Semicontinuity Research Articles

Probabilistic representation of the parabolic stochastic variational inequality with Dirichlet–Neumann boundary and variational generalized backward doubly stochastic differential equations

We derive the existence and uniqueness of the generalized backward doubly stochastic differential equation with subdifferential of a lower semicontinuous convex function under a non-Lipschitz condition. This study allows us to give a probabilistic representation (in the stochastic viscosity sense) to the parabolic variational stochastic partial differential equations with Dirichlet–Neumann conditions.

An accelerated mirror descent algorithm for constrained nonconvex problems

This paper considers the composite optimization problem where the sum of a smooth nonconvex function and a proper lower semicontinuous function has to be minimized over a closed convex set. The mirror descent algorithm and its acceleration variants can be considered as a series of popular methods for solving this problem. Meanwhile, some recent contributions try to circumvent the global Lipschitz condition of smooth part in objective, and establish their global convergence by replacing it with a local one. But, to the best of our knowledge, they still require some extra conditions, including but not limited to the assumption that the abstract set is the whole space. The aim of this paper is to design an accelerated mirror descent algorithm, which uses a convex combination of essentially the whole iteration trajectory, and to show that the local Lipschitz assumption of associated Legendre function together with the Kurdyka-Łojasiewicz property is sufficient to recover its convergence results. In addition to the theoretical improvement in the convergence analysis, it also showcases possible computational advantages which provides an interesting option for practical problems.

LP well-posedness for multidimensional bilevel controlled variational inequalities

In this paper, we introduce a new class of multidimensional bilevel controlled variational inequality problems (BVIP) and analyse their LP well-posedness and LP well-posedness in generalized sense by using the monotonicity and hemicontinuity of the involved functionals. We also derive a sufficient condition for the generalized LP well-posedness by considering the nonemptiness and boundedness of the approximating solution set. After that, we show the generalized LP wellposedness with the help of the upper semicontinuity of the approximating solution set of the problem (BVIP). Additionally, some examples are constructed to demonstrate the theoretical results.

The upper semicontinuity and Hölder stability of optimal solution sets for semidefinite programming

We consider the qualitative and quantitative stability of parametric semidefinite programming problems which are more general than C 2 -smooth parameterization. We do not require the differentiability of problem functions. We show that under metric regularity and Lipschitz-like property of problem functions, the optimal solution set and optimal value function of a restricted perturbed problem are upper semicontinuous, and the modulus of upper semicontinuity for the solution set can be described by a growth function. In addition, we prove that the 1/ k -order Hölder stability of the restricted optimal solution set can be obtained by adding the k -order growth condition of the unperturbed problem and illustrating its applications with examples. Especially, when k = 1, the Hölder stability is just the calmness.

Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data

AbstractWe establish lower semicontinuity results for perimeter functionals with measure data on $${\mathbb {R}}^n$$ R n and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In other words, we lay foundations of a perimeter-based variational approach to mean curvature measures on $${\mathbb {R}}^n$$ R n capable of proving existence in various prescribed-mean-curvature problems with measure data. As crucial and essentially optimal assumption on the measure data we identify a new condition, called small-volume isoperimetric condition, which sharply captures cancellation effects and comes with surprisingly many properties and reformulations in itself. In particular, we show that the small-volume isoperimetric condition is satisfied for a wide class of $$(n{-}1)$$ ( n - 1 ) -dimensional measures, which are thus admissible in our theory. Our analysis includes infinite measures and semicontinuity results on very general domains.

Fixed Point Results in Archimedean Type Neutrosophic b-Metric Spaces

This paper defines Archimedean, Caristi-Krirk balls, and Neutrosophic b metric space (Nb-MS). Usingthe Archimedean idea and complete Nb-MS, we have shown the existence of a shared fixed point usingtwo self-mappings and an upper semicontinuous function. We have demonstrated the existence of a fixed point by employing a k-continuous self-map and an upper semi-continuous function in conjunction with the Archimedean notion and complete Nb-MS. Furthermore, we have proven the completeness of the space by using Archimedean, Nb-MS, and a k-continuous self-map.MSC 2010: 47H10.

Lower semi-continuity of Lagrangian volume

AbstractWe study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and γ-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is γ-lower semi-continuous in two cases. In the first one the volume form comes from a Kähler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous.

Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains

Abstract This study deals with the limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays and multiplicative noise on unbounded thin domains. We first prove the existence and uniqueness of pullback tempered random attractors for the systems and then establish the upper semicontinuity of these attractors when the thin domains collapse onto R {\mathbb{R}} .

Convergence of bi-spatial pullback random attractors and stochastic Liouville type equations for nonautonomous stochastic p-Laplacian lattice system

We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic p-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity of pullback random (ℓ2, ℓq)-attractor is proved for each q ∈ [1, +∞). Then, a convergence result of the time-dependent invariant sample Borel probability measures is obtained in ℓ2. Next, we show that the invariant sample measures satisfy a stochastic Liouville type equation and a termwise convergence of the stochastic Liouville type equations is verified. Furthermore, each family of the invariant sample measures is turned out to be a sample statistical solution, which hence also fulfills a convergence consequence.

The first exit time of fractional Brownian motion from an unbounded domain

Consider a fractional Brownian motions starting at the interior point x0,h‖x0‖+2K∈Rd+1 with the constant K>1, for some fixed x0∈Rd, of an unbounded domain D=x,y∈Rd+1:y>h‖x‖, The function h is a nondecreasing, lower semicontinuous, and convex function on [0,∞) with h(0) being finite. Here we take h−1x=Axαlogxβwith a positive constant A for x>K. It is evident that h−1(x) exhibits monotonic behavior for sufficiently large values of x. Let τD denote the first time that the fractional Brownian motion exits from D. In most cases, we give the asymptotically equivalent estimate of logPτD>t. The proof methods are based on the earlier works of Li, Shi, Lifshits, and Aurzada.

Lower semicontinuity of monotone functionals in the mixed topology on Cb

The main result of this paper characterises the continuity from below of monotone functionals on the space Cb of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair (Cb,ca), where ca denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps Cb→R equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on L∞ is transferred to risk measures on Cb. In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on Cb taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets.

Fast convergence of the primal-dual dynamical system and corresponding algorithms for a nonsmooth bilinearly coupled saddle point problem

AbstractThis paper studies the convergence rate of a second-order dynamical system associated with a nonsmooth bilinearly coupled convex-concave saddle point problem, as well as the convergence rate of its corresponding discretizations. We derive the convergence rate of the primal-dual gap for the second-order dynamical system with asymptotically vanishing damping term. Based on an implicit discretization scheme, we propose a primal-dual algorithm and provide a non-ergodic convergence rate under a general setting for the inertial parameters when one objective function is continuously differentiable and convex and the other is a proper, convex and lower semicontinuous function. For this algorithm we derive a $$O\left( 1/k^2 \right) $$ O 1 / k 2 convergence rate under three classical rules proposed by Nesterov, Chambolle-Dossal and Attouch-Cabot without assuming strong convexity, which is compatible with the results of the continuous-time dynamic system. For the case when both objective functions are continuously differentiable and convex, we further present a primal-dual algorithm based on an explicit discretization. We provide a corresponding non-ergodic convergence rate for this algorithm and show that the sequence of iterates generated weakly converges to a primal-dual optimal solution. Finally, we present numerical experiments that indicate the superior numerical performance of both algorithms.

S STAR p STAR Continuous Maps in Topological Spaces

Goal of this present study seeks to discover and to study the different kind of continuous map and irresolute map is known as semi star pre star continuous map (briefly s*p*continuous) and semi star pre star irresolute map (briefly s*p* irresolute) in topological spaces. As well as discuss a few fundamental characteristics of this continuous map. Further investigate the relationship between the newly defined map and the existing continuous map and irresolute map with suitable examples.

Splitting Method for Support Vector Machine in Reproducing Kernel Banach Space with a Lower Semi-continuous Loss Function

Splitting Method for Support Vector Machine in Reproducing Kernel Banach Space with a Lower Semi-continuous Loss Function

Continuity of the solution set mappings to parametric unified weak vector equilibrium problems via free-disposal sets

In this paper, the lower and upper semicontinuity of the solution mappings for parametric unified weak vector equilibrium problem (PUWVEP) are established, via free-disposal set and non-linear scalarization function. Moreover, example is given to illustrate the obtained results. The results improve the corresponding ones in the literature [Z.Y. Peng and S.S. Chang, Optim. Lett. (2014) 8 159–169.], [Z.Y. Peng, X.B. Li, X.J. Long and X.D. Fan, Optim. Lett. (2018) 12 1339–1356.], [Z.Y. Peng, J.W. Peng, X.J. Long and J.C. Yao, J. Global Optim. 70 (2018) 55–69.].

Spatially adaptive oscillation total generalized variation for image restoration with structured textures

Cartoon and texture are the main components of an image, and decomposing them has gained much attention in various image restoration tasks. In this paper, we propose a novel infimal convolution type model based on total generalized variation (TGV) and spatially adaptive oscillation TGV to address cartoon-texture restoration problems. The proposed spatially adaptive oscillation TGV regularizer is capable of capturing structured textures with different orientations and frequencies in localized regions. Additionally, we incorporate the second-order tensor TGV to regularize the orientation and frequency parameter. The lower semicontinuity of the new functional is established and the existence of the solutions to the proposed model is analyzed. Furthermore, we discuss suitable discretizations of the proposed model, and introduce the alternating minimization algorithm where each subproblem can be implemented by the primal-dual method. Numerical experiments on image decomposition, denoising and inpainting demonstrate that the proposed model excels in preserving textures and is competitive against existing variational and learning-based models.

On semi-continuity and continuity of solution maps of parametric generalized multiobjective games in fuzzy environments

The purpose of this article is to study new results on the continuity of solution maps for parametric generalized multiobjective games in fuzzy environments. Firstly, we revisit parametric fuzzy generalized multiobjective games. Secondly, we establish some sufficient conditions for upper semi-continuity, Hausdorff upper semi-continuity, closedness and compactness of solution maps for such problem under suitable conditions. Thirdly, we introduce the auxiliary problem of parametric fuzzy generalized multiobjective games and the concept of strong Ki-convexity of objective functions for these problems. Finally, results on the lower semi-continuity, Hausdorff lower semi-continuity, continuity and Hausdorff continuity of solution maps to parametric generalized multiobjective games in fuzzy environments are established and studied. Many examples are given to illustrate our results.

Wong-Zakai approximations and random attractors for nonlocal stochastic Schrödinger lattice systems in weighted spaces

This paper deals with the long-term behavior of nonlocal Schrödinger lattice systems with multiplicative white noise and their Wong-Zakai approximate systems in weighted spaces. We first prove the existence and uniqueness of tempered pullback attractors for the original stochastic systems and the Wong-Zakai approximations. Then, we establish the upper semicontinuity of these attractors for the Wong-Zakai approximate systems as the step length of the Wiener shift approaches zero.

Asymptotic behavior of non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by linear mixed white noise

This paper is concerned with the asymptotic behavior of the non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by the linear mixed white noise, which simultaneously contains linear additive noise and multiplicative noise. For the sake of the long-term behavior of the system we considered, we need to utilize a different Ornstein–Uhlenbeck transformation than the general one. First, the existence and uniqueness of pullback random attractors are demonstrated. Then, we prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.

Analysis and evaluation of the effectiveness of program codes for particle swarm and ant colony methods for lower semi-continuous functions

In this paper, a comparative analysis of two stochastic methods of global optimization is carried out: the Particle Swarm Optimization method and the Ant Colony Optimization method. The analysis and evaluation of efficiency were carried out using parameters such as the number of calls to the objective function, the rate of convergence, resources, computational efficiency, sensitivity to parameters, and resistance to initial conditions. The article presents the program codes of these methods in the Python environment, designed to calculate the global extremum of semi-continuous functions from below. Well- known test functions were used to construct (by gluing) semi-continuous functions at different values of input parameters.