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In this paper, we present a thorough study of the regularized submodular maximization problem, in which the objective $ f:=g-\ell $ f := g − ℓ can be expressed as the difference between a submodular function and a modular function. This problem has drawn much attention in recent years. While existing works focuses on the case of g being monotone, we investigate the problem with a nonmonotone g. The main technique we use is to introduce a distorted objective function, which varies weights of the submodular component g and the modular component ℓ during the iterations of the algorithm. By combining the weighting technique and measured continuous greedy algorithm, we present an algorithm for the matroid-constrained problem, which has a provable approximation guarantee. In the cardinality-constrained case, we utilize random greedy algorithm and sampling technique together with the weighting technique to design two efficient algorithms. Moreover, we consider the unconstrained problem and propose a much simpler and faster algorithm compared with the algorithms for solving the problem with a cardinality constraint. [ABSTRACT FROM AUTHOR]

The present paper deals with new results in the study of a class of elliptic quasivariational inequalities in Hilbert spaces. We present a regularization method which consist to consider a sequence of regularized elliptic quasivariational inequalities and prove a convergence result when the parameter of regularization is very small using arguments of monotonicity, convexity, and lower semicontinuity. The mathematical tools developed in this paper are useful in the analysis of a large class of contact problems which lead, in a weak formulation, to elliptic quasivariational inequalities. To provide an example, we illustrate our results in the study of a nonlinear antiplane problem which models the deformation of an antiplane frictional contact for elastic body. The contact is modeled with the static version of the slip-dependent Coulomb's law. [ABSTRACT FROM AUTHOR]

In this article, we focus on the Tikhonov–Browder regularization scheme for both weak and strong vector equilibrium problems. Under suitable coercive conditions, the existence of solutions to weak vector equilibrium problems and their regularized problems are established in both monotone and nonmonotone cases, and furthermore we show that the conditions, which guarantee the boundedness of regularized solutions, become sufficient conditions for the solvability of the original problems. For the strong problems, we use generalized convexities and/or coercivity properties concerning a vector optimization problem to establish convergence results of the regularization method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a frictional contact problem between an elastic body and a foundation. We focus on the optimal control of the model which consists of acting with a control on a portion of the boundary of the body and leading the stress tensor as close as possible to a given target. We state an optimal control problem for which we establish an existence theorem of the solution. We then introduce a sequence of regularized problems depending on a positive parameter α and we study the convergence of the sequence of the solutions when the parameter α tends to zero. Finally, an optimality condition is established for this problem.

In this work, we give a simple approach to a priori estimates for the singular parabolic p(x,t)-Laplace Dirichlet problem in domain Χ with the boundary ∂Χ ∈ C1+β with β ∈ (0, 1). [ABSTRACT FROM AUTHOR]

In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.

The paper surveys articles that construct and investigate direct and iterative methods for computing weighted pseudoinverses and weighted normal pseudosolutions with singular weights. The methods considered in the paper are mainly constructed based on the authors’ articles devoted to the development of the theory of weighted pseudoinversion aimed at investigating the characteristics of both weighted pseudoinverses and weighted normal pseudosolutions with singular weights. The paper uses the following results obtained and investigated by the authors: expansions of weighted pseudoinverses into matrix power series and products, limit representations of such matrices, and determination of decompositions of weighted pseudoinverses based on weighted singular value decompositions of matrices with singular weights. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of an elastic body which is in frictional contact with a foundation on a part of the boundary, and which can come into contact with a rigid obstacle on another part of the boundary. We associate this physical setting with two mechanical models. Every model is mathematically described by a boundary value problem which consists of a system of partial differential equations associated with a displacement condition, a traction condition, a frictional contact condition and a frictionless unilateral contact condition. In both models the unilateral contact is described by Signorini’s condition with non-zero gap. The difference between the models is given by the frictional condition we use. In the first model we use a condition with prescribed normal stress. In the second one, we use a frictional bilateral contact condition. The weak solvability of the boundary value problems we propose herein relies on an abstract generalized saddle point problem. Abstract existence, uniqueness and boundedness results as well as abstract convergence results of a regularization are established. Then, we discuss the existence, the uniqueness, the boundedness and the approximation of the weak solutions based on the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we consider local Dirichlet problems driven by the (r (u) , s (u)) -Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r , s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument. [ABSTRACT FROM AUTHOR]