# An ADMM Based Parallel Approach for Fund of Fund Construction
## 1 Introduction  
Fund of funds (FOF) has become a hot topic during the past several years. As a mutual fund scheme, FOF uses other funds as investment targets and achieve the purpose of indirectly holding securities such as stocks and bonds [1]. The first target of FOF construction is to obtain optimal portfolio among different funds based on the tradeoff between return and risk. To meet this goal, one of the main research activities for the past few years has been FOF modeling. It is worth mentioning that many existing FOF optimization models are based on the Mean-Variance (MV) framework. In 1952, [PERSON] introduced the seminal work, the MV model [2], which is regarded as the fundamental finding of modern portfolio theory (MPT). Since then, it has become the most essential theory to study the FOF asset allocation problem. Despite huge attention, the MV model still has some obvious shortcomings. One drawback which many studies emphasize is that the optimal portfolio can be extremely sensitive to input parameters [3; 4]. Another drawback of the MV model is that this approach sometimes provides over-concentrated portfolios, which would probably cause huge losses if a financial crisis were to happen. Moreover, the MV model ignores the subjective views of investors. These shortcomings limit the MV models' application in the FOF construction. As a result, it is not surprising that many research efforts were placed on extending the MV model [5; 6; 7; 8; 9; 10; 11]. From a practical perspective, there exist many trading restrictions in the real-world financial market. Transaction cost is one of which that should be paid attention. In fact, transaction costs in the financial market is usually treated as non-linear functions, which makes the FOF optimization model a nonlinear programming problem. The interior point method (IPM) [12; 13], sequence quadratic programming (SQP) method [14], parallel variable distribution (PVD) [15] and line search algorithm [16] are all popular methods for nonlinear programming problems. However, the aforementioned methods can be inefficient since they does not explore the structure of the objective function. For example, the generalSQP procedure uses the derivative vector and the Hessian matrix of the Lagrangian function to construct quadratic approximation at each iteration [17]. The Hessian matrix of FOF optimization models may not necessarily be positive semi-definite, so an approximation of the Hessian matrix is needed. Constructing a Hessian approximation by quasi-Newton methods may be poorly performed and time-consuming. The IPM maintains the feasibility during iterations but it is a centralized and computationally expensive method. Moreover, it is hard to parallelize. The PVD method is suitable for parallelization but needs to solve a difficult convex subproblem when applied to FOF optimization model, which makes it hard to be extended to large-scale problems. Most of algorithms that developed in recent years have concentrated on structured nonlinear programming, which aims to characterize a range of nonlinear problems. In 1977, [PERSON] and [PERSON] [18] proposed the alternating direction method of multiplies (ADMM) based on the decomposition. It is widely used in large-scale nonlinear optimization problems thanks to the advantage of being easily extended to parallel and distributed systems. However, it still faces the challenge of slow convergence when applied in the FOF optimization model. [PERSON] et al. [19] first proposed a Newton-based efficient algorithm; however, it only works when facing a non-constrained FOF model, which is unrealistic in practice. A least-squares method and an alternating direction method were developed by [PERSON] et al. [20] and [PERSON] et al. [21], respectively. However, they still consume a lot of time as the scale of the problem grows. The main contribution of this paper is to propose a parallel algorithm general enough to characterize most of the existing specific FOF optimization models.