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Abstract

With the development of optimization theory and the application of computer technology, some new intellective optimization algorithms are developed quickly and applied widely, which is becoming the most important method for optimization problems. In this article, a new intellective optimization algorithm—Specular Reflection Algorithm—is proposed by authors who are inspired by the physical function of mirror. The traditional mathematical theory is used to prove the global convergence of this new algorithm. In order to validate the performance of this algorithm, three classical testing functions are adopted; then the algorithm is used to solve discrete engineering optimization problems and the results indicate that the Specular Reflection Algorithm has higher computation efficiency and extensive prospect for engineering application.

Keywords

Intellective optimization algorithm Specular Reflection Algorithm global convergence engineering optimization

Preface

Intelligence Optimization Algorithm (such as Genetic Algorithm^1–3 (GA), Particle Swarm Optimization^4–7 (PSO), and Simulated Annealing^8,9 (SA)), which stems from the simulation of biological behavior, physical process, or chemical properties, is an effective method for solving complex optimization problems because it not only solves objective function regardless of continuity conditions but also has simple calculation principle and high-efficiency computing power.

Previous researches indicated that each method has its own advantages and disadvantages, for example, the SA can get a global convergence with probability of 100%, but the slow physical process of solid annealing directly affects its efficiency. The GA has excellence in rapid convergence and global search capacity, and it can adjust the search direction adaptively, but the existed premature convergence is the bottleneck which limits its development. Therefore, researchers have been working on researching and developing a new and ideal optimum method. In 2011, a new heuristic global optimization algorithm, Fruit Fly Optimization Algorithm (FOA),¹⁰ is proposed by Wen-Tsao Pan who is inspired by foraging behavior of drosophila;¹¹ the FOA is easy to understand and implement and can be applied in many fields,^12–14 while the shortcomings of FOA is that it traps into local optimal easily and has slow convergence speed.¹⁵ The Cuckoo Search (CS) optimization method¹⁶ is proposed by Amir Hossein Gandomi and Xin-She Yang in 2013, who are inspired by the brood parasitism behavior of cuckoo. The CS is a simple and effective global optimization method¹⁷ and it has been successfully applied to solve a large amount of real-world optimization problems, while the performance of CS can be further improved.¹⁸ In 2014, a biologically inspired optimization algorithm which is used to solving fuzzy shortest path problems is proposed by Zhang Xiaoge.¹⁹ In addition, there are large amounts of improvement schemes about the original algorithm, which are widely used in engineering and science. B. Yang and L. Cheng²⁰ develop a new global optimization algorithm which is named Particle Swarm Optimization combined with Particle Generator (PSO-PG); the calculation accuracy of the proposed algorithm and its optimization efficiency are greatly improved. A. Hosseini and S.M. Hosseini²¹ introduced a differential inclusion steepest decent neural network optimization method which is used to solving the general nonsmooth convex problems; the computed results show that nothing penalty parameter is needed by this method and it can easily deal with infeasible starting points.

The mirror can reflect lights; the object which cannot be observed directly can be seen by people with the help of mirror. Inspired by this phenomenon, this article develops a simple and efficient optimization method—Specular Reflection Algorithm (SRA). SRA is a new kind of artificial intelligent algorithm which is functionally similar to GA, SA, PSO, and FOA. Any optimization model can be calculated by SRA which can be used in machinery, materials, electronics, construction, tourism, military, economic, management, and other fields; it can also be used in combination with other algorithms.

SRA

Introduction of SRA

As an important tool, the mirror is not only closely related with people’s work and life but also widely used in most areas of society. Reflecting light is the basic function of mirror, the object which lurks around the corner or shaded by another object can be observed with the help of mirror, such as periscope—if there is no periscope in service, people inside submarine will be unable to know the condition above the water. The physical model which is simulated by SRA is built (as shown in Figure 1), light rays which are sheltered by something cannot travel alone the direction of Route 1, however, they can be accepted by eyes alone in Route 2, when the object can be observed, the goal of finding the object is easy to achieve. The proposed algorithm, SRA, is an optimal algorithm with the goal of optimization that simulates the process of this phenomenon.

Figure 1.

Optimization way of SRA.

In this section, we will focus on the problem of object function minimization. According to the characteristics of SRA in searching the target object, the process of optimization can be summarized as the following steps:

Initialize the position of “eye,”“mirror,” and “image” randomly, the relationship among the three elements is $Cost (eye) \leq Cost (mirror) \leq Cost (image),$ where $Cost ()$ is the optimized objective function; the lower the value is, the closer the distance is to the target object.

Close to $image$ have a greater probability to get the accurate position of the target object because the accurate position of the target object is unknown and the position of $image$ is closer to it

\begin{matrix} New_image = image + (2 \cdot rand - 1) \\ [c 1 \cdot (image - mirror) + c 2 \cdot (image - eye) + c 3] \end{matrix}

(1)

where $c 1$ , $c 2$ , and $c 3$ are step size for searching; values of the three parameters are assigned in Table 1; and $rand$ is a random number which is larger than 0 and smaller than 1.

Adjust the position of the three elements ( $eye$ , $mirror$ , and $object$ ) as given in Table 2.

Evaluate whether the iteration meets the terminal condition, if it does, stop iteration and the optimization results can be output; otherwise, return to step 2.

Table 1.

Values of $c 1$ , $c 2$ , and $c 3$ .

Values
If	$c 1$	$c 2$	$c 3$
$\begin{matrix} image \neq mirror \\ image \neq eye \end{matrix}$	$c 1 / 2$	$c 2 / 2$	0
$\begin{matrix} image \neq mirror \\ image = eye \end{matrix}$	$c 1$	0	0
$\begin{matrix} image = mirror \\ image \neq eye \end{matrix}$	0	$c 2$	0
$\begin{matrix} image = mirror \\ image = eye \end{matrix}$	0	0	$c 3$

Table 2.

The way to adjust the position of elements.

New position
If	$eye$	$mirror$	$image$
$Cost (New_image) \leq Cost (image)$	$eye = mirror$	$mirror = object$	$image = New_image$
$\begin{matrix} Cost (New_image) \leq Cost (mirror) \\ Cost (New_image) > Cost (image) \end{matrix}$	$eye = mirror$	$mirror = New_image$	Invariant
$Cost (New_image) > Cost (mirror)$	$eye = image$	Invariant	Invariant

Global convergence analysis of SRA

As a novel intelligence optimization algorithm, formula (1) is adopted by SRA to generate the new solution. Irrespective of whether it is good or not, the strategy for the new solution is accepted. And the SRA is a strictly decreased algorithm, that is

C o s t (i m a g e^{(1)}) \geq C o s t (i m a g e^{(2)}) \geq \dots \geq C o s t (i m a g e^{(k)}) \geq \dots

where ${image}^{(k)}$ is the optimal solution for the $k th$ $(k = 1, 2, \dots, s)$ iteration and s (large enough) is the number of iteration.

In view of SRA is different from other intelligence algorithm in the way of generating and accepting new solutions, it is necessary to verify the global convergence of the algorithm using traditional mathematical theory:

Theorem 1

The constraint overall optimization problem presented by formula (2) can converge to the global extreme with 100% probability by SRA

\begin{matrix} min f (x), x \in R^{n} \\ s . t . s_{i} (x) = 0, i = 1, 2, \dots, l \\ h_{j} (x) \geq 0, j = 1, 2, \dots, m \end{matrix}

(2)

Testify

Provided that $f (x^{*}) = min f_{x \in D} (x)$ , where $x^{*}$ is the global optimal solution, $f (x^{*})$ is the optimal value, and $D$ is feasible region. $D = {x | s_{i} (x) = 0, i = 1, 2, l; h_{j} (x) \geq 0, j = 1, 2, m; | x \in R^{n}}$ , and $D \subset R^{n}$ , for any positive number $ξ$ , $N_{ξ} = {x | x \in D, | f (x) - f (x^{*}) | < ξ}$ is neighborhood of $x^{*}$ , because the SRA is a strictly decreased algorithm, so

f (x^{(1)}) - f (x^{*}) \geq f (x^{(2)}) - f (x^{*}) \geq \dots \geq f (x^{(k)}) - f (x^{*}) \geq \dots

Hypothesis that $A_{m} = {ω | x^{(m)} \in N_{ξ}, m \in (1, 2, \dots, s)}$ is the probability of iterative sequence drop into $N_{ξ}$ for the mth time, according to the characteristics of the algorithm SRA, $A_{1} \leq A_{2} \leq \dots \leq A_{m} \leq \dots$ , and $0 \leq A_{m} \leq 1$ , so $A_{m}$ exists can be testified. Next, $lim_{m \to \infty} A_{m} = 1$ will be proved.

During the process of iteration, the probability that the initial feasible solution just drops into $N_{ξ}$ is $A_{1}$ , and the probability that the initial feasible solution just does not drop into $N_{ξ}$ is $B_{1}$

{\begin{matrix} A_{1} = P {x^{(1)} \in N_{ξ}} = A_{1} \\ B_{1} = P {x^{(1)} \notin N_{ξ}} = 1 - A_{1} \end{matrix}

(3)

At the second time, the probability of the generated feasible solution that still does not drop into $N_{ξ}$ is $B_{2}$

B_{2} = B_{1} (1 - A_{2}) = (1 - A_{1}) (1 - A_{2})

(4)

{\begin{matrix} A_{2} = 1 - (1 - A_{1}) (1 - A_{2}) \\ B_{2} = (1 - A_{1}) (1 - A_{2}) \end{matrix}

(5)

And by this analogy, after $m$ time iterations process, the probability of the optimal solution can be acquired

\begin{matrix} A_{m} = 1 - Π_{i = 1}^{m} (1 - A_{i}) = 1 - Π_{i = 1}^{m} (1 - \frac{2 ξ}{x_{max}^{(i)} - x_{min}^{(i)}}) \\ \geq 1 - {(1 - \frac{2 ξ}{x_{max} - x_{min}})}^{m} \end{matrix}

(6)

Calculate the extreme value of formula (6)

\begin{matrix} \lim_{n \to \infty} A_{m} = \lim_{m \to \infty} [1 - Π_{i = 1}^{m} (1 - \frac{2 ξ}{x_{max}^{(i)} - x_{min}^{(i)}})] \\ \geq \lim_{m \to \infty} [1 - {(1 - \frac{2 ξ}{x_{max} - x_{min}})}^{m}] = 1 \end{matrix}

(7)

Since $0 \leq A_{m} \leq 1$ , then $lim_{m \to \infty} A_{m} = 1$ .

Numerical analysis

First, SRA is used to deal with a three-dimensional unconstrained optimization problem; in the process of iteration, each coordinate of solutions is output, so that the features of this algorithm can be illustrated by pictures and text. Next, three classical optimization functions are used to test the efficiency of the algorithm.

Numerical example 1

The three-dimensional test function is known as $f (x, y) = x^{2} + y^{2}$ , whose global minimum is 0(0,0). Coordinate of the initialized points is able to be acquired by searching randomly within [−10, 10]. After 500 iterations, the space curve of the entire searching process and its corresponding flatted view are shown in Figures 2 –5. Features of the algorithm are illustrated in Figure 3, which indicates that the algorithm is capable of searching in global scope; the perturbation is powerful at the early stage, with the iteration going on; the searching scope is gradually narrowing; and finally the algorithm converges to the global extremum. Figures 3 –5 demonstrate that the algorithm is irregular and random during the early stage of iteration, while the SRA will be close to an extremum gradually. The SRA has smaller amount of computations than Swarm Optimization Algorithm because the SRA searches and calculates only one solution for each iteration. And the searching step size can change by iteration, so the computational precision can improve. The ultimate object value and its corresponding coordinate of the test function $f (x, y)$ are 5.7074 × 10⁻²⁴ (−2.389 × 10⁻¹², −2.2539 × 10⁻¹⁷) after 500 times iteration.

Figure 2.

Space curve of test function $f (x, y)$ .

Figure 3.

Flat view of X–Y plane.

Figure 4.

Flat View of X–Z plane.

Figure 5.

Flat view of Y–Z plane.

Numerical example 2

Three classical test functions f ₁, f ₂, and f ₃ which come from the work of Qingbo et al.²² are used to test the performance of the new algorithm, and the ability of SRA is evaluated by comparing with the relevant statistical data which are obtained from reference.²² The names, mathematical expressions, feasible regions, dimensions, and theoretically optimum values of the three test functions are summarized in Table 3. The functions f ₁ and f ₂ have the property of multi-peak and easily go in local optimal value caused by adding cosine modulation; f ₃ is a pathological function whose global extreme is difficult to be searched.

Table 3.

Three test functions for detecting the performance of SRA.

Name	Mathematical expression	Feasible region	Dimension
Rastrigin	$f_{1} (x) = \sum_{i = 1}^{n} (x_{i}^{2} - 10 \cdot \cos (2 π \cdot x_{i}) + 10)$	(−5.12, 5.12)	30,100
Griewank	$f_{2} (x) = 1 + \sum_{i = 1}^{n} \frac{x_{i}^{2}}{4000} - Π_{i = 1}^{n} \cos (\frac{x_{i}}{\sqrt{i}})$	(−100, 100)	30,100
Rosenbrock’s valley	$f_{3} (x) = \sum_{i = 1}^{n - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$	(−100, 100)	30,100

SRA: Specular Reflection Algorithm.

The three test functions are optimized with 30 and 100 design variables, respectively; the maximum iterations MAX_I and the convergence accuracy $δ$ are defined as the terminal condition of iteration, where MAX_I = 60,000 and $δ = 10^{- 6}$ . Every function will run 100 times independently, and the average value which is used for comparing the SAR and other optimization algorithms are listed in Table 4; iterative curves of the SRA are shown in Figures 6 –8. Comparing with other two algorithms, we can come to a conclusion from Table 4 that the SRA has a distinct advantage over them, which not only improves the precision of the optimal value but also decreases the computational complexity. So the SRA is a reliable and efficient new method for global optimization.

Table 4.

Optimization results calculated by SRA and compared with other algorithms.

Function	Dimensions	Algorithm type	Best value	Worst value	Average value	Average number of iterations
Rastrigin	30	DE	164.343411	198.539065	183.475304	–
		MDE	0.0000001	10.944519	4.000655	–
		SRA	6.3808 × 10⁻⁸	9.9986 × 10⁻⁷	7.9941 × 10⁻⁷	4176.8
	100	SRA	9.9987 × 10⁻⁷	6.2246 × 10⁻⁷	9.694 × 10⁻⁷	16,935
Griewank	30	DE	0.00000079	0.000001	0.0000009	–
		MDE	0.00000071	0.009857	0.0005750	–
		SRA	1.7986 × 10⁻¹³	9.9891 × 10⁻⁷	1.9995 × 10⁻⁷	1696.4
	100	SRA	1.3622 × 10⁻¹²	9.9789 × 10⁻⁷	5.2347 × 10⁻⁷	993.85
Rosenbrock’s valley	30	DE	0.011101	1.284672	0.112432	–
		MDE	0.000001	10.675506	4.919782	–
		SRA	8.8744 × 10⁻⁷	0.0099042	0.00086893	38,937
	100	SRA	9.6441 × 10⁻⁷	0.15014	0.056854	56,346

DE: differential evolution; MDE: modified differential evolution; SRA: Specular Reflection Algorithm.

Figure 6.

Iterative curve of Rastrigin.

Figure 7.

Iterative curve of Griewank.

Figure 8.

Iterative curve of Rosenbrock’s valley.

Discrete optimization design for engineering structure

In order to verify the SRA can be applied to engineering design and analyze its ability in solving the discrete combinatory problems, box girder of general-purpose overhead crane is used, whose structure diagram is shown in Figure 9. The research results show that the SRA is an efficient and reliable method for engineering optimization, and it is able to handle discrete optimization problems efficiently.

Figure 9.

Structure diagram of general-purpose overhead crane.

Design variable

The simplified cross-section of box structure shown in Figure 10 is mostly used as girder of bridge crane. The goal of lightweight design is that to ensure the mass of structure minimized under the premise of design requirements. The six design variables used in this article are thickness of top flange plate X ₁(4 ≤ X ₁ ≤ 40), thickness of lower flange plate X ₂(4 ≤ X ₂ ≤ 40), thickness of principal web plate X ₃(4 ≤ X ₃ ≤ 40), thickness of deputy web plate X ₄(4 ≤ X ₄ ≤ 40), height of web plate X ₅(100 ≤ X ₅ ≤ 4000), and internal interval of web plate X ₆(100 ≤ X ₆ ≤ 4000). Influenced by the condition of manufacture, all the design variables are discretized into integral multiple of 1 mm.

Figure 10.

Sectional view of box girder.

Objective function

The span of crane is defined as an invariant parameter, so the objective function can be simplified to design the minimum area of cross-section. Therefore, the objective function can be calculated according to the following formula

f (X) = (X_{1} + X_{2}) \cdot X_{6} + X_{5} \cdot (X_{3} + X_{4})

Constraint

According to the operation characteristics of general bridge crane which is under working condition, the mechanical model of general bridge crane can be simplified as a simply supported beam. And the loads exerted on the crane will be decomposed as shown in Figure 11. $q_{y}$ (inertial load which is caused by the running of crane), $q_{z}$ (gravity load of main girder), $F_{y}$ (initial load of trolley when the crane is braking suddenly), $F_{z}$ (wheel pressure when the trolley is fully loaded) are the loads exerted on the crane, which causes the structure deformation and material yield. Therefore, the strength, rigidity, and stability of structure must meet the requirements of design, for the safety of crane.

Figure 11.

Simplified mechanical model of crane main box beam.

Strength requirement

Calculated value of stress must be smaller than the allowable stress

\sqrt{σ^{2} + σ_{m}^{2} - σ σ_{m} + 3 τ^{2}} \leq [σ]

where $σ$ is the maximum normal stress, and $σ = M_{x} / W_{x} + M_{y} / W_{y}$ ; $M_{x}$ is the maximum bending moment in horizontal direction, $M_{x} = (1 / 8) q_{y} \cdot S^{2} + (1 / 2) F_{y} \cdot S$ ; $M_{y}$ is maximum bending moment in vertical direction, $M_{y} = (1 / 8) q_{z} \cdot S^{2} + (1 / 2) F_{z} \cdot S$ ; $W_{x}$ is section modules in horizontal bending; $W_{y}$ is section modules in vertical bending; $S$ is the span of crane (see Figure 9); $σ_{m}$ is the average value of local stress, $σ_{m} = F_{z} / (290 \times X_{3})$ ; $τ$ is the shear stress, $τ = (F_{z} S_{x}) / (I_{x} \cdot X_{3})$ ; $S_{x}$ is the maximum static moment of gross section; $I_{x}$ is inertia moment of section; and $[σ]$ is allowable stress of steel.

Rigidity requirement

Vertical static deflection of main box beam cannot be larger than the value of allowable deflection

\frac{2 F_{z} \cdot S^{3}}{48 {EI}_{x}} + \frac{5 q_{z} \cdot S^{4}}{384 {EI}_{x}} \leq [Y_{s}]

where ${EI}_{x}$ is bending rigidity of main box beam and $[Y_{s}]$ is known as allowable deflection.

Stability requirement

Box beam has high stiffness; if the height–width ratio of main box beam is smaller than 3, then the global stability of the beam is not needed to check²³

\frac{X_{5}}{X_{6}} \leq 3

Optimization of general bridge crane

The selected general bridge crane can lift 38 ton; its hoisting height and span are 10 and 25.5 m, respectively; and the design variables are defined as $X = [12, 12, 10, 10, 1600, 746]$ , whose geometric meaning is shown in Figure 10. Optimized parameters of SRA for girder structure are defined as following: the maximum number of iteration is MAX_I = 150, and coefficient of step expansion is $c_{1} = c_{2} = c_{3} = 1.68$ . The optimization model is calculated and simulated by SRA for 20 times. The initial parameters before optimization and the optimization results are summarized in Table 5, and Figure 12 shows the iterative curve of objective function.

Table 5.

Comparison computation results before and after optimization.

Name	Value of design variable (unit: mm)						Value of objective function f(X) (unit: mm²)
Name	X ₁	X ₂	X ₃	X ₄	X ₅	X ₆	Value of objective function f(X) (unit: mm²)
Initial parameters	12	12	10	10	1600	746	49,904
Parameters after optimization	11	4	7	4	1576	785	29,111

Figure 12.

Iterative curve of objective function.

By analyzing the data in Table 5 and the iterative curve in Figure 12, the conclusions can be reached: under the condition of the structure constraint, the most reasonable combinations of design variables can be acquired by SRA, and the lightweight design of the metal structure of the crane can be achieved. From 49,904 to 29,111 mm², the weight of the entire structure is decreased to 41.66%, which reduces not only the dead weight of crane nearly 4 ton but also the cost which is used for building construction and equipment transportation. The successful application of SRA in the field of optimization design for crane’s metal structure indicates that the SRA can be widely used in mechanical design and create the considerable economic and social benefits.

Conclusion

The major contribution of this article is to propose an optimization algorithm (SRA) which is a new developed algorithm for global optimization and simulates the phenomenon of light reflection. A new way is designed and adopted to accept the new solution by this algorithm, whether the new solution is the better one or not; by this means, the capacity of global convergence of the algorithm will be improved. Meanwhile, the traditional mathematical theory is used to prove the global convergence of this new algorithm.

Three classical test functions are used to detect the performance of SRA. This article presents some results of numerical experiments which show the new algorithm is more universal, effective, and robust than other common algorithms. The proposed algorithm is used to find the optimized solution of engineering project, which proves that the SRA can solve the practical problems well and have a vast application prospect in engineering.

Footnotes

Academic Editor: Michal Kuciej

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This article is supported by National Nature Science Foundation of China (No. 51275329).

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