Computing the nearest low-rank correlation matrix by a simplified SQP algorithm

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Abstract

In this paper, we propose a numerical method for computing the nearest low-rank correlation matrix (LRCM). Motivated by the fact that the nearest LRCM problem can be reformulated as a standard nonlinear equality constrained optimization problem with matrix variables via the Gramian representation, we propose a new algorithm based on the sequential quadratic programming (SQP) method. On each iteration, we do not solve the quadratic program (QP) corresponding to the exact Hessian, but a modified QP with a simpler Hessian. This QP subproblem can be solved efficiently by equivalently transforming it to a sparse linear system. Global convergence is established and preliminary numerical results are presented to demonstrate the proposed method is potentially useful.

Introduction

In this paper, we propose a numerical algorithm for the problem of computing the nearest low-rank correlation matrix:minXSn12X-CF2,subject todiag(X)=e,X0,rank(X)r,where Sn denotes the space of n×n symmetric matrices, C is a given matrix in Sn,e represents the vector of all ones in Rn,r is an arbitrarily given integer satisfying rn,·F refers to the Frobenius norm, diag(·) returns the main diagonal of a matrix, and X0 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: rank(X)r, problem (1) is a nonconvex problem, so that it is much harder than the nearest correlation matrix problem:minXSn12X-CF2,subject todiag(X)=e,X0.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:X=RRT,where RRn×r, so that (1) can be reformulated as:minRRn×r12RRT-CF2,subject todiag(RRT)=e.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 r×r. 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 [R]{RQ|Qis ar×rorthogonal matrix} 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 X¯=R¯R¯T with R¯Rn×r,rn. Then X¯ is a local minimum of (1) if and only if R¯ 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 definef(X)12XXT-CF2for any XRn×r, andA(X)diag(X)for any XRn×n. So problem (3) can be restated compactly asminRRn×rf(R),subject toA(RRT)=e.The Lagrangian function for (3) is defined byL(R,v)f(R)+vT(A(RRT)-e),where vRn is the vector of Lagrangian multipliers.

Now we calculate the second-order Taylor expansion of the objective and the constraints.

Theorem 1

Let RRn×r and PRn×r. Thenf(R+P)=f(R)+2RRTR-CR,P+RRT-C,PP

Solving subproblems

In this section we analyze some details about the QP subproblem (9).

Differentiating the Lagrangian of QP (9)Lq(P,y)qk(P)+yT(A(RkRkT)+2A(RkPT)-e)=2RkRkTRk-CRk,P+Hk,PPT+yT(A(RkRkT)+2A(RkPT)-e),with respect to the variable P directly, and using relationsHk,PPT=2HkPand[yT(A(RkRkT)+2A(RkPT)-e)]=2[yTA(RkPT)]=2A(y)Rk,we obtain the KKT conditions for (9):HkP+A(y)Rk=CRk-RkRkTRk,A(RkRkT)+2A(RkPT)=e.Taking the vec-operator, we can restate (14a), (14b) as:(IHk)vec(P)+JkTy=-gk,A(RkRkT)+2Jkvec(P)

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

  • (a)

    {Rk} is contained in a compact and convex region.

  • (b)

    The matrices Hk are uniformly positive definite, i.e., λmin(Hk)μ with some constant μ>0 for all k.

  • (c)

    min1inRk(i,:)2ν with some constant ν>0 for all k.

  • (d)

    πkyˆk+1+ρ with some constant ρ>0 and πk is fixed for all k sufficiently large.

  • (e)

    {τk} 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.

References (31)

  • I. Grubišić et al.

    Efficient rank reduction of correlation matrices

    Linear Algebra Appl.

    (2007)
  • D. Simon et al.

    A majorization algorithm for constrained correlation matrix approximation

    Linear Algebra Appl.

    (2010)
  • Z. Zhang et al.

    Optimal low-rank approximation to a correlation matrix

    Linear Algebra Appl.

    (2003)
  • P.A. Absil et al.

    Optimization Algorithms on Matrix Manifolds

    (2008)
  • P.T. Boggs et al.

    Sequential quadratic programming

    Acta Numer.

    (1996)
  • R. Borsdore et al.

    A preconditioned Newton algorithm for the nearest correlation matrix

    IMA J. Numer. Anal.

    (2010)
  • A. Brace et al.

    The market model of interest rate dynamics

    Math. Finance

    (1997)
  • S. Burer et al.

    A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

    Math. Program.

    (2003)
  • S. Burer et al.

    Local minima and convergence in low-rank semidefinite programming

    Math. Program.

    (2005)
  • R. Fletcher, The sequential quadratic programming method, Lecture Notes in Mathematics, Nonlinear Optimization, vol....
  • Y. Gao, D. Sun, A majorized penalty approach for calibrating rank constrained correlation matrix problems, Tech. Rep.,...
  • P.E. Gill, E. Wong, Sequential quadratic programming methods, The IMA Volumes in Mathematics and Its Applications,...
  • G.H. Golub et al.

    Matrix Computations

    (1996)
  • N.I.M. Gould

    On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

    Math. Program.

    (1985)
  • L. Grippo, L. Palagi, V. Piccialli, An unconstrained minimization method for solving low rank SDP relaxations of the...
  • Cited by (0)

    View full text