# A Hybrid Optimization Framework with Dynamic Transition Scheme for Large-Scale Portfolio Management
## 3 Methodology
### Dynamic Information Transition
To convert the optimized solutions to investment weights after the search, the transformation rule between the solution \(x\) of algorithms and actual weight \(w\) of assets in a portfolio is described as Equation (11). Besides, a specific example in Figure 5 is given to illustrate the transformation rule:

\[w_{i}=\frac{|x_{i}|}{\sum_{j=1}^{N}\left|x_{j}\right|}, \tag{11}\]

where \(x_{i}\in[-1,1]\) is the value of \(i^{th}\) dimension of solution \(x\), \(w_{i}\) is the weight of \(i^{th}\) asset in a portfolio, and \(N\) is the total number of assets in a portfolio. Furthermore, to convert the problem from multi-objective optimization to single objective optimization, a popular measurement indicator called Sharpe Ratio (\(SR\)) is applied to consider both the return and risk of a portfolio. The definition of \(SR\) is shown below:

\[SR=\frac{E(R_{P})-R_{f}}{\sigma_{p}}, \tag{12}\]

where \(E(R_{P})\) is the annually expected return of a portfolio (see Equation (13)), \(R_{f}\) is the risk-free rate of that market, and \(\sigma_{p}\) is the standard deviation (i.e., risk) of a portfolio (see Equation (14)). \[E(R_{P})=\frac{T}{Y}\sum_{i=1}^{N}w_{i}r_{i}, \tag{13}\]

\[\sigma_{p}=\sqrt{\frac{T}{Y}WMW^{T}}, \tag{14}\]

where \(T\) is the trading days, \(Y\) is the trading years, \(N\) is the number of assets in a portfolio, \(w_{i}\) is the normalized weight of \(i^{th}\) asset, \(r_{i}\) the average daily return of \(i^{th}\) asset during the whole trading period, \(W\) is the weight matrix of a portfolio, and \(M\) is the covariance of the returns among assets in a portfolio. Since a higher \(SR\) value indicates a better portfolio, the fitness function shown in Equation (15) is the reciprocal of \(SR\) for formulating a minimization problem. In addition, each experiment will independently run 30 times, and the number of function evaluations (\(FE\)) in each run is calculated as Equation (16). \[f(x)=\left\{ \begin{array}{ll}\frac{1}{SR},&\text{ if }SR>0\\ 10^{10},&\text{ otherwise }\end{array} \right., \tag{15}\]

\[FE=\left(\left[\frac{N}{100}+0.5\right]+1\right)\times 5\times 10^{4}. \tag{16}\]

#### 4.2.1 Numerical Results

Tables 6-8 show the summary of the results on all three markets of different dimensions in which the mean and standard deviation of \(SR\) of each algorithm is computed. The HYPO framework attains the best performance of all relatively large-scale portfolios except the smallest portfolio of those markets, including 100 assets on SP500, 50 assets on CSI300 and 100 assets on the mixed market. Specifically, the proposed framework can obtain the

Figure 5: The Example of Transformation between the Solution of Algorithms and Weights of Assets in a Portfolio. higher \(SR\) greater than 2 on all portfolios which hold more than 200 assets at the same time. For the largest portfolio on the mixed market, the HYPO approach excels all algorithms at least 0.5 \(SR\), for which less risk will be taken while obtaining the same returns. More importantly, the stability of algorithms decides whether they can provide a confident trading strategy at each run so that the unpredictable investment can be reduced. This clearly demonstrates that the HYPO algorithm attains very small standard deviation values in which consistent investment decisions are performed at each run, especially on dealing with more complicated and large portfolios with the assets from multiple financial markets. This will effectively extend the scalability and flexibility of a portfolio to consider multiple markets for diversifying the risk into more assets. Similar to the CPU time performance on the CEC2019 functions, the HYPO method spends more computing resources on executing the dynamic information transition for adapting to the problems.