Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions

In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results

Gap functions and error bounds for variational-hemivariational inequalities

In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results

Nonlinear geometric analysis on Finsler manifolds

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the $\Gamma$-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.Comment: 37 pages, to appear in a topical issue of European Journal of Mathematics "Finsler Geometry: New Methods and Perspectives". arXiv admin note: text overlap with arXiv:1602.0039

The Cartan-Hadamard conjecture and The Little Prince

The generalized Cartan-Hadamard conjecture says that if $\Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le \kappa \le 0$, then the boundary of $\Omega$ has the least possible boundary volume when $\Omega$ is a round $n$-ball with constant curvature $K=\kappa$. The case $n=2$ and $\kappa=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $\kappa=0$, and a special case of the conjecture for \kappa \textless{} 0 and a version for \kappa \textgreater{} 0. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $\kappa=0$. The generalization to $n=4$ and $\kappa \ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le \kappa$ to a weaker candle condition $Candle(\kappa)$ or $LCD(\kappa)$.We also find counterexamples to a na\"ive version of the Cartan-Hadamard conjecture: For every \varepsilon \textgreater{} 0, there is a Riemannian 3-ball $\Omega$ with $(1-\varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $\varepsilon$ and with arbitrarily large volume.We begin with a pointwise isoperimetric problem called "the problem of the Little Prince." Its proof becomes part of the more general method.Comment: v3: significant rewritting of some proofs, a mistake in the proof of the ball counter-example has been correcte

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Some recent developments on the Steklov eigenvalue problem

The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.Comment: 157 pages, 7 figures. To appear in Revista Matem\'atica Complutens

Computation of Generalized Averaged Gaussian Quadrature Rules

The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie [2] developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, the author derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. In [2], [5], [3] stable numerical procedures for computation of the corresponding averaged Gaussian rules are proposed. An analogous procedure can be applied also for a more general class of weighted averaged Gaussian rules introduced in [1]. Those results are presented in [4]. Here we we give a survey of the quoted results, which are obtained jointly with L. Reichel (Kent State Univ., OH (U.S.)

Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds

Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before

Geometrie

The program covered a wide range of new developments in geometry. To name some of them, we mention the topics “Metric space geometry in the style of Alexandrov/Gromov”, “Polyhedra with prescribed metric”, “Willmore surfaces”, “Constant mean curvature surfaces in three-dimensional Lie groups”. The official program consisted of 21 lectures and included four lectures by V. Schroeder (Zürich) and S. Buyalo (Sankt-Petersburg) on “Asymptotic geometry of Gromov hyperbolic spaces”