# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 2 Preliminaries  
### The Overview of Meta-Heuristic Algorithms  
In terms of nature-inspired views, population-based meta-heuristic algorithms can be roughly categorized into Evolutionary Computation (EC) and Swarm Intelligence (SI). As inspired by the natural selection and biological evolution theory, evolution-based approaches like Genetic Algorithms (GA) [17] maintain superior populations through natural inheritance processes, basically consisting of several classical operations including mutation, crossover, and selection to perform updates and elimination at each generation. Differential Evolution (DE) algorithms [18] update their offspring according to the difference among parent individuals. Intrinsically, the choice of reference individuals to generate new trial vectors directly decides the search direction performing the exploration or exploitation. There are many mutation strategies such as DE/Rand/1, DE/Best/1, DE/Current-to-best/1, DE/rand-to-best/1 and DE/current-to-rand/1 having been proposed in past studies, some of which are trying to further integrate them to a combined strategy. To balance the exploration and exploration during the search process, Refs. [19; 20; 21] present several efficient self-adaptive mechanisms to dynamically select mutation strategies from the strategy pool and also adjust hyper-parameters such as scaling factors and crossover rates by reviewing historical performance.  
By mimicking biological swarm behaviors, SI techniques have attracted much attention in the past few decades. After being affected by the environment in natural evolution, many kinds of social animals have maintained an efficient survival strategy for foraging, mating and enemy defending. The core components in swarm cooperative mechanisms including organization, communication and assignment are considered as the key features to obtain highly competitive advantages of those species in nature, some of which have inspired many powerful SI-based algorithms. Whale Optimization Algorithms (WOA) [22] are based on the two hunting behaviors to attain both exploration and exploitation capabilities while Particle Swarm Optimization (PSO) [23] algorithms imitate the swarm behaviors of fishes or birds to update the solutions through considering group and individual information.  
Despite the effectiveness of meta-heuristic approaches having been successfully demonstrated in many practical problems, those algorithms following a single search pattern are limited to specific kinds of tasks, and then cannot flexibly adapt to more challenging problems. Accordingly, in recent years, some studies have tried to hybridize different meta-heuristics. The authors of [24] try to integrate tabu search and simulated annealing into the Ant Colony System (ACS) to improve the local search ability, while [25] presents an Electrical Harmony-based Hybrid Meta-heuristic (EHHM) algorithm to optimize feature selection on machine learning tasks. In addition, Ref. [26] dynamically assigns agents to the bat algorithm and krill herd algorithm by introducing a logistic function to calculate the selection probability at each iteration in terms of the performance rank of agents whereas [27] generates a new trial vector from both the cuckoo search and PSO algorithm, and a control factor is introduced to balance the weights of the two methods. However, although the proposed frameworks are easy to implement, the continuity of search of each meta-heuristic mechanism is broken up, for which the search advantages of each meta-heuristic cannot be demonstrated through iterations. Meanwhile, there are limited studies applying hybridized meta-heuristics to large-scale problems, especially in handling the problem with dependent decision variables.