Computing the nearest low-rank correlation matrix by a simplified SQP algorithm
Introduction
In this paper, we propose a numerical algorithm for the problem of computing the nearest low-rank correlation matrix:where denotes the space of symmetric matrices, C is a given matrix in represents the vector of all ones in is an arbitrarily given integer satisfying refers to the Frobenius norm, returns the main diagonal of a matrix, and means X is positive semidefinite.
Problem (1) is a very important matrix optimization model from finance, since it occurs as part of the calibration of the multi-factor London inter-bank offer rate (LIBOR) market model [4], an interest rate derivatives pricing model used in some financial institutions for violation and risk management of their interest rate derivatives portfolio. The reader is referred to [26] for a literature review of interest rate models.
Because of the rank constraint: , problem (1) is a nonconvex problem, so that it is much harder than the nearest correlation matrix problem:Therefore, numerical methods for (2) such as the methods in [3], [16], [18], [24] cannot be applied to (1).
The method proposed in this paper tackles problem (1) based on the widely adopted approach which uses the Gramian representation:where , so that (1) can be reformulated as:This reformulation immediately brings two evident benefits. First, the number of variables is notably reduced when r is much smaller than n. Second, we do not need to worry about the rank constraint and the positive semidefinite constraint since they have already vanished. However, this reformulation has the major drawback that it still contains infinitely many local minima. Indeed, if R is a local minimum of problem (2), then RQ would be another local minimum for any orthogonal matrix Q of order . This actually yields an orbit in the symmetric matrix space, leading to efficient optimization methods based on manifolds. In fact, according to [14], the equivalence class leads to an important submanifold of the products of n unit spheres called the Cholesky manifold. For a comprehensive introduction of optimization algorithms on manifold, the reader is referred to [1], [29]. However, our method does not belong to the category of optimization on manifolds. For more background on using the Gramian representation to solve the low-rank semidefinite programming problem, the reader is referred to Burer and Monteiro [5], [6] and Grippo, Palagi and Piccialli [12], [13]. The following proposition reveals that the local minima of (1) and the local minima of (3) are equivalent. Proposition 1 Suppose with . Then is a local minimum of (1) if and only if is a local minimum of (3). Proof The proof follows word by word from Proposition 2.3 of [6]. □
Since (3) is a standard nonlinear constrained optimization problem, we propose a sequential quadratic programming (SQP) method to solve it. The powerful SQP method is currently one of the most efficient methods for nonlinearly constrained optimization problems. It finds an approximate solution of a sequence of quadratic programs where a quadratic model of the Lagrangian function is minimized subject to the linearized constraints. For a literature review of the SQP method, the reader is referred to [2], [7], [9].
There exist various efficient numerical algorithms for solving problem (1). Here we list some of them. The geometric method in [14], the majorization method in [23], the trigonometric parameterization method (TPM) in [25], and the Lagrangian method in [31] are some impressive methods in previous years. They also utilize the Gramian representation. In addition, The geometric method [14] is based on manifolds. In recent years, there has been a good research progress on the studied problem. For example, Gao and Sun [8] proposed a penalized majorization method which is really efficient and can deal with problems with inequality constraints. The sequential semismooth Newton approach proposed by Li and Qi [17] solves (1) by a nonconvex reformulation based on the well-known result in [21] that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming (SDP) problem. More recently, Wen and Yin [30] has proposed a feasible method using the Cayley transform on the Stiefel manifold for problems with orthogonality constraints. In fact, problem (1) is a special case of this type.
This paper is organized as follows. In Section 2, we describe a framework of our SQP algorithm for the nearest LRCM problem. In Section 3, we discuss some theoretical and computational details about QP subproblems. Global convergence of the proposed algorithm is proved in Section 4. Section 5 is devoted to some practical issues and numerical results. Concluding remarks are made in the final section.
Section snippets
An SQP approach
In this section, we present an SQP algorithm for (3). For notational convenience, we definefor any , andfor any . So problem (3) can be restated compactly asThe Lagrangian function for (3) is defined bywhere is the vector of Lagrangian multipliers.
Now we calculate the second-order Taylor expansion of the objective and the constraints. Theorem 1 Let and . Then
Solving subproblems
In this section we analyze some details about the QP subproblem (9).
Differentiating the Lagrangian of QP (9)with respect to the variable P directly, and using relationsandwe obtain the KKT conditions for (9):Taking the vec-operator, we can restate (14a), (14b) as:
Global convergence
Based on the well-studied global convergence property for SQP methods, we can guarantee global convergence of Algorithm 1 under the following assumptions. Assumption 1 is contained in a compact and convex region. The matrices are uniformly positive definite, i.e., with some constant for all k. with some constant for all k. with some constant and is fixed for all k sufficiently large. is bounded.
Now we state the global convergence result as
Numerical results
In this section, we report numerical results of the proposed algorithm and comparisons with classical majorization method (Major) [23] and the feasible optimization method (FOptM) [30]. All the experiments were performed on a PC with Intel Core i7–3770 Processors of 3.40 GHz and 8 GB of RAM, and all the codes are implemented in MATLAB.
Algorithmic issues. We first discuss several algorithmic issues related to stopping criteria, starting points, updating rules for the penalty parameter, strategies
Concluding remarks
In this paper, a numerical approach for computing the nearest low-rank correlation matrix is analyzed. This approach exploits the sequential quadratic programming method to find a local minimum of an equivalent nonlinear reformulation of the nearest low-rank correlation matrix problem. Numerical results show that the proposed method is quite useful for such problem, when the dimension n of the matrix is large and the rank r of the matrix is not much smaller than n. The key of the success of our
Acknowledgments
I am very grateful to the two anonymous referees for their helpful comments and suggestions which helped improve the quality of this paper.
This work is supported by Talent Introduction Foundation of Shanghai University of Electric Power, No. K2013–018.
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