| The quantum mechanics is one of the most exciting theories in the field of physics in the last century; it provides new theories and ideas for the continued innovation and development of information science. Quantum computation combines quantum mechanics with information science; and it has the characteristics of high parallelism, exponential storage capacity and acceleration effect for classical heuristic algorithm, which make it become the focus of various scientific research people. Meanwhile, as a hot research topic, the basic idea of Evolutionary computation is "survival of the fittest", and then it makes great achievement in solving the complicated optimization problems.According to the aforementioned research, some scholars had begun to propose a new theoretical framework-quantum inspired evolutionary algorithm (QEA) which combined quantum computation and evolutionary computation. The conventional QEA uses a new representation, called a Q-bit, for the probabilistic representation, and Q-bit representation can improve the diversity of population compared with other representations; by observing the statues of Q-bit, it can be converted into binary representation; quantum rotation gate is used as the update operator which can drive the search direction of individual to the optimal area instead of traditional mutation operator, crossover operator, etc.. These features make QEA has stong global searching ability, a fast convergence speed and good robustness. Exsiting research results have shown that the performance of QEA is superior to that of the conventional evolutionalry algorithm. At the same time, considering the advantages of QEA, it has also been widely used in several areas such as engineering construction, optimization management, etc., and one of the most successful applications is combinatorial optimization. However, the types of combinatorial optimization problems which QEA can solve are limited. Existing research mainly focuses on knapsack problems, TSP problems, and job shop scheduling problems. So it is necessary to extend QEA to other types of combinatorial optimization problems, which can enrich the research content of QEA. Meanwhile, research on quantum evolutionary theory and its corresponding algorithm is of great significance in theory and practice.Based on synthetically analysis of all kinds of researched results, this paper proposes two new QEAs in order to solve single-objective optimization problems (SOPs) and multi-objective optimization problems (MOPs) respectively. The new QEAs propose a novel real-coded method which includes a new trigonometric quantum individual (TQI). The TQI consists of two parts:the real numbers part and the corresponding probability amplitudes part. These two parts have gone through a process of evolution in two different ways, and then these two ways react upon each other in order to improve the diversity of population and efficiency of the new QEAs. Subsequently, the improved strategies for optimization are proposed in order to build the new real-coded single-objective quantum evolutionary algorithm (SQEA) and real-coded multi-objective quantum evolutionary algorithm (MQEA) respectively.Finally, these new QEAs will be applied into portfolio optimization problems. Considering the features of portfolio optimization problems, there are lots of fuzzy and uncertainty factors which affect investors' decision. These factors are mainly on the forms of subjective uncertainty, and they can not reflected by traditional mathematical tools. Combineing the uncertain programming theory which is proposed by Professor Liu Baoding, this paper gives two new portfolio optimization models. The first model is single objective fuzzy portfolio optimization model, and the other is multi-objective uncertainty portfolio optimization model, and then the new QEAs will be used to solve the corresponding portfolio optimization model.The main place of innovation of this paper lies in the following four respects:Firstly, a new real-coded single objcet quantum evolutionary algorithm based on trigonometric functions transform is proposed. It uses the prior information which includes optimal solution and gradient information of objective function, and defines a linear crossover operator with acceleration mechanism in order to implement the evolution operation. These new evolution strategies can reduce the possibility of sinking into local optima, and improve the accuracy of solution.Secondely, the new single-objective portfolio optimization models with fuzzy returns are proposed. These portfolio optimization models use fuzzy entropy and its corresponding theory to build a new risk measurement. Subsequentyly, the new real-coded SQEA is mixed with fuzzy simulation method to solve these fuzzy single-objective portfolio optimization models;Thirdly, a new real-coded multi-objective quantum evolutionary algorithm is constructed. Based on NSGA-II, this new algorithm defines self-adaptive clone mechanism and contains a dynamic population set in order to increase the selection pressure for the excellent individuals;Fourthly, on the basis of uncertain measure and following Markowitz's idea, the security returns are assumed to be uncertain variables that are neither random nor fuzzy, and then four uncertain multi-objective portfolio optimization models are presented;. In these models, the investment return is quantified by the expected value of a portfolio, and the risk is quantified by variance, entropy, and quadratic entropy in uncertainty theory respectively. Lastly, the new real-coded MQEA is used to solve these uncertain multi-objective portfolio optimization models.
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