# Support Vector Machine-Assisted Importance Sampling for Optimal Reliability Design
## 1 Introduction  
There are various approaches to increase system reliability [1], for example, raising component reliability, increasing redundancy level, exchanging positions of important components, selection of different redundancy methods (e.g., active vs. standby), maintenance, etc. These measures undoubtedly will increase the system budget. Therefore, there needs to be a trade-off between reliability improvement and system budget [2]. No matter which measure we choose to enhance system reliability, the corresponding problem can be abstracted into a similar mathematical model, i.e., either maximizing system reliability under various resource constraints, or minimizing resource requirement under minimum reliability requirements. Of course, different methods of increasing system reliability correspond to different design variables, and the expression of system reliability is also different. Practically, we have to choose the most appropriate of one or several measures to improve system reliability according to the problem at hand. Another problem associated with this is how to procure the optimal decision scheme according to the established mathematical model. There is a common sense that almost all optimal reliability designs (ORDs) are nondeterministic polynomial hard (NP-hard) problems [3]. Canonical methods for optimization, such as the linear/dynamic programming techniques, often fail or reach local optima in solving high-dimensional and complex problems. These are usually referred to as the exact methods. Although much more computational complexity is involved, exact methods are able to provide precise optimal solutions. These approaches are particularly advantageous for small-scale systems. More importantly, their solutions can be used to measurethe performance of newly developed optimization strategies. Hence, there are also some improved versions of this kind of algorithm [4]. The difficulties associated with applying the mathematical programming on large-scale engineering systems have contributed to the development of alternative solutions. One of the alternatives is the heuristic approach. Heuristics do not guarantee precise optimal solutions, but are highly recommended for solving ORD. This is because heuristics achieve reasonable solution quality for large-scale systems within relatively short periods [5]. Interestingly, heuristics usually leverage the sort information obtained by importance measures to guide the search direction. Importance measures, evaluating the relative importance of different components/positions in a system, can be used to prioritize components/positions in a system by quantitatively measuring their impact amount on the system reliability, whose value may not be as useful as their relative ranking [6]. The application of importance measures will guide the convergent direction and limit the randomness in heuristics, so that they can reach the (near) global optimal solution sooner. The role of importance measures in system design has been proved to be crucial. The heuristic approach is more efficient than the exact method, but it still takes a long time if the problem's scale is very large. To further accelerate the convergence speed, researchers have turned their attention to intelligent algorithms (i.e., metaheuristics). These are generally stochastic search methods that mimic the metaphor of natural biological evolution, social behavior of species, and natural/physical phenomena. Metaheuristics are currently considered to be the most promising solutions, because they can find (near) optimal solutions within reasonable CPU time. An old intelligent algorithm is simulated annealing (SA), which is invented on the basis of annealing (slow cooling after heating) of melted metals to crystallize their structures [7]. SA can jump out of the local optimum with a certain probability, and eventually tends to the global optimum. For applications of SA to solve ORD, we refer to [8; 9]. Another well-known intelligent algorithm is the genetic algorithm (GA).