Information Geometry and Interior-Point Algorithms in Semidefinite Programs and Symmetric Cone Programs

We develop an information geometric approach to conic programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity, naturally arising in information science including statistics, machine learning and signal processing etc. First we introduce an information geometric framework of conic programming. Then we focus on semidefinite and symmetric cone programs. Recently, we demonstrated that the number of iterations of Mizuno–Todd–Ye predictor–corrector primal–dual interior-point methods is (asymptotically) expressed with an integral over the central trajectory called “the curvature integral”. The number of iterations of the algorithm is approximated surprisingly well with the integral even for fairly large linear/semidefinite programs with thousands of variables. Here we prove that “the curvature integral” admits a rigorous differential geometric expression based on information geometry. We also obtain an interesting information geometric global theorem on the central trajectory for linear programs. Together with the numerical evidence in the aforementioned work, we claim that “the number of iterations of the interior-point algorithm is expressed as a differential geometric quantity.”     

Strong Convergence Theorems for Maximal and Inverse-Strongly Monotone Mappings in Hilbert Spaces and Applications

In this paper, we prove two strong convergence theorems for finding a common point of the set of zero points of the addition of an inverse-strongly monotone mapping and a maximal monotone operator and the set of zero points of a maximal monotone operator, which is related to an equilibrium problem in a Hilbert space. Such theorems improve and extend the results announced by Y. Liu (Nonlinear Anal. 71:4852–4861, 2009). As applications of the results, we present well-known and new strong convergence theorems which are connected with the variational inequality, the equilibrium problem and the fixed point problem in a Hilbert space.     

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.     

Classifying resolving subcategories over a Cohen–Macaulay local ring

Let R be a Cohen–Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving subcategories of mod R in terms of specialization-closed subsets of Spec R. We give a classification of the resolving subcategories closed under tensor products and transposes. Under restrictive hypotheses, we also give better classification results.     

Necessary conditions for constrained optimal control problems via mathematical programming

Necessary conditions for optimal control problems with state-control variable inequality constraints are obtained via mathematical programming formulation and functional analysis in Banach space. These conditions are general ones that hold without any constraint qualifications but differentiability. Furthermore, these conditions are shown to be equivalent to the classical result in the presence of the linear independence constraint qualification.     

Isometries of some F-algebras of holomorphic functions on the upper half plane

Linear isometries of N ^p (D) onto N ^p (D) are described, where N ^p (D), p > 1, is the set of all holomorphic functions f on the upper half plane D = {z ∈ ?: Im z > 0} such that sup _y >0 ∫_? ln ^p (1 + |(x + iy)|) dx < +∞. Our result is an improvement of the results by D.A. Efimov.     

Some constraint qualifications for quasiconvex vector-valued systems

In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint qualifications, the closed cone constraint qualification for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint qualification for vector-valued quasiconvex programming (the VQ-BCQ). Based on previous results by Benoist et al. (Proc Am Math Soc 13:1109–1113, 2002), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554–563, 2011), and (Nonlinear Anal 74:1279–1285, 2011), we show that the VQ-CCCQ (resp. the VQ-BCQ) is the weakest constraint qualification for Lagrangian-type strong (resp. min–max) duality. As consequences of the main results, we study semi-definite quasiconvex programming problems in our scheme, and we observe the weakest constraint qualifications for Lagrangian-type strong and min–max dualities. Finally, we summarize the characterizations of the weakest constraint qualifications for convex and quasiconvex programming.     

Surface effects on saturation magnetization in nanoporous Ni

Nanoporous Ni specimens with ligament lengths of 10–210 nm and specific surface areas of 0.03–0.58 nm^?1 were fabricated by the dealloying of Ni_0.25Mn_0.75 alloy and annealing at 473–873 K, and saturation magnetization investigated in terms of their size dependence. Saturation magnetization decreased with decreasing ligament length or increasing specific surface area. This trend is the same as that for nanoparticle Ni. However, the saturation magnetization of nanoporous Ni tends to be lower than that of the nanoparticle Ni when their specific surface areas are the same. It is suggested, therefore, that the surface effect due to a noncollinear arrangement is enhanced by the surface defects in the nanoporous Ni.     

Analysing the sintering of heterogeneous powder structures by in situ microtomography

This work analyses the microstructure changes of various copper-based powder systems during sintering from 3D images provided by in situ synchrotron microtomography. The investigated systems include a copper powder with a wide particle size distribution of 0–63 µm poured into a quartz capillary, a pre-sintered copper compact with artificially created large pores and a mixture of copper and alumina particles. The experiments were carried out at the European Synchrotron in Grenoble, France. Powders were sintered up to 1060°C under reducing atmosphere in a furnace located between the X-ray source and the detector. During each experiment, 3D images were taken at various times of the thermal cycle. We have obtained images with a resolution of 1.5 µm and the time of acquisition of every image was ～1 min. Quantitative analysis of these images allowed the changes of various important parameters to be followed. Such parameters characterise the sintering process at the particle length scale: interparticle coordination, pore size distribution and particle centre-to-centre distance. Moreover, by tracking the displacement of each particle centre and comparing it to the displacement predicted by classical mean field assumption, we have been able to assess the magnitude of particle rearrangement occurring during sintering. From these data, the sintering behaviour of heterogeneous powder systems is discussed with particular emphasis of collective particle phenomena.     

Unwanted combustion enhancement by C6F12O fire suppressant

Several agents are under consideration to replace CF₃Br for use in suppressing fires in aircraft cargo bays. In a Federal Aviation Administration (FAA) performance test simulating the explosion of an aerosol can, however, the replacements, when added at sub-inerting concentrations, have all been found to create higher pressure rise than with no agent, hence failing the test. Thermodynamic equilibrium calculations as well as perfectly-stirred reactor (PSR) simulations with detailed reaction kinetics, are performed for one of these agents, C₆F₁₂O (Novec 1230), to understand the reasons for the unexpected enhanced combustion rather than suppression. The high pressure rise with added agent is shown to depend on the amount of agent, and can only occur if a large fraction of the available oxidizer in the chamber is consumed, corresponding to stoichiometric proportions of fuel, oxygen, and agent. A kinetic model for the reaction of C₆F₁₂O in hydrocarbon–air flames has been developed. Stirred-reactor simulations predict that at higher agent loadings, the inhibition effectiveness of C₆F₁₂O is relatively insensitive to the overall stoichiometry, and the marginal inhibitory effect of the agent is greatly reduced, so that the mixture remains flammable over a wide range of conditions corresponding to those of the FAA test. The present findings are consistent with and support the earlier analyses for C₂HF₅ and CF₃Br, which were also evaluated in the FAA test.