Vicinity Of Saddle Point Research Articles

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

AbstractWe extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP.

Generalized Devil's staircase and RG flows

We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, τ, with modular properties. Throughout the work we consider a special limit when the imaginary part of τ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group SL(2,Z). In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at Imτ→0. We show that the RG-like dynamics in the fundamental domain of SL(2,Z) for asymmetric lattices asymptotically approaches the “Silver ratio”. For a Hubbard model of particles on a ring interacting via long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.

Evolution in the weak-mutation limit: Stasis periods punctuated by fast transitions between saddle points on the fitness landscape

A mathematical analysis of the evolution of a large population under the weak-mutation limit shows that such a population would spend most of the time in stasis in the vicinity of saddle points on the fitness landscape. The periods of stasis are punctuated by fast transitions, in lnNe/s time (Ne , effective population size; s, selection coefficient of a mutation), when a new beneficial mutation is fixed in the evolving population, which accordingly moves to a different saddle, or on much rarer occasions from a saddle to a local peak. Phenomenologically, this mode of evolution of a large population resembles punctuated equilibrium (PE) whereby phenotypic changes occur in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but the phenotype does not change substantially. Theoretically, PE has been linked to self-organized criticality (SOC), a model in which the size of "avalanches" in an evolving system is power-law-distributed, resulting in increasing rarity of major events. Here we show, however, that a PE-like evolutionary regime is the default for a very simple model of an evolving population that does not rely on SOC or any other special conditions.

Analyzing highly uncertain source regions in the Ex-UTLS and their effects on small-scale atmospheric composition using probabilistic retroplume calculations

Forecasts of atmospheric composition in the Ex-UTLS (extratropical upper troposphere – lower stratosphere) are highly sensitive to uncertainties in the source regions of air masses. This study investigates the reliability of calculated source regions and their effects on trace gas distributions in this region. For this purpose, an ensemble-based probabilistic retroplume approach is developed to track source regions of air masses. This approach provides the ability to account for both, (i) the contribution of different source regions to an air mass by adjoint diffusion and (ii) the uncertainty of theses source regions with respect to forecast dynamics. Probabilistic retroplume calculations are applied to an air mass in the vicinity of a weak saddle-point in the Ex-UTLS in order to investigate its effect on source regions. The results indicate high sensitivity of source regions to uncertainties in atmospheric transport revealed by perturbed meteorological forecasts. Additionally considering the sources of surrounding air masses leads to even larger spread of source regions. Spreading over large parts of Europe and altitudes between 8 and 14 km within two days, probable source regions comprise completely opposite transport-directions. This large uncertainty of source regions show significant effect on atmospheric composition in the vicinity of the saddle-point on a short timescale. Evaluating the effect of meteorological uncertainties on local distributions of trace gases, ozone concentrations range from 50 to 350 ppbv at the same location. Probabilistic retroplume calculations suggest a clear connection between these concentrations and related source regions.

Magnetic spectrum of trigonally warped bilayer graphene: Semiclassical analysis, zero modes, and topological winding numbers

We investigate the fine structure in the energy spectrum of bilayer graphene in the presence of various stacking defaults, such as a translational or rotational mismatch. This fine structure consists of four Dirac points that move away from their original positions as a consequence of the mismatch and eventually merge in various manners. The different types of merging are described in terms of topological invariants (winding numbers) that determine the Landau-level spectrum in the presence of a magnetic field as well as the degeneracy of the levels. The Landau-level spectrum is, within a wide parameter range, well described by a semiclassical treatment that makes use of topological winding numbers. However, the latter need to be redefined at zero energy in the high-magnetic-field limit as well as in the vicinity of saddle points in the zero-field dispersion relation.

Spontaneous breaking of fourfold rotational symmetry in two-dimensional electron systems as a topological phase transition

Motivated by recent observations of $C_4$ symmetry breaking in strongly correlated two-dimensional electron systems on a square lattice, we analyze this phenomenon within an extended Fermi liquid approach. It is found that the symmetry violation is triggered by a continuous topological phase transition associated with exchange of antiferromagnetic fluctuations. In contrast to predictions of mean-field theory, the structure of a part of the single-particle spectrum violating $C_4$ symmetry is found to be highly anisotropic, with a peak located in the vicinity of saddle points.

Efficient and Adaptive Lagrange-Multiplier Methods for Nonlinear Continuous Global Optimization

Lagrangian methods are popular in solving continuous constrained optimization problems. In this paper, we address three important issues in applying Lagrangian methods to solve optimization problems with inequality constraints. First, we study methods to transform inequality constraints into equality constraints. An existing method, called the slack-variable method, adds a slack variable to each inequality constraint in order to transform it into an equality constraint. Its disadvantage is that when the search trajectory is inside a feasible region, some satisfied constraints may still pose some effect on the Lagrangian function, leading to possible oscillations and divergence when a local minimum lies on the boundary of the feasible region. To overcome this problem, we propose the MaxQ method that carries no effect on satisfied constraints. Hence, minimizing the Lagrangian function in a feasible region always leads to a local minimum of the objective function. We also study some strategies to speed up its convergence. Second, we study methods to improve the convergence speed of Lagrangian methods without affecting the solution quality. This is done by an adaptive-control strategy that dynamically adjusts the relative weights between the objective and the Lagrangian part, leading to better balance between the two and faster convergence. Third, we study a trace-based method to pull the search trajectory from one saddle point to another in a continuous fashion without restarts. This overcomes one of the problems in existing Lagrangian methods that converges only to one saddle point and requires random restarts to look for new saddle points, often missing good saddle points in the vicinity of saddle points already found. Finally, we describe a prototype Novel (Nonlinear Optimization via External Lead) that implements our proposed strategies and present improved solutions in solving a collection of benchmarks.

Anisotropic renormalization of quasiparticles in cuprate-oxide superconductors

The Hubbard model on the simple square lattice in the vicinity of the Mott transition is considered by the 1/ d expansion method, with d being the spatial dimensionality. The superexchange interaction is enhanced by antiferromagnetic spin fluctuations, and its enhanced portion is similar to an interaction due to the so-called spin-fluctuation mechanism. Gutzwiller's heavy electrons are renormalized to be light by the enhanced superexchange interaction, which is responsible for high-temperature superconductivity; the formation of light quasiparticles occurs, in particular, for k s in the vicinity of saddle points in two dimensions. This anisotropic renormalization of quasi-particles can explain not only the anisotropic pseudogap in normal states but also other anomalous normal-state properties such as the large anisotropy of resistivity between the parallel and the perpendicular directions to CuO 2 planes.

Effect of Phonons on Optical Transitions near Critical Points in the Band Structure

The effect of phonons on the density of states near a critical point in the band structure of an insulator or semiconductor is investigated. Dyson's equation is solved assuming that the electron-phonon coupling parameter is independent of wave vector. Electron phonon-vertex and central-cell corrections are negligible. It is concluded that so long as the critical point is not near threshold, the phonons broaden the structure associated with the critical point, but introduce no structure of their own. Therefore, electron-phonon self-energies cannot explain the anomalous structure observed in the vicinity of saddle points in a number of semiconductors.