学教育论文代写 The past ten years have been characterized by different algorithms evolution as suggested in the literature review for solving many- and…

### Abstract 数学教育论文代写

The past ten years have been characterized by different algorithms evolution as suggested in the literature review for solving many- and multi-objective optimization functions or problems. Besides, the results of the stated algorithms is based on their potential to generate a diversity front. Which is near the Pareto optimal proximity front. Thus, proximity together with diversity significantly or strongly rely on the Pareto front’s geometry. Or whether it generates a Euclidean, hyperbolic, or spherical hyper-surface. Current many- and multi-objective evolutionary algorithms, however, indicate poor different geometries’ versatility.

Addressing the issue requires a proposition for a novel evolutionary algorithm, which will adapt the proximity. And diversity metrics accordingly and estimates the generated front’s geometry based on a fast process under O(M × N) sophistication in computation. Whereby M is the objectives size and volume of the population represented by N. Hence, to create the sample population for the future. Outcomes are chosen relying on their support to proximity and diversity, especially for the non-dominated front based on geometry estimation. Empirical computation indicates that the suggested algorithms outperform many- and multi-objective state-of-the-art evolutionary algorithms based on the test problems under different objectives and geometries (M=3,5, and 10).

**Introduction 数学教育论文代写**

Algorithms’ evolutions have been extensively utilized in literature to mitigate many- and Multi-objective Optimization Problems. Or the MOPs in that the goals are usually contrary to each other during the solution. The aim of many- and multi-objective EA or the Evolutionary Algorithms. Or the MOEAs is to estimate the PF or the optimal Pareto Front based on the solution under non-dominated sets. The achievement of the aim requires that MOEAs focus on creating non-dominated fronts that are nearby the Pareto Front (convergence or proximity). Which are well-distributed on the involved diversity or optimal Pareto Front. For instance, In different models, the accomplishment of proximity is based on utilizing diversity. And Pareto-dominance that is preserved based on crowding distance (Panichella, 2019).

Many- and multi-objective evolutionary algorithms ensure diversity based on well-diverse vectors’ weight while focusing on attaining PF. Or Pareto Front following the selection of the aggregation function-based approach (Özdemir, et al., 2013). Scholars have noted that MOEAs perform well based on multi-objective optimization problems (Tanabe & Oyama, 2017). In that the involved problems have more than three objectives or goals. Therefore, the degradation takes place since the solutions’ number that is non-dominated in the involved populations increases exponentially based on M or the objective number. The results illustrated by the literature are that there is a dramatic loss of selection stress regarding the Pareto Front (Tanabe & Oyama, 2017). 数学教育论文代写

#### The mitigation of the involved limitations has led to different practitioners suggesting different approaches to increase either the diversity or proximity of the involved populations.

The involved strategies could be categorized into different groups like indicator-based strategies, dominance-relation-based. And reference-point based. The example has replaced the crowding distance following the number of operators in the point of reference that supports both diversity. And proximity of population members (He, et al., 2018). Practitioners have suggested the utilization of other techniques that utilize a new dominance connection that has more robust selection pressure than Pareto-based dominance (Yuan, et al., 2016). 数学教育论文代写

Besides, other utilizes the grid difference and grid dominance to maintain population members. And support the selection pressure of diversity based on a grid-based approach (Li, 2021). Besides, Zhao et al. (2020) indicated that in some MOEA, the strategy of selection is dependent on the improved indicator in the involved case. The calculation of the indicator maintains a reference point set, which is adaptively updated and maintained.

The many- and multi-objective evolutionary algorithms stated in the above context utilize heuristics. And strategies that are created based on the implicit assumption that the Pareto Front (approximation of non-dominated front) has been found to possess a Euclidean orientation or curvature. For instance, in the cases of MOEA, the points of reference are established based on Dennis. And Das’s systematic approach or strategy (Das & Dennis, 1998) that placed points on a flat hyper-surface or normalized hyper-plane.

#### Besides, a model has categorized objectives space in equal-sized grids. 数学教育论文代写

But most PF or Pareto Front of multi-objective problems is concave (spherical geometry). Or convex (hyperbolic geometry) according to Jiang & Yang (2017). Therefore, current studies have implemented non-linear adjustment approaches to create the shape of the generated front by the many- and multi-objective evolutionary algorithms (Falcón-Cardona & Coello, 2019). Front modeling approaches, however, incur a significant cost considering the sophistication of the computation of the fitting approaches applied.

The involved study has introduced robust and appropriate MOEA approaches. For example, adaptive geometry estimation-based MOEA utilized for different optimization functions or problems. Also, the suggested model fails to assume the Pareto Front geometry and its non-dominated. Or approximation front produced in the respective generation. The overall framework has been inherited by models while replacing the crowding distance with various types of scores collaborating both proximity and diversity of the non-dominated categories.

Therefore, MOEA utilizes a rapid heuristic based on computational complexity to predict the front’s geometry in the involved generations. The proximity is evaluated as the distance between the ideal point. And population member while the diversity is evaluated based on the population members’ distance. Also, the utilized distance to compute diversity and proximity corresponds to the Lp sequence linked to the estimated geometry.

**Importance of the Topic 数学教育论文代写**

The involved study has focused on the various spaces like the M-dimensional space, given that the vectors’ norm. Or length which is determined based on the following Euclidean expression. For the involved space (Euclidean), the space between two points (say B and A) is the straight-line norm linking the stated points. The Euclidean norm, however, sometimes fails to offer the most precise. Or accurate measure of the space between the stated points in M-dimensional space generic (Panichella, 2019). A p-norm or an Lp norm is a typical Euclidean norm expressed in the different equations according to Panichella (2019).

The involved Euclidean norm, for example, the norm illustrates a unique setting, for example, Lp norm whereby p is equal to two. Various measures of p represent different values of the space between the two points say B and A. Hence, the points’ set is represented provided that they are equidistant to an original point. For example, axes’ origin, but changes based on the utilized Lp norm (Özdemir, et al., 2013). The all points’ set based on distance from the axes’ origin establishes a hyper-surface unit. The curvature or geometry of the hyper-surface critically relies on the exponent value p according to Falcón-Cardona & Coello (2019). Further explanations have been offered regarding the concept whereby it is considered that the unit hyper-surface will represent unit curves of the bi-dimensional space.

Therefore, in the case of p = 1, the curve of a unit will be flat while corresponding to the straight line linking various points, for example, (1,0) or (0,1). Hence, the involved points equidistant (a unit) to the origin of the plan lie on the stated line. Besides, in the case of p > 1, the involved curve unit will be concave. And whenever the value of p is 2, then it represents a circle’s unit, whereby the circle has a radius equal to 1 unit. However, in the case of p < 1 the equidistance points are located on the curve in hyperbolic geometry.

#### The value of p, therefore, determines the curvature or geometry of the hypersurfaces unit concerned with the Lp norm. 数学教育论文代写

Past assessments have proposed to leverage the connection between the curvature. And Lp of the involved hypersphere’s unit to predict the geometry of the non-dominated front generated by a many- and multi-objective evolutionary algorithm. Özdemir et al. (2013) suggested an indicator-based multi-. And many-objective evolutionary algorithm that utilizes a Lo curves’ family to generate a relevant referent set that is utilized to evaluate the indicator or ∆p. Currently, practitioners like Panichella (2022) have utilized a typical variant of the different norm (Lp). And utilized other algorithms to generate the parameters of the typical simplex function that reduces fitting errors.

Methods utilized in past studies, however, to create the established categories are usually expensive. Besides, front establishment needs to mitigate a non-linear fitting issue based on iterative numerical approaches, resulting in a huge computation cost. For instance, the fitting steps outlined by Li (2021) have general sophistication, given that the iteration value of the involved algorithm. The study has introduced a quicker procedure to predict the curvature of the non-dominated front whereby the general sophistication will be represented. Given the objectives’ number and population size represented. Also, it has been vital to involve the quick approach in the suggested model while replacing the distance (crowding) with diversity and proximity heuristics.

**Findings **

The results indicate that the 1^{st} non-dominated is normalized. And rescaled by the application of the formula based on Euclidean geometry or curvature. Given the objective whereby the provided with the solution. And the least measure of the i^{th} objective in the involved solutions for the front under consideration. Besides, based on the numerator, the involved goals are interpreted to possess a theoretical point similar to the axes’ origin. The involved denominator is the M-dimensional intercept based on the objective axis. Besides, the involved M-dimensional hyper-plane comprises the extreme vectors. Based on the largest values of the objectives in the concerned front, upon the translation inclined to the axes’ origin.

The intercepts are acquired based on the solutions of the system of the linear expression given that ‘a’ is equal to 1 implying that the hyperplane’s M-dimensional created by the extreme points. Therefore, the results in the involved non-dominated categories are normalized. And scaled depending on the ai values for the 1^{st} non-dominated front under investigation. Hence, concurrently with the objectives of the non-dominated category taking values in (0,1). The other non-dominated front objectives could possess greater values than 1. 数学教育论文代写

#### Evaluation of the curvature for the 1^{st} non-dominated category, there will be a fitting error that will require finding a hyper-surface unit that fits best the normalized objectives.

The involved curvature or geometry of the hyper-surface unit is evaluated based on the Lp norm association. Hence, it will be necessary to find the p value for the norm corresponding to the hypersurfaces unit best fitting the front’s normalized objectives. Besides, an optimal hyper-surface fitting has a norm in that the involved points in the non-dominated front are equidistance to the theoretical point coinciding. With the axes’ origin upon normalization of the values under consideration. Moreover, the extreme positions in the front represent the intersections of the non-dominated front based on the axes’ objectives. Also, upon normalization, the non-dominated front intersection point will be with objective equal to one. And solution is equal to zero for the other points. Besides, its distance to the axes’ origin or the theoretical point will be equidistant to the exponent point selected.

The score of proximity for a generic solution will be evaluated as a distance based on the norm of the vector’s objective to the theoretical position. Therefore, the solution will be oriented to the concerned population. Which is located on the hypersurfaces unit linked to the predicted norm with solution being the proximity equal to one.

Besides, a solution based on the stated proximity will be less than one for the dominated front values of the unit hyper-surface of norm Lp. Consequently, the solution with a proximity less than one will be further from the theoretical position than to the unit hyper-surface points linked to the norm Lp. The solution’s diversity has been represented and has been evaluated as the least distance of the norm based on the other results in the non-dominated category. 数学教育论文代写

#### The score of survival of the involved solutions is in the factor of population collaborating both proximity (to minimize) and diversity (to maximize) as.

Therefore, the involved extreme solutions have been assigned the maximum possible score of survival based on preserving the elements in the population for the other generations. Therefore, the steps initialize different sets explained. Which retains the track of the involved solutions based on the assigned score values. And comprising of the solutions that are to be scored whereby the proximity scores will be computed based on constraints. Therefore, it will be essential to adopt the MOEA in MATLAB based on empirical models. And frameworks available to the public. Hence the MOEA source code (together with results from experiments) are available to the public. Also, empirical approaches offer source code for the involved initial problems and algorithms utilized as the study’s benchmark.

The involved many- and multi-objective evolutionary algorithms have focused on similar elements. And parameters set based on past literature (Li, 2021; Tanabe & Oyama, 2017). However, the results indicate that in MOEA the size of the population (N) relies on the reference points’ number created with the adopted systematic approach (Panichella, 2022). The population size, instead, could be arbitrary in various models considering that it fails to utilize reference points. To ensure a fair comparison, however, the study adopted similar sizes of populations. And a similar fitness number of evaluations for the involved algorithms. Particularly, the number was set at a value of about N = 275,210, and 91. While the objectives’ number was about M = 10, 5, and 3 giving evaluations of fitness number at N x 300 iterations.

**Conclusion 数学教育论文代写**

The involved study has introduced a robust many- and multi-objective evolutionary algorithm like MOEA whereby the selection steps considers diversity for both the proximity. And the size of the population to the theoretical point. The difference between the approach and the state-of-the-art approaches in MOEA is that the former fails to assume the Pareto Front curvature or geometry. Rather, the geometry is estimated based on the front following a fast procedure whereby the complexity of computational is less than the complexity of the sorting algorithm in the non-dominated front. The study has evaluated the performance of MOEA based on different empirical models with different objective problems based on varying properties and different objectives’ numbers.

Moreover, the study has compared the performance of many- and multi-objective evolutionary algorithms with other state-of-the-art algorithm models like MOEA. The acquired outcomes indicated that the many- and the multi-objective evolutionary algorithm are better in terms of performance than the other models based on the objectives’ number. It is recommended that future research explore options in heuristics to predict the curvature of the non-dominated categories or Pareto Fronts. Besides, the studies should focus on applying MOEA to complex multi-objective problems, more test benchmarks, and real-world problems.

**Reflection 数学教育论文代写**

There are different areas of reflection during the course based on teaching practices employed by different teachers throughout the course. Communication has been a core element of consideration for successful teaching practices for the education of this complex matter. Language is crucial when learning Euclidean since wrong interpretation. And audience could lead to different understanding and presentation of the multi-objective problems and solutions. Therefore, the curriculum unit development has developed from a general issue to a specific one based on different limiting elements. Robust understanding and experience by the involved teachers have organized the involved course into understandable sections for easier comprehension and interpretation by the involved students.

The lesson schedule involved during the course illustrated different strengths. And opportunities that could be tapped by the students for their continuous professional development. There has been a comprehensive flow of lessons during the course from easier units to more complex functions and multi-objective problems. The involved teachers have experience in this field considering the number of years they have taught the same topics and units. Therefore, they have amassed detailed information about the involved topic.

#### Whenever the involved concepts are taught by less experienced tutors, the understanding level of the students reduces.

The involved topics are complex and take time for students to fully understand. And formulate successful models to solve the involved mathematical problems. The approach taken during the course considered moving from less difficult or complex problems to more sophisticated multi-objective problems. This approach allowed the students to gain more confidence when tackling MOPs based on Euclidean geometry. 数学教育论文代写

The teaching practice experienced throughout the course illustrated the detailed experience of the tutors in terms of using the geometrical concept. Besides, the approaches utilized to solve the involved problems illustrated informed steps for reaching the solution. The literature has proposed different programs that could be utilized to understand. And solve the geometric problems under consideration. Some programs have more advantages than others. But they tend to complement each other based on the assumption made in establishing the solution. The technical expertise utilized in the literature is based on further. And detailed research regarding Euclidean and non-Euclidean geometry.

The curvature of functions can be evaluated and determined based on simple steps provided by the involved literature. Besides, the validity and reliability of the results have been determined by past practitioners. And scholars through empirical studies and experiments. The involved teaching practice involved in the literature is successful since it has offered adequate pieces of literature vital for the involved topic. However, poor application of different teaching practices could lead to detrimental. Or adverse results of the involved geometrical concept. Besides, more programs should be established and created to compliments past. And current models considering future constraints.

**References 数学教育论文代写**

Das, I. & Dennis, J. E., 1998. Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. * SIAM Journal on Optimization, *8(3), p. 631–657.

Falcón-Cardona, J. & Coello, C., 2019. Indicator-based Multi-Objective Evolutionary Algorithms: A Comprehensive Survey. * ACM Comput. Surv., *9(4), pp. 39.1-39.34.

He, X., Zhou, Y., Chen, Z. & Zhang, Q., 2018. * Evolutionary Many-objective Optimization based on Dynamical Decomposition. *s.l., IEEE Transactions on Evolutionary Computation.

Jiang, S. & Yang, S., 2017. A Strength Pareto Evolutionary Algorithm Based on Reference Direction for Multiobjective and Many-Objective Optimization. * IEEE Transactions on Evolutionary Computation, *Volume 21, pp. 329–346.

Li, M., 2021. Is our archiving reliable? Multiobjective archiving methods on “simple” artificial input sequences. * ACM Transactions on Evolutionary Learning and Optimization, *1(3,9), pp. 1-19. 数学教育论文代写

Özdemir, S., Bara’a, A. A. & Khalil, Ö. A., 2013. Multi-objective evolutionary algorithm based on decomposition for energy efficient coverage in wireless sensor networks. * Wireless personal communications, *71(1), p. 195–215.

Panichella, A., 2019. * An Adaptive Evolutionary Algorithm based on Non-Euclidean Geometry for Many-objective Optimization. *Prague, Czech Republic, Association for Computing Machinery.

Panichella, A., 2022. * An Improved Pareto Front Modeling Algorithm for Large-scale ManyObjective Optimization. In The Genetic and Evolutionary Computation Conference Association for Computer Machinery. *Boston, MA, USA, ACM.

Tanabe, R. & Oyama, A., 2017. * Benchmarking MOEAs for multi-and manyobjective optimization using an unbounded external archive. In Proceedings of the Genetic and Evolutionary Computation Conference. *s.l., ACM, p. 633–640.

Yuan, Y., Xu, H., Wang, B. & Yao, X., 2016. A new dominance relationbased evolutionary algorithm for many-objective optimization. * IEEE Transactions on Evolutionary Computation, *20(1), p. 16–37.

Zhao, H. et al., 2020. Decomposition and Dominance Relation Based Many-objective Evolutionary Algorithm[J]. * Journal of Electronics & Information Technology, *42(8), pp. 1975-198.