## 3 Proposed MPUSMs-IMFEAO for Personalized Recommendation
### Construction of Multidimensional Preference User Surrogate Models and Their Derivatives
#### 3.1.2 Training and Usage of MPUSMs and Their Derivatives  
The middle part of Figure 2 and the training part of Algorithm 2 clearly depict the construction process of MPUSMs, partial-MPUSMs, and \(K\) probability models. As shown on the right in Figure 2, the proposed interactive multifactorial evolutionary optimization algorithm (IMFEAO) performs the MFEA evolution process for certain generations to obtain some relatively superior candidates and then eventually provides a Top-N list for interaction with the user. As the evolving individuals of population may not be real items, social knowledge derived from existing rated items cannot be used for this process. Hence, the evolving individuals are only evaluated with the partial-MPUSMs. After the evolution loop ends, the pre-recommended list will be filled with superior individuals and matched to real items, and the corresponding social knowledge could be added to provide more adequate evaluation information. To this end, two types of surrogates are designed for separately evaluating evolving individuals (or virtual items) and pre-recommended items.  
Firstly, we train \(K\) dual-RBM models with \(K\) clustered data. Then, partial-MPUSMs consist of only \(K\) dual-RBMs to approximate the user's \(K\) dimensional preferences. Their formula for assessing individuals belonging to the \(k\)-th dimension of preferences is shown in (7) and is only used to evaluate the individuals \(x^{\prime}\) in the MFEA evolutionary process. For evaluating the items \(x\) from the pre-recommended list, social knowledge is used together with \(K\) Dual-RBM models to form MPUSMs, as shown in (8). Finally, the probability models of MPUSMs derived from each positive RBM are calculated with (5) and an initial population that better matches the user preferences could be generated by (6). In these two types of models, the components \(\max(E_{\theta_{p}})\), \(\min\) (\(E_{\theta_{p}}\)), \(\max(E_{\theta_{n}})\) and \(\min(E_{\theta_{n}})\) of the \(preference(x)\) as defined in (4) are calculated based on the samples of each cluster, which is quite different from that in our previous work [13]:  
\[\hat{f}_{k}^{p}(x^{{}^{\prime}})=\sigma(preference(x^{{}^{\prime}})) \tag{7}\]  
\[\hat{f}_{k}^{c}(x)=\sigma(a*social(x)+preference(x)) \tag{8}\]  
During the MFEA evolution process, (7) is used to directly evaluate the individuals with skill factor \(k\). For the pre-recommended list, superior individuals are drawn from every skill group in the population, but their cluster features may have changed significantly as a result of evolution. To facilitate a reliable evaluation, these individuals should be converted to corresponding real items before finding the most appropriate preference surrogates for them to evaluate. Here, the Euclidean distance is used to find the real items for individuals in the set of items to be searched, as well as the most appropriate surrogate in the MPUSMs for the items in the pre-recommended list, for which the process is shown in (9):  
\[r(x_{i})=\hat{f}_{k}^{c}(x_{i}) \tag{9}\]  
where \(r(x_{i})\) is the rating of the \(i\)-th item \(x\); \(k=\arg\min D_{j}\) denotes that the \(i\)-th item is best suited to be evaluated by the \(k\)-th surrogate model in the MPUSMs, of which \(D_{j}=||x_{i}-c_{k}||\), \((j=1,2,\cdots K)\) is the Euclidean distance between the \(i\)-th item and the \(k\)-th clustering center.