Fengmin Xu

A mixed 0–1 LP for index tracking problem with CVaR risk constraints

Index tracking problems are concerned in this paper. A CVaR risk constraint is introduced into general index tracking model to control the downside risk of tracking portfolios that consist of a subset of component stocks in given index. Resulting problem is a mixed 0–1 and non-differentiable linear programming problem, and can be converted into a mixed 0–1 linear program so that some existing optimization software such as CPLEX can be used to solve the problem. It is shown that adding the CVaR constraint will have no impact on the optimal tracking portfolio when the index has good (return increasing) performance, but can limit the downside risk of the optimal tracking portfolio when index has bad (return decreasing) performance. Numerical tests on Hang Seng index tracking and FTSE 100 index tracking show that the proposed index tracking model is effective in controlling the downside risk of the optimal tracking portfolio.

THE EIGENVALUE DISTRIBUTION OF SCHUR COMPLEMENTS OF NONSTRICTLY DIAGONALLY DOMINANT MATRICES AND GENERAL H−MATRICES ∗

The paper studies the eigenvalue distribution of Schur complements of some special matrices, including nonstrictly diagonally dominant matrices and general H matrices. Zhang, Xu, and Li (Theorem 4.1, The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl., 422:250-264, 2007) gave a condition for an n×n diagonally dominant matrix A to have |JR+ (A)| eigenvalues with positive real part and |JR− (A)| eigenvalues with negative real part, where |JR+(A)| (|JR−(A)|) denotes the number of diagonal entries of A with positive (negative) real part. This condition is applied to establish some results about the eigenvalue distribution for the Schur complements of nonstrictly diagonally dominant matrices and general H matrices with complex diagonal entries. Several conditions on the n×n matrix A and the subset � � N = {1,2,···,n} are presented so that the Schur complement A/� of A has |JR+ (A)| | J �+ (A)| eigenvalues with positive real part and |JR− (A)| | J � R− (A)| eigenvalues with negative real part, where |J � R+ (A)| (|J � R− (A)|) denotes the number of diagonal entries of the principal submatrix A(�) of A with positive (negative) real part.

An efficient optimization approach for a cardinality-constrained index tracking problem

In the practical business environment, portfolio managers often face business-driven requirements that limit the number of constituents in their tracking portfolio. A natural index tracking model is thus to minimize a tracking error measure while enforcing an upper bound on the number of assets in the portfolio. In this paper we consider such a cardinality-constrained index tracking model. In particular, we propose an efficient nonmonotone projected gradient (NPG) method for solving this problem. At each iteration, this method usually solves several projected gradient subproblems. We show that each subproblem has a closed-form solution, which can be computed in linear time. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the NPG method is a local minimizer of the cardinality-constrained index tracking problem. We also conduct empirical tests to compare our method with the hybrid evolutionary algorithm [P.R. Torrubiano and S. Alberto. A hybrid optimization approach to index tracking. Ann Oper Res. 166(1) (2009), pp. 57–71] and the hybrid half thresholding algorithm [F. Xu, Z. Xu and H Xue. Sparse index tracking: an regularization based model and solution, Submitted, 2012] for index tracking. The computational results demonstrate that our approach generally produces sparse portfolios with smaller out-of-sample tracking error and higher consistency between in-sample and out-of-sample tracking errors. Moreover, our method outperforms the other two approaches in terms of speed.

A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems

In this paper, a discrete filled function algorithm embedded with continuous approximation is proposed to solve max-cut problems. A new discrete filled function is defined for max-cut problems, and properties of the function are studied. In the process of finding an approximation to the global solution of a max-cut problem, a continuation optimization algorithm is employed to find local solutions of a continuous relaxation of the max-cut problem, and then global searches are performed by minimizing the proposed filled function. Unlike general filled function methods, characteristics of max-cut problems are used. The parameters in the proposed filled function need not to be adjusted and are exactly the same for all max-cut problems that greatly increases the efficiency of the filled function method. Numerical results and comparisons on some well known max-cut test problems show that the proposed algorithm is efficient to get approximate global solutions of max-cut problems.

A new Lagrangian net algorithm for solving max-bisection problems

The max-bisection problem is an NP-hard combinatorial optimization problem. In this paper, a new Lagrangian net algorithm is proposed to solve max-bisection problems. First, we relax the bisection constraints to the objective function by introducing the penalty function method. Second, a bisection solution is calculated by a discrete Hopfield neural network (DHNN). The increasing penalty factor can help the DHNN to escape from the local minimum and to get a satisfying bisection. The convergence analysis of the proposed algorithm is also presented. Finally, numerical results of large-scale G-set problems show that the proposed method can find a better optimal solutions.

Sparse portfolio rebalancing model based on inverse optimization

This paper considers a sparse portfolio rebalancing problem in which rebalancing portfolios with minimum number of assets are sought. This problem is motivated by the need to understand whether the initial portfolio is worthwhile to adjust or not, inducing sparsity on the selected rebalancing portfolio to reduce transaction costs (TCs), out-of-sample performance and small changes in portfolio. We propose a sparse portfolio rebalancing model by adding an l1 penalty item into the objective function of a general portfolio rebalancing model. In this way, the model is sparse with low TCs and can decide whether and which assets to adjust based on inverse optimization. Numerical tests on four typical data sets show that the optimal adjustment given by the proposed sparse portfolio rebalancing model has the advantage of sparsity and better out-of-sample performance than the general portfolio rebalancing model.

A sparse enhanced indexation model with norm and its alternating quadratic penalty method

Abstract Optimal investment strategies for enhanced indexation problems have attracted considerable attentions over the last decades in the field of fund management. In this paper, a featured difference from the existing literature is that our main concern of the investigation is the development of a sparse-enhanced indexation model to describe the process of assets selection by introducing a sparse regularization instead of binary variables, which is expected to avoid the over-fitting and promote a better out-of-sample performance for the resulting tracking portfolio to some extent. An Alternating Quadratic Penalty (AQP) method is proposed to solve the corresponding nonconvex optimisation problem, into which the Block Coordinate Descent (BCD) algorithm is integrated to solve a sequence of penalty subproblems. Under some suitable assumptions, we establish that any accumulation point of the sequence generated by the AQP method is a KKT point of the proposed model. Computational results on five typical data-sets are reported to verify the efficiency of the proposed AQP method, including the superiority of the sparse model with the AQP method over one cardinality constrained quadratic programming model with one of its solution methods in terms of computational costs, out-of-sample performances, and the consistency between in-sample and out-of-sample performances of the resulting tracking portfolios.

The Local Linear

This paper studies the nonparametric regressive function with missing response data. Three local linear -estimators with the robustness of local linear regression smoothers are presented such that they have the same asymptotic normality and consistency. Then finite-sample performance is examined via simulation studies. Simulations demonstrate that the complete-case data -estimator is not superior to the other two local linear -estimators.

M

A portfolio rebalancing model with self-finance strategy and consideration of V-shaped transaction cost is presented in this paper. Our main contribution is that a new constraint is introduced to confirm that the rebalance necessity of the existing portfolio needs to be adjusted. The constraint is constructed by considering both the transaction amount and transaction cost without any additional supply to the investment amount. The V-shaped transaction cost function is used to calculate the transaction cost of the portfolio, and conditional value at risk (CVaR) is used to measure the risk of the portfolios. Computational tests on practical financial data show that the proposed model is effective and the rebalanced portfolio increases the expected return of the portfolio and reduces the CVaR risk of the portfolio.

-Estimation with Missing Response Data

The special importance of Difference of Convex (DC) functions programming has been recognized in recent studies on nonconvex optimization problems. In this work, a class of DC programming derived from the portfolio selection problems is studied. The most popular method applied to solve the problem is the Branch-and-Bound (B&B) algorithm. However, “the curse of dimensionality” will affect the performance of the B&B algorithm. DC Algorithm (DCA) is an efficient method to get a local optimal solution. It has been applied to many practical problems, especially for large-scale problems. A B&B-DCA algorithm is proposed by embedding DCA into the B&B algorithms, the new algorithm improves the computational performance and obtains a global optimal solution. Computational results show that the proposed B&B-DCA algorithm has the superiority of the branch number and computational time than general B&B. The nice features of DCA (inexpensiveness, reliability, robustness, globality of computed solutions, etc.) provide crucial support to the combined B&B-DCA for accelerating the convergence of B&B.