Compact convergence

Abstract: We show that if f n is a sequence of uniformly L p-bounded functions on a measure space, and if f n → f pointwise a.e., then lim for all 0 < p < ∞. This result is also generalized in Theorem 2 to some functional other than the L p norm, namely → 0 for suitable j: C → C and a suitable sequence f n. A brief discussion is given of the usefulness of this result in variational problems.

Abstract: The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm d where q and ql are uniquely determined orthonormal vectors. The parameter 1/ is . ntially arbitrary in that it depends upon p. It was suggested in Part 1 that a suitable ice for I] would be zero since if it were negative, or large and positive, the matrix KI hence HI might become needlessly badly conditioned. It was noted moreover that osing I] in this way gives rise to a new algorithm. the two algorithms in this class already published, that due to Davidon (1959) modified by Fletcher & Powell (1963) is obtained by putting P equal to zero and s shown in Part I that this led, in general, to negative values of 1]. We thus expect quence of matrices {HI} obtained by that algorithm to exhibit a tendency to arity and this tendency has been noted by, among others, Broyden (1967) and on (1969). In a more recent algorithm, due to Greenstadt (1967), if H is positive ite the values of 1] are even more negative than those occurring in the DFP ithm. One result of this is that for this algorithm the matrices H cannot, unlike for the DFP algorithm, be proved to be positive definite and this has serious tions when considering numerical stability. this paper we show theoretically that the new algorithm is stable and we prove is the only member of the class considered for which a certain matrix error is reduced strictly monotonically when minimizing quadratic functions. We the effect of rounding and of poor conditioning of H on the attainable accuracy solution and conclude by presenting the results of a numerical survey in he performance of the new algorithm for a variety of test problem is compared t of the DFP algorithm. C. G. BROYDEN Computing Centre, University of Essex, Wivenhoe Park, Colchester, Essex

Abstract: Applications and issues application to learning, state dependent noise and queueing applications to signal processing and adaptive control mathematical background convergence with probability one - Martingale difference noise convergence with probability one - correlated noise weak convergence - introduction weak convergence methods for general algorithms applications - proofs of convergence rate of convergence averaging of the iterates distributed/decentralized and asynchronous algorithms.