Stability of the Duality Gap in Linear Optimization

In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + ∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-01 and SECTyP-UNCuyo Res. 4540/13-R

Analytic central path, sensitivity analysis and parametric linear programming

In this paper we consider properties of the central path and the analytic center of the optimal face in the context of parametric linear programming. We first show that if the right-hand side vector of a standard linear program is perturbed, then the analytic center of the optimal face is one-side differentiable with respect to the perturbation parameter. In that case we also show that the whole analytic central path shifts in a uniform fashion. When the objective vector is perturbed, we show that the last part of the analytic central path is tangent to a central path defined on the optimal face of the original problem

Parametric LP Analysis

Parametric linear programming is the study of how optimal properties depend on data parametrizations. The study is nearly as old as the field of linear programming itself, and it is important since it highlights how a problem changes as what is often estimated data varies. We present what is a modern perspective on the classical analysis of the objective value\u27s response to parametrizations in the right-hand side and cost vector. We also mention a few applications and provide citations for further stud

Ising model on 3D random lattices: A Monte Carlo study

We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to 128,000 approx. 503 sites which are linked together according to the Voronoi/Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and \phi^4 field theory on three-dimensional regular cubic lattices.Comment: 35 pages, LaTex, 8 tables, 8 postscript figure

Critical Phenomena and Renormalization-Group Theory

We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O($N$)-symmetric universality classes, including the $N\to 0$ limit that describes the critical behavior of self-avoiding walks. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as $N$-component systems with cubic anisotropy, O($N$)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the $\beta$-functions. Finally, we consider a Hamiltonian with symmetry $O(n_1)\oplus O(n_2)$ that is relevant for the description of multicritical phenomena.Comment: 151 pages. Extended and updated version. To be published in Physics Report

Applications of No-Collision Transportation Maps in Manifold Learning

In this work, we investigate applications of no-collision transportation maps introduced in [Nurbekyan et. al., 2020] in manifold learning for image data. Recently, there has been a surge in applying transportation-based distances and features for data representing motion-like or deformation-like phenomena. Indeed, comparing intensities at fixed locations often does not reveal the data structure. No-collision maps and distances developed in [Nurbekyan et. al., 2020] are sensitive to geometric features similar to optimal transportation (OT) maps but much cheaper to compute due to the absence of optimization. In this work, we prove that no-collision distances provide an isometry between translations (respectively dilations) of a single probability measure and the translation (respectively dilation) vectors equipped with a Euclidean distance. Furthermore, we prove that no-collision transportation maps, as well as OT and linearized OT maps, do not in general provide an isometry for rotations. The numerical experiments confirm our theoretical findings and show that no-collision distances achieve similar or better performance on several manifold learning tasks compared to other OT and Euclidean-based methods at a fraction of a computational cost

Discrete Geometric Singular Perturbation Theory

We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold $S$. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of $S$. The persistence of the critical manifold $S$, local stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of $W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincar\'e maps, the geometry and dynamics of which can be described in detail using DGSPT.Comment: Updated to include minor corrections made during the review process (no major changes