Adaptive Preconditioned Gradient Descent with Energy

We propose an adaptive time step with energy for a large class of preconditioned gradient descent methods, mainly applied to constrained optimization problems. Our strategy relies on representing the usual descent direction by the product of an energy variable and a transformed gradient, with a preconditioning matrix, for example, to reflect the natural gradient induced by the underlying metric in parameter space or to endow a projection operator when linear equality constraints are present. We present theoretical results on both unconditional stability and convergence rates for three respective classes of objective functions. In addition, our numerical results shed light on the excellent performance of the proposed method on several benchmark optimization problems.Comment: 32 pages, 3 figure

On Robustness in the Gap Metric and Coprime Factor Uncertainty for LTV Systems

In this paper, we study the problem of robust stabilization for linear time-varying (LTV) systems subject to time-varying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, we compute an upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the time-varying stability margin and TV controller. A connection between robust stabilization for LTV systems and an Operator Corona Theorem is also pointed out.Comment: 20 page

On the Strong Coupling Limit of the Faddeev-Hopf Model

The variational calculus for the Faddeev-Hopf model on a general Riemannian domain, with general Kaehler target space, is studied in the strong coupling limit. In this limit, the model has key similarities with pure Yang-Mills theory, namely conformal invariance in dimension 4 and an infinite dimensional symmetry group. The first and second variation formulae are calculated and several examples of stable solutions are obtained. In particular, it is proved that all immersive solutions are stable. Topological lower energy bounds are found in dimensions 2 and 4. An explicit description of the spectral behaviour of the Hopf map S^3 -> S^2 is given, and a conjecture of Ward concerning the stability of this map in the full Faddeev-Hopf model is proved.Comment: 21 pages, 0 figure

Gravitational corrections to Higgs potentials

Understanding the Higgs potential at large field values corresponding to scales in the range above $10^{10}{\rm GeV}$ is important for questions of vacuum stability, particularly in the early universe where survival of the Higgs vacuum can be an issue. In this paper we show that the Higgs potential can be derived in away which is independent of the choice of conformal frame for the spacetime metric. Questions about vacuum stability can therefore be answered unambiguously. We show that frame independence leads to new relations between the beta functions of the theory and we give improved limits on the allowed values of the Higgs curvature coupling for stability.Comment: 21 pages, 5 figures, jhep style, v