Minimax systems

The variational approach to solving nonlinear problems eventually leads to the search for critical points of related functionals. In case of semibounded functionals, one can look for extrema. Otherwise, one is forced to use minimax methods. There are several approaches to such methods. In this paper we unify these approaches providing one theory that works for all of them. The usual approach has used Palais-Smale sequences. We show that all of them lead to Cerami sequences as well. Applications are given.     

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Numerical treatment of an asset price model with non-stochastic uncertainty

In contrast to stochastic differential equation models used for the calculation of the term structure of interest rates, we develop an approach based on linear dynamical systems under non-stochastic uncertainty with perturbations. The uncertainty is described in terms of known feasible sets of varying parameters. Observations are used in order to estimate these parameters by minimizing the maximum of the absolute value of measurement errors, which leads to a linear or nonlinear semi-infinite programming problem. A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is suggested. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. A special deleting rule permits one to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convexC ¹-functions are studied. On the basis of the solutions of the semi-infinite programming problems a technical trading system for future contracts of the German DAX is suggested and developed. Supported by the Stiftung Rheinland/Pfalz für Innovation, No. 8312-386261/307.     

Tikhonov Functionals Incorporating Tolerances

Tikhonov functionals are a well known method for solving inverse problems. They consist of a discrepancy and a penalty term. The first term evaluates the deviation of simulated data from measured data. We alternate this term by incorporating tolerances, which neglects small deviations from the data within a prescribed tolerance. This approach adapts ideas from support vector regression, which utilizes such a tolerance for identity operators and semi discrete problems. Furthermore, the application for inverse problems is motivated by applications where such tolerances naturally occur, e.g. application with multiple measurements. In this case instead of one measurement a confidence interval for the measurement can be used. In this work we provide an overview on the necessary analysis and alternation of Tikhonov functionals incorporating tolerances. In addition, an example of applications are shown and discussed. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Weighted energy-dissipation functionals for doubly nonlinear evolution

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.     

Efficient numerical methods for entropy-linear programming problems

Entropy-linear programming (ELP) problems arise in various applications. They are usually written as the maximization of entropy (minimization of minus entropy) under affine constraints. In this work, new numerical methods for solving ELP problems are proposed. Sharp estimates for the convergence rates of the proposed methods are established. The approach described applies to a broader class of minimization problems for strongly convex functionals with affine constraints.     

RN上一类拟线性椭圆型方程广义解的多重性

本文在\mathbb{R}^{N上研究一类拟线性椭圆型方程广义解的多重性.借助下半连续泛函的不光滑临界点理论, 得到了方程的解集是无穷和无界的.     

Method of interior variations and existence of S-compact sets

The variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions. The variational approach is based on the relationship between the variation of the equilibrium energy and the equilibrium measure. In all three cases the following result is obtained: for the energy functional and the class of admissible compact sets corresponding to the problem, the arising stationary compact set is fully characterized by a certain symmetry property.     

General Regularizing Functionals for Solving Ill-Posed Problems in Lebesgue Spaces

We study sufficient conditions for general integral functionals in Lebesgue spaces to possess regularizing properties required for solving nonlinear ill-posed problems. We select special classes of such functionals: uniformly convex and quasiuniformly convex (in the extended sense). We give a series of examples and, in particular, a functional that can be used in a generalized version of the maximum entropy method in Lebesgue spaces.     

Existence of three weak solutions for a perturbed anisotropic discrete Dirichlet problem

In this paper, we study the existence of at least three distinct solutions for a perturbed anisotropic discrete Dirichlet problem. Our approach is based on recent variational methods for smooth functionals defined on reflexive Banach spaces. Some examples are presented to demonstrate the application of our main results.     

Semilinear problems in unbounded domains

As formulated by Silva [E.A. de B.e. Silva, Linking theorems and applications to semilinear elliptic problems at resonance, Nonlinear Anal. 16 (1991) 455-477] and Schechter [M. Schechter, A generalization of the saddle point method with applications, Ann. Polon. Math. 57 (3) (1992) 269-281; M. Schechter, New saddle point theorems, in: Generalized Functions and Their Applications, Varanasi, 1991, Plenum, New York, 1993, pp. 213-219], the sandwich theorem has become a very useful tool in finding critical points of functionals leading to solutions of partial differential equations. In the present paper, this theorem is strengthened to apply to more general situations. We present some applications.     

Identification of distributed systems and the theory of the regularization

The identification problems, i.e., the problems of finding unknown parameters in distributed systems from the observations are very important in modern control theory. The solutions of these identification problems can be obtained by solving the equations of the first kind. However, the solutions are often unstable. In other words, they are not continuously dependent on the data. The regularization or Tihonov's regularization is known as one of the stabilizing algorithms to solve these non well-posed problems. In this paper is studied the regularization method for identification of distributed systems. Several approximation theorems are proved to solve the equations of the first kind. Then, identification problems are reduced to the minimization of quadratic cost functionals by virtue of these theorems. On the other hand, it is known that the statistical methods for identification such as the maximum likelihood lead to the minimization problems of certain quadratic functionals. Comparing these quadratic cost functionals, the relations between the regularization and the statistical methods are discussed. Further, numerical examples are given to show the effectiveness of this method.     

Templates for convex cone problems with applications to sparse signal recovery

This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal first-order method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the total-variation norm, ||Wx||₁ where W is arbitrary, or a combination thereof. In addition, the paper introduces a number of technical contributions such as a novel continuation scheme and a novel approach for controlling the step size, and applies results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with state-of-the-art methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient large-scale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms.     

Sandwich pairs in critical point theory

Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional is semibounded, one can find a Palais-Smale (PS) sequence

These sequences produce critical points if they have convergent subsequences (i.e., if satisfies the PS condition). However, there is no clear method of finding critical points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for functionals that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.

     

On the acceleration of the double smoothing technique for unconstrained convex optimization problems

In this article, we investigate the possibilities of accelerating the double smoothing (DS) technique when solving unconstrained nondifferentiable convex optimization problems. This approach relies on the regularization in two steps of the Fenchel dual problem associated with the problem to be solved into an optimization problem having a differentiable strongly convex objective function with Lipschitz continuous gradient. The doubly regularized dual problem is then solved via a fast gradient method. The aim of this article is to show how the properties of the functions in the objective of the primal problem influence the implementation of the DS approach and its rate of convergence. The theoretical results are applied to linear inverse problems by making use of different regularization functionals.     

Locally extra-optimal regularizing algorithms and a posteriori estimates of the accuracy for ill-posed problems with discontinuous solutions

Local a posteriori estimates of the accuracy of approximate solutions to ill-posed inverse problems with discontinuous solutions from the classes of functions of several variables with bounded variations of the Hardy or Giusti type are studied. Unlike global estimates (in the norm), local estimates of accuracy are carried out using certain linear estimation functionals (e.g., using the mean value of the solution on a given fragment of its support). The concept of a locally extra-optimal regularizing algorithm for solving ill-posed inverse problems, which has an optimal in order local a posteriori estimate, was introduced. A method for calculating local a posteriori estimates of accuracy with the use of some distinguished classes of linear functionals for the problems with discontinuous solutions is proposed. For linear inverse problems, the method is bases on solving specialized convex optimization problems. Examples of locally extra-optimal regularizing algorithms and results of numerical experiments on a posteriori estimation of the accuracy of solutions for different linear estimation functionals are presented.     

Iterative processes of Fejér type in ill-posed problems with a priori information

In the latter thirty years, the solution of ill-posed problems with a priori information formed a separate field of research in the theory of ill-posed problems. We mean the class of problems, where along with the basic equation one has some additional data on the desired solution. Namely, one states some relations and constraints which contain important information on the object under consideration. As a rule, taking into account these data in a solution algorithm, one can essentially increase its accuracy for solving ill-posed (unstable) problems. It is especially important in the solution of applied problems in the case when a solution is not unique, because this approach allows one to choose a solution that meets the reality. In this paper we survey the methods for solving such problems. We briefly describe all relevant approaches (known to us), but we pay the main attention to the method proposed by us. This technique is based on the application of iterative processes of Fejér type which admit a flexible and effective realization for a wide class of a priori constraints.     

Enumerative technique for an extreme point fractional program

This paper presents a technique for solving a linear fractional functionals program whose optimum is supposed to occur at one of the extreme points of a convex polyhedron. This polyhedron is defined by a system of linear inequalities with non-negative restrictions on the variables. The approach is a dual method type in the sense that feasibility is achieved only at the optimal solution.     

Existence results for obstacle problems for nonlinear hemivariational inequality at resonance

In this paper, we are interested in studying the existence of solutions to obstacle problems for nonlinear hemivariational inequality at resonance driven by the p$p$-Laplacian. Using a variational approach based on the nonsmooth critical point theory for nondifferentiable functionals. We prove two existence theorems.