SQP algorithms for solving Toeplitz matrix approximation problem

The problem we are interested in is the best approximation of a given matrix by a positive semi–definite symmetric Toeplitz matrix. Toeplitz matrices appear naturally in a variety of problems in engineering. Since positive semi–definite Toeplitz matrices can be viewed as shift invariant autocorrelation matrices, considerable attention has been paid to them, especially in the areas of stochastic filtering and digital signal processing applications [7] and [12]. Several problems in digital signal processing and control theory require the computation of a positive definite Toeplitz matrix that closely approximates a given matrix. For example, because of rounding or truncation errors incurred while evaluating F , F does not satisfy one or all conditions. Another example in the power spectral estimation of a wide–sense stationary process from a finite number of data, the matrix F formed from the estimated autocorrelation coefficients, is often not a positive definite Toeplitz matrix [11]. In control theory, the Gramian assignment problem for discrete–time single input system requires the computation of a positive definite Toeplitz matrix, which also satisfies certain inequality constraints [9]. We consider the following problem: Given a data matrix F ∈ IRn×n, find the nearest symmetric positive semi-definite Toeplitz matrix T to F and rank T = m. Use of the Frobenius norm as a measure gives rise to