This paper introduces the idea of fractional programming in a portfolio model and builds four fractional programming portfolio optimization models based on various risk measures in a fuzzy environment, with the aim of addressing the complexity of historical data of real securities markets and the fundamental form of existing portfolio models. The four models build on the mean–variance(MV) model by adding a number of useful limitations, such as restrictions on short selling, proportionate investment boundary restrictions, and portfolio cardinality constraints, to better suit the requirements of genuine currency-related stock markets. For the portfolio optimization problem, which is a 0-1 mixed-integer fractional programming problem, a dual-loop hybrid heuristic algorithm is proposed. This algorithm incorporates the constraints of the model into the algorithm, thereby avoiding the drawbacks of the penalty function method. The empirical analysis part uses historical data to simulate investments and compare portfolio strategies under various risk metrics in order to show how well the models perform. The numerical results of the four models are also compared, showing that the models are suitable for different investors and that they are consistent with actual stock market conditions.

We propose an efficient outer space branch-and-bound algorithm for minimizing linear multiplicative problems (LMP). First, by introducing auxiliary variables, LMP is transformed into an equivalent problem (ELMP), where the number of auxiliary variables is equal to the number of linear functions. Subsequently, based on the properties of exponential and logarithmic functions, further equivalent transformation of ELMP is performed. Next, a novel linear relaxation technique is used to obtain the linear relaxation problem, which provides a reliable lower bound for the global optimal value of LMP. Once more, branching operation takes place in the outer space of the linear function while embedding compression technique to remove infeasible regions to the maximum extent possible, which significantly reduces the computational cost. Therefore, an outer space branch-and-bound algorithm is proposed. In addition, we conduct convergence analysis and complexity proof for the algorithm. Finally, the computational performance of the algorithm is demonstrated based on the experimental results obtained by testing a series of problems.

The filled function method is a deterministic algorithm for finding a global minimizer of global optimization problems, and its effectiveness is closely related to the form of the constructed filled function. Currently, the filled functions mainly have three drawbacks in form, namely, parameter adjustment and control (if any), inclusion of exponential or logarithmic functions, and properties that are discontinuous and non-differentiable. In order to overcome these limitations, this paper proposed a parameter-free filled function that does not include exponential or logarithmic functions and is continuous and differentiable. Based on the new filled function, a filled function method for solving unconstrained global optimization problems was designed. The algorithm selected points in the feasible domain that were far from the global minimum point as initial points, and improved the setting of the step size in the stage of minimizing the filled function to enhance the algorithm's global optimization capability. In addition, tests were conducted on 14 benchmark functions and compared with existing filled function algorithms. The numerical experimental results showed that the new algorithm proposed in this paper was feasible and effective.