Projectile-target search algorithm: a stochastic metaheuristic optimization technique

ABSTRACT


INTRODUCTION
Optimization techniques, especially stochastic nature-inspired metaheuristic algorithms have become the important and popular tools to deal with complex high dimensional global optimization problems in many real-life applications. The global optimization problems can be multimodal with a huge number of local optima and non-differentiable which cannot be solved by using traditional numerical optimization methods [1]. Therefore, many metaheuristic optimization algorithms have been proposed to alleviate the problems.
The nature of a stochastic metaheuristic optimization algorithm is employing the random-search mechanism to visit different parts of the search space and then approaches as close as possible to global optimum point [2,3]. Important problem of the metaheuristic algorithms is how to increase the probability to overcome the local optima and find the better global optimum value.
In this paper, a new metaheuristhic algorithm is introduced. The proposed algorithm is based on kinematics of projectile motion. The projectile is launched from a point at the ground level with a given velocity, moved in various directions under a uniform gravity, and landed on a target at a surface. However, the proposed projectile-target search (PTS) algorithm does not emphasize on the projectile trajectory that launched from an angle. The algorithm tends to utilize the tracking of the envelope of projectile trajectory which encloses all possible points in the search space. The envelope of projectile trajectory could reach points in the search space which are out of reach from any projectiles moving with any initial points and velocities [4]. The information of the trajectory could be usefull for a projectile to hit a target from its starting point. In the proposed PTS algorithm, the beneficial property of the envelope of projectile trajectory is applied to find a target on a linear shape surface. Futhermore, the PTS algorithm has two main control parameters in order to accelerate convergence and searching processes and hence, it can find the better global optimum value faster than any other optimization algorithms. In order to examine the general performance of the proposed algorithm, it is tested on six benchmarks and their shifted functions. The performance is compared to other algorithms' results which have been reported in [5][6][7][8][9][10].

PROJECTILE MODEL
The projectile is defined to be launched from a ground level (h = 0), with an initial velocity v, and at an angle of inclination θ measured with respect to the horizontal as shown in Figure 1. The path function of the projectile (y) as a function of horizontal distance (x) is specified as follows: where: g = 9.81 m/s 2 The size and shape of the projectile trajectories vary according to the launch angles at an initial velocity v above the horizontal, as seen in Figure 2. These trajectories have an envelope of projectile trajectory. The envelope of projectile trajectory (φ) is a path that encloses and intersects all possible projectile paths to find its target onto a hill. The shape of the hill surface is defined as the impact function ψ. The equation for the envelope of projectile trajectory [4,11] is determined by For a continuous impact function ψ(x) on 0 ≤ x <  with ψ (0) = 0, the surface of ψ and the path of φ have exactly one point of intersection. It is noted that there exists a unique target for which ψ(x) = φ(x).
The main goal of the proposed PTS algorithm is to minimize the difference between ψ and φ in order to ensure a projectile reach the target precisely. A function γ(x) is then implemented to search the target on the impact surface and it is given by

PROJECTILE-TARGET SEARCH ALGORITHM
The initial parameters of the PTS algorithm are the population size N, the number of variables D, the lower bound xmin, the upper bound xmax, and the maximum iteration Imax. The lower and upper bounds of variables are expressed by The initial projectile population as candidate solutions is randomly generated by assigning random values (rand j ) within [0, 1] to each boundary as follows: The target population is randomly created by perturbing a randomly selected projectile with the difference of the two other randomly selected projectiles. The target population is generated by where the indices a 1 , a 2 , and a 3  {1,2,…,N} are randomly chosen integers and must be different from each other and all are also different from the base index.
Evaluating the fitness values of the projectiles and the targets are carried out by using (7) and (8) as follows: The best projectile x bestj (I) and its best value f best (I) are then selected by comparing the fitness values of each x i,j (I) and t i,j (I) , as follows: where I = 1,2,…, I max which denotes the subsequent generation created for each iteration.
The main process of PTS optimization is iterating the projectile in order to reach its best fitness value. The projectile search model for converging towards to the target is as follows: ) (12) end end end In this PSA optimization, a linear shaped hill with a slope of m is used as the impact function. The impact function is As the impact function has been specified, the function γ(x) could be defined here by and The distance between projectile and target at the current iteration is calculated as follows: , There are 2 parameters for controlling the distance (d), those are r and s, where (r, s) > 0. Other control parameter for (11) and (12) is defined by The new target population (t k,j (I+1) ) is also created in here. Finally, the best projectile for the next iteration is The processes are repeated until f best (I+1) meets its desired accuracy level (ɛ) or the iteration has reached I max and results in x bestj (I+1) as the satisfied projectile landing onto the target.

RESULTS AND ANALYSIS
There are six benchmarks and their shifted functions used to evaluate performance of the proposed PTS algorithm. The mathematical formulation of the benchmark functions are given in Table 1. In order to verify the performance of the proposed PTS algorithm, it is compared to other algorithms which have been reported in [5][6][7][8][9][10]. To carry out the comparison of algorithm performance, the approach using is to compare the accuracies for a fixed number of iterations. In the experiment, 100 dimensions (D = 100), 30 search agents (N = 30), and 1000 iterations (I max = 1000) are implemented for each algorithm. The statistical results after 30 independent experiments are evaluated. The mean and standard deviation (SD) values of the best solutions from the last iterations are put as the metrics to assess the performance of algorithms.  The parameters needed to drive the PTS algorithm are: gravitational acceleration (g), initial velocity of projectile (v), slope of target's surface (m), q, r, and s. Gravitational acceleration is a constant of g = 9.81. The initial velocity of projectile, the slope of the target's surface, and q give important contributions to the convergence processes, while r and s are controlled the searching process of the PTS algorithm. To facilitate the analysis, selected functions are used as experiment objects. Figure 3 describes the influence of v on convergence process of the Ackley function under fixed m condition, while Figure 4 is its performance with variations of m at v fixed condition. As seen in Figure 3, v has a strong effect to convergence speed of the PTS algorithm. The convergence process runs relative slowly when v has large values. On the other hand, m has slightly effect to the convergence speed. However, the final fitness values of the searching process vary significantly with both v and m as seen in Table 2.
Other control parameter of the PTS algorithm is q, as shown in Figure 5. Increasing the value of q will speed up the convergence process of the PTS algorithm. This parameter also gives very high contribution to the best final fitness value which could be reached by the algorithm. Table 3 presents the best final fitness value of Sphere function according to its q values. However, not all problems behave the same way as Ackley and Sphere functions. For example, Schwefel function will suffer from stagnation condition and lose its convergence to the global optimum if q is set to a high value, as shown in Figure 6. The parameter q speeds up convergence process for all test functions, but Schwefel function needs relative small speed in order to go to the global optimum path.    It has been mentioned that parameters v, m, and q influence to the convergence speed and the closest optimum value that could be reached by the PTS searching processes. However, the convergence towards global optimum will be failed if the projectiles are trapped into local optima in the search space. Furthermore, the projectiles generated by iteration processes are also dependent to distance control parameters: r and s. These parameters are very important in order to enhance probability of the new projectiles to avoid from local optima.
Comparisons of algorithms' performances are summarized in Table 4 and Table 5. The obtain results for benchmarks F 1 -F 6 are shown in Table 4. As summarized in Table 4, PTS outperforms hybrid firefly algorithm (HFA), velocity-based artificial bee colony algorithm (VABC), alternative differential evolution algorithm (ADE) and opposition-based magnetic optimization algorithm (FMOA), excluding modified monkey algorithm (MMA), on F 1 . Once again, PTS has the best performance on F 2 and F 3 . Investigation on F 4 shows that ADE is the best, but PTS and VABC are almost as good. On function F 5 , PTS is the best, but MMA, FA, and ADE perform almost as well as PTS. The comparative results also demonstrate that PTS, firefly algorithm (FA), and ADE have the best performances on F 6. Experiments on F 7 and F 12 show that PTS could find their best global optima points as seen in Table 5. For shifted functions F 7 -F 12 , PTS performs much better than moth-flame optimization algorithm (MFO) [6].

CONCLUSION
From experiments on selected well-known benchmarks and their shifted functions, it has been demonstrated that PTS algorithm is an effective optimization algorithm to deal with high dimensions global optimization problems. It is also proven to be a very competitive algorithm compared to other well-known metaheuristic algorithms  ISSN: 2088-8708