On global optimality of gradient descent algorithms for fixed-rate scalar multiple description quantizer design

We prove that Trushkin´s (1982) sufficient conditions for the global optimality of a locally optimal fixed-rate scalar quantizer also ensure the global optimality of a locally optimal fixed-rate multiple description scalar quantizer of convex codecells, with respect to a fixed index assignment. This result also holds for the fixed-rate multiresolution scalar quantizer of convex codecells. As a consequence the well-known log-concave pdf condition can be extended to the multiple description and multiresolution case.