The upper semi-continuity of the solution map to the extended homogeneous complementarity problem with the R0R_0-condition

In this paper we introduce a concept of the $R_0$-condition for the extended homogeneous complementarity problem, and show that the upper semi-continuity of the solution map is equivalent to the $R_0$-condition in the extended homogeneous complementarity problem

A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization

We propose a novel trust region method for solving a class of nonsmooth and nonconvex composite-type optimization problems. The approach embeds inexact semismooth Newton steps for finding zeros of a normal map-based stationarity measure for the problem in a trust region framework. Based on a new merit function and acceptance mechanism, global convergence and transition to fast local q-superlinear convergence are established under standard conditions. In addition, we verify that the proposed trust region globalization is compatible with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence results. We further derive new normal map-based representations of the associated second-order optimality conditions that have direct connections to the local assumptions required for fast convergence. Finally, we study the behavior of our algorithm when the Hessian matrix of the smooth part of the objective function is approximated by BFGS updates. We successfully link the KL theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type condition to show superlinear convergence of the quasi-Newton version of our method. Numerical experiments on sparse logistic regression and image compression illustrate the efficiency of the proposed algorithm.Comment: 56 page

Differential variational inequalities

International audienceThis paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems , and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensiona

Conjugate two-dimensional electric potential maps

Two dimensional electric potential maps based on voltage detection in conducting paper are common practice in many physics courses in college. Most frequently, students work on `capacitor-like' geometries with current flowing between two opposite electrodes. A `topographical' investigation across the embedding medium (map of equipotential curves) allows to reassure a number of physical properties. This paper focuses on some less common configurations that bear pedagogical interest. We analyze `open-geometries' with electrodes in the form of long strips with slits. They provide a natural groundwork to bring the student to complex variable methods. Aided by this, we show that shaping the conducting paper board one may analyze finite size effects, as well as some meaningful discontinuities in the measured potential. The concept of conjugate electric potentials is exploited. Equipotentials and electric field lines acquire interchangeable roles and may be obtained in complementary `dual' experiments. A feasible theoretical analysis based on introductory complex variables and standardized numerics gives a remarkable quantification of the experimental results.Comment: 15 pages, 8 figure

Morceaux Choisis en Optimisation Continue et sur les Systèmes non Lisses

MasterThis course starts with the presentation of the optimality conditions of an optimization problem described in a rather abstract manner, so that these can be useful for dealing with a large variety of problems. Next, the course describes and analyzes various advanced algorithms to solve optimization problems (nonsmooth methods, linearization methods, proximal and augmented Lagrangian methods, interior point methods) and shows how they can be used to solve a few classical optimization problems (linear optimization, convex quadratic optimization, semidefinite optimization (SDO), nonlinear optimization). Along the way, various tools from convex and nonsmooth analysis will be presented. Everything is conceptualized in finite dimension. The goal of the lectures is therefore to consolidate basic knowledge in optimization, on both theoretical and algorithmic aspects

Comments on the Necessity and Implications of State-Dependence in the Black Hole Interior

We revisit the "state-dependence" of the map that we proposed recently between bulk operators in the interior of a large AdS black hole and operators in the boundary CFT. By refining recent versions of the information paradox, we show that this feature is necessary for the CFT to successfully describe local physics behind the horizon --- not only for single-sided black holes but even in the eternal black hole. We show that state-dependence is invisible to an infalling observer who cannot differentiate these operators from those of ordinary quantum effective field theory. Therefore the infalling observer does not observe any violations of quantum mechanics. We successfully resolve a large class of potential ambiguities in our construction. We analyze states where the CFT is entangled with another system and show that the ER=EPR conjecture emerges from our construction in a natural and precise form. We comment on the possible semi-classical origins of state-dependence.Comment: 136 pages, 16 figure

Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions

In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the solution of an infinite dimensional analogue of a semi-definite program. This embedding is adaptive and non-linear. A main feature of our approach is the existence of a non-linear out-of-sample extension formula of the embedding coordinates, called a projected Nystr\"om approximation. This extrapolation formula yields an extension of the kernel matrix to a data-dependent Mercer kernel function. Our empirical results indicate that this embedding method is more robust with respect to the influence of outliers, compared with a spectral embedding method.Comment: 16 pages, 5 figures. Improved presentatio

Firewall argument for acoustic black holes

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. June 8, 2015.We investigate the rewall paradox proposed by AMPS [1] by rst explaining the Information Paradox together with Hawking's derivation of the thermal radiation emitted from a evaporating black hole [28]. We then ask if one can apply arguments similar to that of Hawking and AMPS in the regime of uid mechanics, which was rst considered by Unruh [59]. We assume that a black hole, with a geometry conformal to the Schwarzschild metric, can be formed in a uid. The sonic hole or \dumb" hole, which is characterized by an acoustic event horizon, is the locus of points at which the background uid is traveling at the local speed of sound. Since sound disturbances are coupled to the background uid and travel at the speed of sound, the acoustic event horizon a ects sound disturbances in a manner analogous to how gravitational black holes a ect light [62]. Like a gravitational black hole, which evaporates by emitting Hawking radiation, we check if an acoustic black hole will emit in a similar kind of radiation in the form of phonons. This is done by constructing a massless scalar eld describing phonon propagation and treating the acoustic black hole just like a gravitational black hole. We apply the arguments put forth by Hawking and AMPS and see if there is any validity to an \acoustic rewall" as this would require certain physical phenomena emerging from sub-atomic scales

Firewall argument for acoustic black holes

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. June 8, 2015.We investigate the rewall paradox proposed by AMPS [1] by rst explaining the Information Paradox together with Hawking's derivation of the thermal radiation emitted from a evaporating black hole [28]. We then ask if one can apply arguments similar to that of Hawking and AMPS in the regime of uid mechanics, which was rst considered by Unruh [59]. We assume that a black hole, with a geometry conformal to the Schwarzschild metric, can be formed in a uid. The sonic hole or \dumb" hole, which is characterized by an acoustic event horizon, is the locus of points at which the background uid is traveling at the local speed of sound. Since sound disturbances are coupled to the background uid and travel at the speed of sound, the acoustic event horizon a ects sound disturbances in a manner analogous to how gravitational black holes a ect light [62]. Like a gravitational black hole, which evaporates by emitting Hawking radiation, we check if an acoustic black hole will emit in a similar kind of radiation in the form of phonons. This is done by constructing a massless scalar eld describing phonon propagation and treating the acoustic black hole just like a gravitational black hole. We apply the arguments put forth by Hawking and AMPS and see if there is any validity to an \acoustic rewall" as this would require certain physical phenomena emerging from sub-atomic scales