Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

In this paper we study distributionally robust constraints on risk measures (such as standard deviation less the mean, Conditional Value-at-Risk, Entropic Value-at-Risk) of decision-dependent random variables. The uncertainty sets for the discrete probability distributions are defined using statistical goodness-of-fit tests and probability metrics such as Pearson, likelihood ratio, Anderson-Darling tests, or Wasserstein distance. This type of constraints arises in problems in portfolio optimization, economics, machine learning, and engineering. We show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. We also show how the counterpart can be constructed for risk measures that are nonlinear in the probabilities (for example, variance or the Conditional Value-at-Risk). We provide the computational tractability status for each of the uncertainty set-risk measure pairs that we could solve. Numerical examples including portfolio optimization and a multi-item newsvendor problem illustrate the proposed approach

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

In this paper we focus on robust linear optimization problems with uncertainty regions defined by ø-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on ø-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with ø-divergence uncertainty is tractable for most of the choices of ø typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.robust optimization;ø-divergence;goodness-of-fit statistics

Magnetic and Thermodynamic Properties of the Collective Paramagnet-Spin Liquid Pyrochlore Tb2Ti2O7

In a recent letter [Phys. Rev. Lett. {\bf 82}, 1012 (1999)] it was found that the Tb$^{3+}$ magnetic moments in the Tb$_2$Ti$_2$O$_7$ pyrochlore lattice of corner-sharing tetrahedra remain in a {\it collective paramagnetic} state down to 70mK. In this paper we present results from d.c. magnetic susceptibility, specific heat data, inelastic neutron scattering measurements, and crystal field calculations that strongly suggest that (1) the Tb$^{3+}$ ions in Tb$_2$Ti$_2$O$_7$ possess a moment of approximatively 5$\mu_{\rm B}$, and (2) the ground state $g-$tensor is extremely anisotropic below a temperature of $O(10^0)$K, with Ising-like Tb$^{3+}$ magnetic moments confined to point along a local cubic $$diagonal (e.g. towards the middle of the tetrahedron). Such a very large easy-axis Ising like anisotropy along a$$ direction dramatically reduces the frustration otherwise present in a Heisenberg pyrochlore antiferromagnet. The results presented herein underpin the conceptual difficulty in understanding the microscopic mechanism(s) responsible for Tb$_2$Ti$_2$O$_7$ failing to develop long-range order at a temperature of the order of the paramagnetic Curie-Weiss temperature $\theta_{\rm CW} \approx -10^1$K. We suggest that dipolar interactions and extra perturbative exchange coupling(s)beyond nearest-neighbors may be responsible for the lack of ordering of Tb$_2$Ti$_2$O$_7$.Comment: 8 POSTSCRIPT figures included. Submitted to Physical Review B. Contact: [email protected]

Some new evidence on the determinants of large- and small-firm innovation

Empirical analyses presented by Acs and Audretsch suggest differences in the market structure determinants of innovation between large and small firms in U.S. manufacturing. The evidence they offer is ambiguous. By using data for a different country (The Netherlands), a different measure of innovation and a different aggregation level, we offer new evidence, allowing a revaluation of the findings for the U.S. material. Moreover, the influence of the market structure determinants does not appear to differ between a period of sluggish growth (1983) and one of relatively high growth (1989)

Long Range Order at Low Temperature in Dipolar Spin Ice

Recently it has been suggested that long range magnetic dipolar interactions are responsible for spin ice behavior in the Ising pyrochlore magnets ${\rm Dy_{2}Ti_{2}O_{7}}$ and ${\rm Ho_{2}Ti_{2}O_{7}}$. We report here numerical results on the low temperature properties of the dipolar spin ice model, obtained via a new loop algorithm which greatly improves the dynamics at low temperature. We recover the previously reported missing entropy in this model, and find a first order transition to a long range ordered phase with zero total magnetization at very low temperature. We discuss the relevance of these results to ${\rm Dy_{2}Ti_{2}O_{7}}$ and ${\rm Ho_{2}Ti_{2}O_{7}}$.Comment: New version of the manuscript. Now contains 3 POSTSCRIPT figures as opposed to 2 figures. Manuscript contains a more detailed discussion of the (i) nature of long-range ordered ground state, (ii) finite-size scaling results of the 1st order transition into the ground state. Order of authors has been changed. Resubmitted to Physical Review Letters Contact: [email protected]

Local moment formation in zinc doped cuprates

We suggest that when zinc is substituted for copper in the copper oxide planes of high $T_{c}$ superconductors, it does not necessarily have a valency of 2+. Rather, the valency of a zinc impurity should be determined by its surrounding medium. In order to study this hypothesis, we examine the effect of static impurities inducing diagonal disorder within a one band Hubbard model coupled to a localised state. We use this model to discuss the physics of zinc doping in the cuprates. Specifically, we discuss the formation of local moments near impurity sites and the modification of the transverse spin susceptibility in the antiferromagnetic state.Comment: 7 pages RevTex, includes 4 figure

Application of the Cluster Variation Method to Spin Ice Systems on the Pyrochlore Lattice

The cactus approximation in the cluster variation method is applied to the spin ice system with nearest neighbor ferromagnetic coupling. The temperature dependences of the entropy and the specific heat show qualitatively good agreement with those observed by Monte Carlo simulations and experiments, and the Pauling value is reproduced for the residual entropy. The analytic expression of the q-dependent magnetic susceptibility is obtained, from which the absence of magnetic phase transition is confirmed. The neutron scattering pattern is also evaluated and found to be consistent with that obtained from Monte Carlo simulations.Comment: 8 pages, 7 figure