Lower bounds to variational problems with guarantees

Well-posedness for multi-time variational inequality problems via generalized monotonicity and for variational problems with multi-time variational inequality constraints

This paper investigates the boundedness and stability of solutions to the affine variational inequality problem. The concept of a solution ray to a variational inequality defined by an affine, mapping and on a closed convex set is introduced and characterized; the connection of such a ray with the boundedness of the solution set of the given problem is explained. In the case of the monotone affine variational inequality, a complete description of the solution set is obtained which leads to a simplified characterization of the boundedness of this set as well as to a new error bound result for approximate solutions to such a variational problem. The boundedness results are then combined with certain degree- theoretic arguments to establish the stability of the solution set of an affine variational inequality problem.

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

Geometric finite elements (GFE) generalize the idea of Galerkin methods to variational problems for unknowns that map into nonlinear spaces. In particular, GFE methods introduce proper discrete function spaces that are conforming in the sense that values of geometric finite element functions are in the codomain manifold \(\mathcal {M}\) at any point. Several types of such spaces have been constructed, and some are even completely intrinsic, i.e., they can be defined without any surrounding space. GFE spaces enable the elegant numerical treatment of variational problems posed in Sobolev spaces with nonlinear codomain space. Indeed, as GFE spaces are geometrically conforming, such variational problems have natural formulations in GFE spaces. These correspond to the discrete formulations of classical finite element methods. Also, the canonical projection onto the discrete maps commutes with the differential for a suitable notion of the tangent bundle as a manifold, and we therefore also obtain natural weak formulations. Rigorous results exist that show the optimal behavior of the a priori L2 and H1 errors under reasonable smoothness assumptions. Although the discrete function spaces are no vector spaces, their elements can nevertheless be described by sets of coefficients, which live in the codomain manifold. Variational discrete problems can then be reformulated as algebraic minimization problems on the set of coefficients. These algebraic problems can be solved by established methods of manifold optimization. This text will explain the construction of several types of GFE spaces, discuss the corresponding test function spaces, and sketch the a priori error theory. It will also show computations of the harmonic maps problem, and of two example problems from nanomagnetics and plate mechanics.

In this paper, new results, which exhibit some new applications for Mordukhovich's subdifferential in nonsmooth optimization and variational problems, are established. Nonsmooth (fractional) multiobjective optimization problems in special Banach spaces are studied, and some necessary and sufficient conditions for weak Pareto-optimality for these problems are introduced. Through this work, we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-$(p,r)$-invexity. Some optimality conditions regarding the generalized KT-$(p,r)$-invexity notion and Kuhn-Tucker points are provided. Also, we seek a connection between linear (semi-) infinite programming and nonlinear programming. Some sufficient conditions for (proper) optimality under invexity are provided. A nonsmooth variational problem corresponding to a considered multiobjective problem is defined and the relations between the provided variational problem and the considered optimization problem are studied. The final part of the paper is devoted to illustrating a penalization mechanism, using the distance function as a tool, to provide some conditions to the solutions of the nonsmooth variational inequality problems. All results of the paper have been established in the absence of gradient vectors, using the properties of Mordukhovich's subdifferential in Asplund spaces.

On integral principles for nonholonomic systems

Weak Sharp Solutions of Variational Inequalities

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter is studied. The problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms depending nonlinearly on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues that correspond to a normalized system of eigenfunctions. The variational problem is approximated by a finite element mesh scheme on a uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are proved depending on the mesh size and the eigenvalue size. The results obtained are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

In this paper, a new approach to analyze optimality and saddle‐point criteria for a new class of nonconvex variational problems involving invex functions is studied. Namely, the modified objective function method is used for the considered variational problem in order to characterize its optimal solution. In this method, for the considered variational problem, corresponding modified variational problem with the η‐objective function is constructed. The equivalence in the original variational problem and its associated modified variational problem with the η‐objective function is proved under invexity hypotheses. Furthermore, by using the notion of a Lagrangian function, the connection between a saddle‐point in the modified objective function variational problem and an optimal solution in the considered variational problem is presented. Some examples of nonconvex variational problems are also given to verify the results established in the paper.

So far studies of melodic variation in folk song have been few. Contributions of a positive nature have, on the whole, come rather indirectly, in several instances as a result of indexing melodies, or as a by-product of stylistic analysis. When the problem of variation itself is taken up directly a satisfactory methodology may develop. Something of this kind has been achieved in the Deutsche Volkslieder mit ihren Melodien published by the Deutsches Volksliedarchiv.

In this paper, we propose a new class of variational inequality problems, say, uncertain variational inequality problems based on uncertainty theory in finite Euclidean spaces $R^{n}$ . It can be viewed as another extension of classical variational inequality problems besides stochastic variational inequality problems. Note that both stochastic variational inequality problems and uncertain variational inequality problems involve uncertainty in the real world, thus they have no conceptual solutions. Hence, in order to solve uncertain variational inequality problems, we introduce the expected value of uncertain variables (vector). Then we convert it into a classical deterministic variational inequality problem, which can be solved by many algorithms that are developed on the basis of gap functions. Thus the core of this paper is to discuss under what conditions we can convert the expected value model of uncertain variational inequality problems into deterministic variational inequality problems. Finally, as an application, we present an example in a noncooperation game from economics.

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.

We study the -convergence as of a family of integral functionals with integrand , where the integrand oscillates with respect to the space variable . The integrands satisfy a two-sided power estimate on the coercivity and growth with different exponents. As a consequence, at least two different variational Dirichlet problems can be connected with the same functional. This phenomenon is called Lavrent'ev's effect. We introduce two versions of -convergence corresponding to variational problems of the first and second kind. We find the -limit for the aforementioned family of functionals for problems of both kinds; these may be different. We prove that the -convergence of functionals goes along with the convergence of the energies and minimizers of the variational problems. Bibliography: 23 titles.

The variational problem of the form of bodies with minimum drag for given lift force, volume, and other constraints in general leads to a second-order partial differential equation even for the simplest methods of drag calculation (Newton law and averaged friction coefficient). The solution of this equation is not justified; in its place an approximate solution is suggested which consists of: a) selection of a “scheme” characterized by certain parameters which are determined from the solution of the extremal problem, b) determination of the optimal surface form for the selected “scheme” with the aid of the system of ordinary Euler equations. This paper presents a comparison of the body “schemes” with minimum drag and maximum L/D and presents the solution of several variational problems. At the present time we have quite complete information on the form of minimum-drag bodies for zero lift (nonlifting bodies), and both approximate and quite rigorous methods are known for solving the corresponding variational problems. This cannot be said at all of the form of lifting bodies, for which the requirements are numerous, differing essentially for vehicles of different application, and are generally not limited to a single flight regime. Account for all the mandatory requirements in solving the variational problems is not possible; therefore in the majority of cases these solutions do not yield answers which are directly suitable in practice; rather they yield limiting estimates. The natural tendency to utilize for lifting bodies the axisymmetric form which is customary for nonlifting bodies leads to the study of axisymmetric bodies at angle of attack, axisymmetric bodies with skewed base, sections of axisymmetric bodies cut by planes, etc. In order to obtain a broader view of the optimum forms of lifting bodies we must, obviously, drop the limitation to axisymmetric bodies and bodies with similar cross sections. However, in the case of an arbitrary extremal surface the Euler equation is a second-order partial differential equation, and its simple solution is difficult. In practice it seems wise to solve those variational problems whose Euler equation may be reduced to a system of ordinary differential equations. Therefore, we propose the following method for selecting the optimum forms: a) we select a scheme, a form, which is formed by a set of planes and cylindrical, conical, spherical surfaces and which is defined by parameters that are found from the solution of the extremal problem; b) for the selected “scheme” the generators of the “scheme” surface are found from the solution of the variational problem. For the calculation of the air pressure on the body surface we use the empirical Newton law, which yields in the majority of cases results which are very close to the results of the more rigorous methods. It is assumed that the pressure may vanish only at the trailing edge of the body. The frictional drag coefficient, averaged over the body surface, is assumed to be independent of the body shape. In the case of a body of simple form the hottest portion is the frontal portion and account for the thermal protection requirements reduces to the selection of suitable dimensions of this portion of the body. In the general case the problem is stated as follows: find the form of the minimum-drag body for a given lift force, volume, length, and other conditions. To the particular case of the body with maximum L/D corresponds the value of the Lagrange multiplier λ=−1/k. All the results of calculations presented in the paper are intended only to illustrate the method. After the present paper was submitted for publication, another study [3] appeared which also proposes a method for determining the optimal parameters.

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.