Multiplayer Battle Game-Inspired Optimizer for Complex Optimization Problems

Battle royale games[36, 37], such as PUBG, Infinite Law, Warzone, and Call of Duty, typically integrate elements from both survival games, focusing on exploration and equipment collection, and competitive gaming, where the objective is to eliminate opponents until only one person remains. While the rules of these games may not be identical, there are discernible common patterns that characterize their gameplay. For example, a prevalent game mechanic, known as “safe zones”, is widely employed to compel players into more confined areas, fostering encounters and battles among them. Over time, the designated safe area progressively diminishes, and players situated outside this zone face the risk of injury or elimination until only the last individual or team remains as the victor. We acknowledge the crucial role played by the“safe zone” mechanic in these games, significantly shaping player movement and battle strategies. Recognizing the widespread influence and impact of this mechanic, we have intentionally directed our simulation efforts toward the generation and utilization of the safe zone.

Another factor that affects the outcome of a match is the players themselves, as different players tend to choose different strategies when faced with even the same situation. For example, cautious players typically opt to engage in battle when they have a higher likelihood of success and avoid confrontation when the odds are less favorable. On the other hand, aggressive players consistently lean towards engaging in battle upon encountering a new opponent. Furthermore, the quality of equipment significantly influences players’ decision-making. Generally, superior equipment correlates with an elevated probability of winning, instilling greater confidence in players when engaging in battle. To reflect the importance of players, when designing the new algorithm, we especially introduce several different strategies to simulate the behaviors of players under different psychological conditions. Our intention is for these strategies to effectively preserve individual diversity.

Since our primary objective is the development of an effective algorithm rather than blindly pursuing simulation games, we are open to making trade-offs on the rules. Here, we consider each player as a candidate solution within the search space and map their actions to coordinate updates in the solution space. As players are gradually eliminated, the number of individuals decreases in the game. However, to maintain a constant population size, we have chosen not to adopt this rule. Furthermore, we overlook equipment differences between players and instead treat each individual equally. This implies adopting different movement and battle strategies with equal probability for all individuals. In the subsequent subsection, we provide a comprehensive overview of the proposed MBGO algorithm.

The newly proposed MBGO algorithm is a population-based heuristic algorithm that iteratively refines the population by simulating common mechanisms found in battle games and behaviours taken by players under various psychological states. Similar to typical EC algorithms, the MBGO first randomly generates multiple individuals to form an initial population. Then, the Euclidean distance between the best individual and the worst individual in the current population is employed to define the “safe zone”, effectively partitioning the entire search space into the safe area and the non-safe area. Individuals located in different areas use different movement strategies to generate new individuals. Importantly, individuals choose to update their current positions only if the new individual represents an improvement; otherwise, the current individual remains unchanged. After all individuals have gone through this movement phase, they transition into the battle phase. During this stage, each individual randomly chooses another individual as an opponent and determines the battle strategy to generate a new individual. The proposed MBGO still maintains the use of elite selection in the battle phase, accepting the new individual only if it is superior; otherwise, the individual “loses” the battle and is reborn in the same place. Subsequently, individuals re-enter the movement and battle phases again, and this cycle continues iteratively until the termination condition is met. Finally, Fig. 1 provides a simple optimization process of the proposed MBGO algorithm, which consists of initialization, movement phase, and battle phase.

Before delving into the detailed introduction of three core operators, we first define some symbols that will be used in the following to avoid ambiguity. Here, we take the minimization problem as an example, assuming that the optimization problem contains $D$ variables, that is, a $D$-dimensional problem. Thus, for any individual $\mathbf{x}_{i}$ (i.e., candidate solution), it comprises $D$ variables, represented as $\mathbf{x}_{i}=(x_{i}^{1},x_{i}^{2},...,x_{i}^{D})$. For any dimension $k(k\in\{1,2,...,D\})$, its upper bound and lower bound are defined as $UB^{k}$ and $LB^{k}$ respectively. Additionally, $\mathbf{x}_{best}$ and $\mathbf{x}_{worst}$ represent the best individual and the worst individual in the current population, and $Fit()$ is the evaluation function and returns the fitness value of the passed-in individual.

Initialization is executed only once to generate the initial population. Among various initialization methods, we employ the most common approach: random initialization. For any individual $\mathbf{x}_{i}$, Eq.(1) uses the uniform distribution to generate the initial value in the $k$-th dimension:

Suppose the population size, a hyperparameter that needs to be defined in advance, is set to $N$, the initial population can be represented in the form of a matrix, as follows:

Movement Phase is mainly responsible for the exploitation capability and uses the concept of the “safe zone” to guide individuals to converge toward potential areas. Fig. 1 illustrates this mechanism used in two real battle royale games. Although the rules in each game may differ, the general idea is to select a part of the area as a safe area and continuously shrink the subsequent safe prefetching to encourage players to gather together as the game progresses. To simulate this mechanism, a crucial consideration is how to determine the boundaries of the safe area.

As an attempt, we leverage the current optimal individual, $\mathbf{x}_{best}$, and designate it as the center of the safe area. Regarding the determination of the radius covered by the safe area, we use the Euclidean distance between the best and worst individuals in the current population as the baseline. It’s important to note that the baseline distance may not necessarily be equal to the distance between the two furthest individuals in the current population. To increase randomness, this baseline distance is then multiplied by a random factor between 0.8 and 1.2, and the resulting value is utilized as the final safety radius, as shown in Eq.(2). Thus, the entire search space dynamically divides into two distinct areas as the population converges:

For individuals within the safe area, where the distance from the current best individual is less than the safety radius, it suggests that the individual is likely situated in a potential area. Here, we introduce Eq.(3) to use the local information of the current optimal individual for the search. However, for individuals outside the safe area, meaning they are currently distant from the potential area, Eq.(4) is employed to accelerate movement toward the current optimal area. Specifically, for each dimension $k$, one of the two methods is selected for update with equal probability. The last point that needs to be mentioned is that the elite selection is employed to update the population. This implies that an individual will only be updated if the fitness of the new individual, $\mathbf{x}_{new}$, surpasses that of the original individual, $\mathbf{x}_{i}$, otherwise, the current state will be maintained:

Battle Phase is mainly responsible for the exploration capability and simulates diverse battle behaviors that arise when players encounter each other randomly in the game. While players may employ different strategies when facing opponents of varying skill levels, common objectives include avoiding opponent attacks to minimize damage received and inflicting maximum damage on the opponent to secure victory. To simulate the diverse behaviors of players under various psychological conditions, we have simplified the observations from real games and imposed some idealized restrictions. Here, we assume a player will not confront multiple opponents simultaneously and will, instead, randomly designate another individual as the opponent in each battle phase. Additionally, fitness information is employed to gauge the strength of an individual, aligning with factors such as player level.

To simulate the player’s behavior, we have designed two different strategies as outlined in Eqs.(5) and (6) to update the population. For the sake of simplicity, we have standardized the rules for selecting battle strategies for all individuals. Specifically, if the individual encounters a stronger opponent (i.e., an opponent with higher fitness), adopt Eq.(5); otherwise, adopt Eq.(6) to generate the new individual. Similarly, the elite selection is also applied during the battle phase:

The introduction of these core components of the proposed MBGO algorithm has ended. It should be noted that the selection operation is designed into the movement phase and the battle phase, and they both involve generation updates. Algorithm 1 finally gives a detailed overview of the proposed MBGO.

The benchmark functions are carefully designed to compare the performance of different algorithms, with the flexibility to set different dimensions based on specific requirements. In this context, we selected widely used dimensions, specifically 10 dimensions (10-D) and 30-D, for the CEC2017 functions. Additionally, for the CEC2020 functions, we opted for higher dimensions, namely 50-D and 100-D.

The parameter configurations for the involved algorithms are as follows. The common parameters across the nine algorithms are uniformly set as follows: the population size is set to 100, and the maximum number of fitness evaluations is set to $1000\times D$, where $D$ represents the dimensionality of the optimization problem. The parameters unique to each algorithm are aligned with mainstream settings, and their specific settings are summarized in Tab. I.

To investigate whether there are significant differences between the proposed algorithm and other algorithms, we conducted the U-test and Holm’s multiple comparisons at the termination of the algorithm, that is, at the maximum number of fitness evaluations. Additionally, we present the average and standard deviation of the optimal solution found in the final 30 trial runs for all algorithms. Tab. III and Tab. IV summarize the results of 10-D and 30-D on CEC2017 test suites, and Tab. V and Tab. VI summarize the results of 50-D, and 100-D on CEC2020 test suites, respectively. Besides, we present the average convergence curves of all algorithms in both 50-D and 100-D for all CEC2020 functions, as illustrated in Figs. 3 and 4.

Not limited to human-designed benchmark functions, we employed 10 real industrial design problems to evaluate the performance of the proposed MBGO algorithm. These real-world problems are frequently characterized by robust constraints, posing challenges that enhance the complexity of the search for optimal solutions. Since none of the algorithms inherently addresses constraints, here a unified penalty function is employed in this context, which serves to impose an infinite penalty on the fitness of individuals that violate constraints, reflecting the most unfavorable outcomes. The details of the ten engineering design problems are outlined in Tab. II.

The parameter configuration for all algorithms remains identical to the previous experiment, with the sole exception being the maximum number of fitness evaluations. Since the dimensions of these real-world problems vary, we standardized the maximum number of fitness evaluations for all problems to 20,000. To analyze the performance between algorithms, we once again employ the U-test and Holm’s multiple comparisons to determine whether there is a significant difference at the end of all algorithms. Tab. VII summarizes the average and standard deviation of optimal solutions discovered by these algorithms, along with the outcomes of statistical testing.

For the sake of simplicity, we continue to use the symbols introduced in Sec. III and define the maximum number of fitness evaluations as $T$. Therefore, the complexity of the core operations of the MBGO algorithm is as follows:

• •

  initialization: $O(N\times D)$.

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  the selection for current best individual: $O(N)$.

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  the selection for current worst individual: $O(N)$.

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  the movement phase: $O(N\times D)$.

• •

  the battle phase: $O(N\times D)$.

In summary, the computational complexity of the MBGO algorithm can be calculated by Eq.(8).

The proposed MBGO primarily consists of three operations: initialization, the movement phase, and the battle phase. Excluding the initialization operation applied in all comparison algorithms, the remaining two operations significantly contribute to performance improvement. This fundamental distinction is the key reason for the superior performance of the proposed MBGO algorithm compared to others. Here, we conduct an in-depth analysis of the two operations and enumerate the benefits they bring.

The movement phase uses the Euclidean distance between the best individual and the worst individual to divide the entire search space into potential (safe) areas and non-potential (unsafe) areas. Individuals in different areas employ varied strategies to update their current positions, effectively reducing the risk of rapid convergence that may confine the population to local areas, thereby preventing premature maturation. Moreover, as the population converges, individuals gradually converge toward the global area, resulting in the gradual reduction of the distance between the best individuals and the worst individuals. This effectively simulates the widespread adoption of “safe zones” in the game, which gradually shrinks to a smaller area. Furthermore, the optimal individual of each generation is designated as the center of the safe area, offering effective guidance to all other individuals to reduce random searches and accelerate overall processes. We thus can say that the movement phase guarantees the convergence speed of the proposed MBGO algorithm.

The battle phase simulates various player behaviors in response to opponents, aiming to increase the diversity of the population. Each individual randomly selects an opponent and employs different update strategies based on the opponent’s fitness, fostering information exchange among individuals. Furthermore, the adversaries individuals encounter in different generations undergo constant changes, fundamentally heightening the diversity of the population and facilitating the generation of diverse offspring individuals. Additionally, we intentionally use vector information from better individuals to worse individuals to encourage the exploration of unknown potential areas and prevent premature convergence. This approach is inspired by the concept of simulated annealing. However, it is essential to clarify that the pursuit of diversity does not imply the unconditional acceptance of inferior individuals. In this context, the elite selection is implemented to ensure that population diversity is maintained without weakening the population.

Excessive emphasis on either convergence speed or diversity can pose challenges for the population to converge to the global optimum. Striking the right balance between these factors is crucial for the effective optimization of the algorithm. The collaborative synergy between the two stages enables the proposed MBGO algorithm to achieve a harmonious balance between exploration and exploitation, distinguishing itself prominently among all competing algorithms when facing various optimization problems. In addition, the rational use of individual distribution information and fitness landscape information also helps the MBGO algorithm select appropriate search strategies, ensuring sustained high search performance.

The results of statistical tests demonstrate that the proposed MBGO algorithm consistently outperforms others, showcasing superior performance across both manually designed functions and real engineering problems. In the case of the CEC2017 test suite, the MBGO algorithm demonstrates superiority over all but HBA in low-dimensional problems. However, when considering the overall performance, the MBGO algorithm stands out as excellent. Notably, it exhibits a growing advantage as dimensionality increases, leading to the achievement of the best performance on almost all problems. Moreover, in the evaluation across all functions of CEC2020, the MBGO algorithm overwhelmingly outperforms all comparison algorithms. The convergence curves depicted in Fig. 3 and Fig. 4 further show that the proposed MBGO algorithm not only achieves high convergence accuracy but also exhibits the characteristic of rapid convergence speed. Through the analysis and summary of all function characteristics, it becomes evident that the MBGO algorithm performs better on complex problems with high dimensions and multiple peaks.

Despite not meticulously designing the constraint processing module and opting for the widely used penalty function, experimental results indicate that the proposed MBGO algorithm consistently discovers satisfactory feasible solutions when compared with other classic EC algorithms. These strong constraints, result in a substantial number of infeasible solutions and complicate the effectiveness of the search process. Overall, despite the challenges posed by strong constraints, the MBGO algorithm remains competitive and ranks among the best. This observation underscores the potential applicability of the MBGO algorithm to constrained optimization problems.

While we have highlighted the strengths of the MBGO algorithm, it is undeniable that there is still considerable room for improvement. In this context, we present some open topics with the hope of sparking new inspiration for those who follow in our footsteps.

1. The interactive switching between the two stages is a key factor in the success of the MBGO algorithm. Exploring how to make informed and context-specific switches based on the characteristics of the optimization problem is a worthwhile research topic. Instead of treating the two stages equally, tailoring the switching mechanism to the specific nuances of the problem could enhance the MBGO algorithm’s adaptability and performance.

2. While elite selection ensures that the population converges toward the optimal area, it introduces a new challenge: the direct discarding of inferior offspring individuals without any utilization. This operation can lead to a waste of evaluation resources, particularly for expensive problems. Thus, a valuable research topic involves the reuse of these discarded individuals, aiming to further enhance search performance.

3. Real games often feature intricate and engaging gameplay mechanics. In this context, we simplify these complex mechanisms to design optimized game mechanics. Thus, introducing these diverse strategies into subsequent improved versions could be an interesting research topic. Additionally, modeling the behavior of game players, aiming to replicate battle behavior more realistically, is a subject worthy of further study.

4. The hyperparameters of optimization algorithms frequently exert a significant influence on performance. Here, we simplify the process by unifying and fixing these parameters. However, it’s important to note that this strategy may constrain performance to a certain extent, especially when applied to problems with diverse characteristics. Thus, exploring adaptive or problem-specific tuning of hyperparameters could be a valuable avenue for improvement in future iterations of the algorithm.

Certainly, the scope of research is not limited to the topics listed. There are numerous areas worthy of study, such as extending the MBGO algorithm to different optimization problems, including multi-objective optimization, dynamic optimization, and multi-task optimization, among others. Exploring these avenues can contribute to the MBGO algorithm’s versatility and applicability across a broader range of problem domains.