The geometry of linearly and quadratically constrained optimization problems for signal processing and communications

Constrained minimization problems considered in this paper arise in the design of beamformers for radar, sonar, and wireless communications, and in the design of precoders and equalizers for digital communications. The problem is to minimize a quadratic form under a set of linear or quadratic constraints. We present solutions to these problems and establish a connection between them. A majorization result for matrix trace and Poincare's separation theorem play key roles in establishing the connection. We show that our solutions can be formulated as generalized sidelobe cancellers (GSCs), which tie our constrained minimizations to linear minimum mean-squared error (LMMSE) estimations. We then express our solutions in terms of oblique projection matrices and establish the geometry of our constrained minimizations.