A Dual Approach to Multidimensional $L_p$ Spectral Estimation Problems

A complete duality theory is presented for the multidimensional L_p spectral estimation problem. The authors use a new constraint qualification (BWCQ) for infinite-dimensional convex programs with linear type constraints recently introduced in [Borwein and Wolkowicz, Math. Programming, 35 (1986), pp. 83-96]. This allows direct derivation of the explicit optimal solution of the problem as presented in [Goodrich and Steinhardt, SIAM J. Appl. Math., 46 (1986), pp. 417-426], and establishment of the existence of a simple and computationally tractable unconstrained Lagrangian dual problem. Moreover, the results illustrate that (BWCQ) is more appropriate to spectral estimation problems than the traditional Slater condition (which may only be applied after transformation of the problem into an L_p space [Goodrich and Steinhardt, op. cit.] and which therefore yields only necessary conditions)