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In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results.

Fondecyt Postdoc Project 3180080 Basal Program from CONICYT-Chile CMM-AFB 170001 National Foundation for Science & Technology Development (NAFOSTED) 101.01-2017.325

In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an $$R_0$$ tensor. We focus on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the min function, the Fischer–Burmeister (FB) function and the penalized FB function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.

Using a short proof, we show that every set function f can be decomposed into the difference of two monotone increasing and strictly submodular functions g and h, i.e., $$f=g-h$$ , and every set function f can also be decomposed into the difference of two monotone increasing and strictly supermodular functions g and h, i.e., $$f=g-h$$ .

This paper proposes an inertial Bregman proximal gradient method for minimizing the sum of two possibly nonconvex functions. This method includes two different inertial steps and adopts the Bregman regularization in solving the subproblem. Under some general parameter constraints, we prove the subsequential convergence that each generated sequence converges to the stationary point of the considered problem. To overcome the parameter constraints, we further propose a nonmonotone line search strategy to make the parameter selections more flexible. The subsequential convergence of the proposed method with line search is established. When the line search is monotone, we prove the stronger global convergence and linear convergence rate under Kurdyka–Łojasiewicz framework. Moreover, numerical results on SCAD and MCP nonconvex penalty problems are reported to demonstrate the effectiveness and superiority of the proposed methods and line search strategy.

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular value. As an application we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions., 27 pages

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature., Comment: 16 pages, comments are welcomed! The paper is revised according to the comments of the journal referees, and some references are added. Accepted by The Journal of Geometric Analysis

Let E be a real reflexive Banach space with its dual $$E^*$$ and f be a proper, convex and lower-semi-continuous function on E. The purpose of this paper is to introduce and study a new class of mappings from E into $$E^*$$ called f-pseudocontractive mappings with the notion of f-fixed points. In the case that E is a real reflexive Banach space and f is a strongly coercive, bounded and uniformly Frechet differentiable Legendre function which is strongly convex on bounded subsets of E, a sequence is constructed which converges strongly to a common f-fixed point of two f-pseudocontractive mappings. As a consequence, we obtain a scheme which converges strongly to a common zero of monotone mappings. Furthermore, this analog is applied to approximate solutions to convex optimization problems. Our results improve and generalize many of the results in the literature.

The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms.

In this article, we consider the problem of finding zeros of monotone inclusions of three operators in real Hilbert spaces, where the first operator’s inverse is strongly monotone and the third is linearly composed, and we suggest an extended splitting method. This method allows relative errors and is capable of decoupling the third operator from linear composition operator well. At each iteration, the first operator can be processed with just a single forward step, and the other two need individual computations of the resolvents. If the first operator vanishes and linear composition operator is the identity one, then it coincides with a known method. Under the weakest possible conditions, we prove its weak convergence of the generated primal sequence of the iterates by developing a more self-contained and less convoluted techniques. Our suggested method contains one parameter. When it is taken to be either zero or two, our suggested method has interesting relations to existing methods. Furthermore, we did numerical experiments to confirm its efficiency and robustness, compared with other state-of-the-art methods.

In this paper, a full-Newton step Interior-Point Method for solving monotone Weighted Linear Complementarity Problem is designed and analyzed. This problem has been introduced recently as a generalization of the Linear Complementarity Problem with modified complementarity equation, where zero on the right-hand side is replaced with the nonnegative weight vector. With a zero weight vector, the problem reduces to a linear complementarity problem. The importance of Weighted Linear Complementarity Problem lies in the fact that it can be used for modelling a large class of problems from science, engineering and economics. Because the algorithm takes only full-Newton steps, the calculation of the step size is avoided. Under a suitable condition, the algorithm has a quadratic rate of convergence to the target point on the central path. The iteration bound for the algorithm coincides with the best iteration bound obtained for these types of problems.

In this paper, we propose two conjugate gradient methods for solving large-scale monotone nonlinear equations. The methods are developed by combining the hyperplane projection method by Solodov and Svaiter (Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Springer, pp 355–369, 1998) and two modified search directions of the famous Dai and Liao (Appl Math Optim 43(1): 87–101, 2001) method. It is shown that the proposed schemes satisfy the sufficient descent condition. The global convergence of the methods are established under mild conditions, and computational experiments on some benchmark test problems show that the methods are promising.

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $$L_\omega ^p L_t^\infty \dot{H}^{1+\gamma }$$ -norm and a temporal Holder regularity under the $$L_\omega ^p L_x^2$$ -norm for the solution of the proposed equation with an $$\dot{H}^{1+\gamma }$$ -valued initial datum for $$\gamma \in [0,1]$$ . Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $${\mathscr {O}}(h^{1+\gamma }+\tau ^{1/2})$$ and $${\mathscr {O}}(h^{1+\gamma }+\tau ^{(1+\gamma )/2})$$ for the Galerkin-based Euler and Milstein schemes, respectively.

In this paper, we introduce some new approximation projection algorithms for solving monotone equilibrium problems over the intersection of fixed point sets of demicontractive mappings. By combining subgradient projection methods and hybrid steepest descent methods, strong convergence of the algorithms to a solution is shown in a real Hilbert space. Some numerical illustrations and comparisons are also reported to show the efficiency and advantage of the proposed algorithms.

This paper aims to obtain a strong convergence result for a Douglas–Rachford splitting method with inertial extrapolation step for finding a zero of the sum of two set-valued maximal monotone operators without any further assumption of uniform monotonicity on any of the involved maximal monotone operators. Furthermore, our proposed method is easy to implement and the inertial factor in our proposed method is a natural choice. Our method of proof is of independent interest. Finally, some numerical implementations are given to confirm the theoretical analysis.

Let $$10$$ and the Fourier series of $$\frac{1}{f}$$ is p-th power $$\nu$$ -absolutely convergent. If $$\varphi$$ is holomorphic on the range of f, then there exists an almost monotone algebra weight $$\chi$$ on $${\mathbb {Z}}$$ such that $$\chi \le K\omega$$ for some constant $$K>0$$ and the Fourier series of $$\varphi \circ f$$ is p-th power $$\chi$$ -absolutely convergent. This gives p-th power analogue of a Theorem by Bhatt and Dedania; and rectifies Theorems of Kinani and Bouchikhi.

Due to the ubiquitous presence of missing values (MVs) in real-world datasets, the MV imputation problem, aiming to recover MVs, is an important and fundamental data preprocessing step for various data analytics and mining tasks to effectively achieve good performance. To impute MVs, a typical idea is to explore the correlations amongst the attributes of the data. However, those correlations are usually complex and thus difficult to identify. Accordingly, we develop a new deep learning model called MIssing Data Imputation denoising Autoencoder (MIDIA) that effectively imputes the MVs in a given dataset by exploring non-linear correlations between missing values and non-missing values. Additionally, by considering various data missing patterns, we propose two effective MV imputation approaches based on the proposed MIDIA model, namely MIDIA-Sequential and MIDIA-Batch. MIDIA-Sequential imputes the MVs attribute-by-attribute sequentially by training an independent MIDIA model for each incomplete attribute. By contrast, MIDIA-Batch imputes the MVs in one batch by training a uniform MIDIA model. Finally, we evaluate the proposed approaches by experimentation in comparison with existing MV imputation algorithms. The experimental results demonstrate that both MIDIA-Sequential and MIDIA-Batch achieve significantly higher imputation accuracy compared with existing solutions, and the proposed approaches are capable of handling various data missing patterns and data types. Specifically, MIDIA-Sequential performs better than MIDIA-Batch for data with monotone missing pattern, while MIDIA-Batch performs better than MIDIA-Sequential for data with general missing pattern.

The paper concerns with the two numerical methods for approximating solutions of a monotone and Lipschitz variational inequality problem in a Hilbert space. We here describe how to incorporate regularization terms in the projection method, and then establish the strong convergence of the resulting methods under certain conditions imposed on regularization parameters. The new methods work in both cases of with or without knowing previously the Lipschitz constant of cost operator. Using the regularization aims mainly to obtain the strong convergence of the methods which is different to the known hybrid projection or viscosity-type methods. The effectiveness of the new methods over existing ones is also illustrated by several numerical experiments.

In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem.

The chromatic orthogonal art gallery problem is a well-known problem in the computational geometry. Two points in an orthogonal polygon P see each other if there is an axis-aligned rectangle inside P contains them. An orthogonal guarding of P is k-colorable, if there is an assignment between k colors and the guards such that the visibility regions of every two guards in the same color have no intersection. The purposes of this paper are discussing the time complexity of k-colorability of orthogonal guarding and providing algorithms for the chromatic orthogonal art gallery problem. The correctness of presented solutions is proved, mathematically. Herein, the heuristic method is used that leads us to an innovative reduction, some optimal and one approximation algorithms. The paper shows that deciding k-colorability of orthogonal guarding for P is NP-complete. First, we prove that deciding 2-colorability of P is NP-complete. It is proved by a reduction from planar monotone rectilinear 3-SAT problem. After that, a reduction from graph coloring implies this is true for every fixed integer $$k\ge 2$$ . In the third step, we present a 6-approximation algorithm for every orthogonal simple polygon. Also, an exact algorithm is provided for histogram polygons that finds the minimum chromatic number.

Symbolic data is aggregated from bigger traditional datasets in order to hide entry specific details and to enable analysing large amounts of data, like big data, which would otherwise not be possible. Symbolic data may appear in many different but complex forms like intervals and histograms. Identifying patterns and finding similarities between objects is one of the most fundamental tasks of data mining. In order to accurately cluster these sophisticated data types, usual methods are not enough. Throughout the years different approaches have been proposed but they mainly concentrate on the “macroscopic” similarities between objects. Distributional data, for example symbolic data, has been aggregated from sets of large data and thus even the smallest microscopic differences and similarities become extremely important. In this paper a method is proposed for clustering distributional data based on these microscopic similarities by using quantile values. Having multiple points for comparison enables to identify similarities in small sections of distribution while producing more adequate hierarchical concepts. Proposed algorithm, called microscopic hierarchical conceptual clustering, has a monotone property and has been found to produce more adequate conceptual clusters during experimentation. Furthermore, thanks to the usage of quantiles, this algorithm allows us to compare different types of symbolic data easily without any additional complexity.

We study the pointwise partial ordering of representative functions for a monotone operator and in particular we focus on the bigger conjugate representative functions that represent a fixed initial (non-maximal) monotone operator. The first problem considered is that of constructing a new representative function from a given member of this class when wanting to add an additional monotonically related point. This study allows us to prove that all bigger conjugate representable monotone sets are monotonically closed. This result sheds light on the structure of the domains for maximal monotone operators and enables us to study the sum theorem for FPV operators in Banach spaces which posses a dual space that has a strictly convex renorm.

Submodular optimization has been well studied in combinatorial optimization. However, there are few works considering about non-submodular optimization problems which also have many applications, such as experimental design, some optimization problems in social networks, etc. In this paper, we consider the maximization of non-submodular function under a knapsack constraint, and explore the performance of the greedy algorithm, which is characterized by the submodularity ratio $$\beta $$ and curvature $$\alpha $$ . In particular, we prove that the greedy algorithm enjoys a tight approximation guarantee of $$ (1-e^{-\alpha \beta })/{\alpha }$$ for the above problem. To our knowledge, it is the first tight constant factor for this problem. We further utilize illustrative examples to demonstrate the performance of our algorithm.

We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen's theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. As an application of our result in the setting of $II_1$-factors, we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of Vidal, thus yielding a new way to quantify entanglement in that context. Building on previous work in the infinite-dimensional setting, we show that trace vectors play the role of maximally entangled states for general $II_1$-factors. Examples are drawn from infinite spin chains, quasi-free representations of the CAR, and discretised versions of the CCR., 36 pages, v2: journal version, 38 pages

In this paper, we study nonlinear matrix equations $$\begin{aligned} X^p=A+\sum \limits _{i=1}^m M_i^T(X\#B)M_i \end{aligned}$$ and $$\begin{aligned} X^p=A+\sum \limits _{i=1}^j M_i^T(X\#B)M_i+\sum \limits _{i=j+1}^m M_i^T(X^{-1}\#B)M_i, \end{aligned}$$ where p, m, j are positive integers, $$1\le j\le m$$ , A, B are $$n\times n$$ positive definite matrices and $$M_i(i=1,2,3,\ldots ,m)$$ are $$n\times n$$ nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method.

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises., Comment: This paper generalizes the idea in our NOT published paper arXiv:1510.03522. There is a substantial overlap with arXiv:1510.03522

By introducing for monotone recurrence relations pseudo solutions, which are analogues of pseudo orbits of dynamical systems, we show that for general monotone recurrence relations the rotation set is closed, and each element in the rotation set is realized by a Birkhoff orbit. Moreover, if there is an orbit without rotation number, then the system has positive topological entropy, and we can construct orbits shadowing different rotation numbers.

Let $$\sigma$$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $$OM_+^1$$ be the set of normalized positive operator monotone functions on $$(0, \infty )$$ . In this paper, we study the class of $$\sigma$$ -subpreserving functions $$f\in OM_+^1$$ satisfying $$\begin{aligned} f(A\sigma B) \le f(A)\sigma f(B) \end{aligned}$$ for all invertible positive operators A and B. We provide some criteria for f to be trivial, i.e., $$f(t)=1$$ or $$f(t)=t$$ . We also establish characterizations of $$\sigma$$ -preserving functions $$f\in OM_+^1$$ satisfying $$\begin{aligned} f(A\sigma B) = f(A)\sigma f(B) \end{aligned}$$ for all invertible positive operators A and B. In particular, when $$\lim _{t\rightarrow 0} (1\sigma t) =0$$ , the function $$f\in OM_+^1\backslash \{1,t\}$$ preserves $$\sigma$$ if and only if f and $$1\sigma t$$ are representing functions for a weighted harmonic mean.

In this paper, we propose a new sequential quadratic optimization algorithm for solving two-block nonconvex optimization with linear equality and generalized box constraints. First, the idea of the splitting algorithm is embedded in the method for solving the quadratic optimization approximation subproblem of the discussed problem, and then, the subproblem is decomposed into two independent low-dimension quadratic optimization subproblems to generate a search direction for the primal variable. Second, a deflection of the steepest descent direction of the augmented Lagrangian function with respect to the dual variable is considered as the search direction of the dual variable. Third, using the augmented Lagrangian function as the merit function, a new primal–dual iterative point is generated by Armijo line search. Under mild conditions, the global convergence of the proposed algorithm is proved. Finally, the proposed algorithm is applied to solve a series of mid-to-large-scale economic dispatch problems for power systems. Comparing the numerical results demonstrates that the proposed algorithm possesses superior numerical effects and good robustness.

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of $\mathbb R^{n}$ with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from $\mathbb R^{n}$ into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to $-\infty $ and it goes to 1 if all variables go to $+\infty $ . The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

In a series of papers (Solodov and Svaiter in J Convex Anal 6(1):59–70, 1999; Set-Valued Anal 7(4):323–345, 1999; Numer Funct Anal Optim 22(7–8):1013–1035, 2001) Solodov and Svaiter introduced new inexact variants of the proximal point method with relative error tolerances. Point-wise and ergodic iteration-complexity bounds for one of these methods, namely the hybrid proximal extragradient method (1999) were established by Monteiro and Svaiter (SIAM J Optim 20(6):2755–2787, 2010). Here, we extend these results to a more general framework, by establishing point-wise and ergodic iteration-complexity bounds for the inexact proximal point method studied by Solodov and Svaiter (2001). Using this framework we derive global convergence results and iteration-complexity bounds for a family of projective splitting methods for solving monotone inclusion problems, which generalize the projective splitting methods introduced and studied by Eckstein and Svaiter (SIAM J Control Optim 48(2):787–811, 2009).

We study symmetric properties of positive solutions to the Choquard type equation $$ (-\Delta )^{s}_{p} u + |x|^{a} u = \left (\frac{1}{|x|^{n-\alpha }}*u^{q} \right ) u^{r} \quad \text{in}\ \mathbb{R}^{n}, $$ where $0< s1$ , $r>0$ , $a\ge 0$ and $(-\Delta )^{s}_{p}$ is the fractional $p$ -Laplacian. Via a direct method of moving planes, we prove that every positive solution $u$ which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if $a>0$ .

In many real classification problems a monotone relation between some predictors and the classes may be assumed when higher (or lower) values of those predictors are related to higher levels of the response. In this paper, we propose new boosting algorithms, based on LogitBoost, that incorporate this isotonicity information, yielding more accurate and easily interpretable rules. These algorithms are based on theoretical developments that consider isotonic regression. We show the good performance of these procedures not only on simulations, but also on real data sets coming from two very different contexts, namely cancer diagnostic and failure of induction motors.

This paper develops a new approach—based on the majorization theory—to the information design problem in Bayesian persuasion mechanisms, i.e., models in which the sender selects the signal structure of the agent(s) who then reports it to the non-strategic receiver. We consider a class of mechanisms in which the posterior payoff of the sender depends on the value of a realized posterior mean of the state, its order in the sequence of possible means, and the marginal distribution of signals. We provide a simple characterization of mechanisms in which optimal signal structures are monotone partitional. Our approach has two economic implications: it is invariant to monotone transformations of the state and allows to decompose setups with multiple agents into independent Bayesian persuasion mechanisms with a single agent. As the main application of our characterization, we show the optimality of monotone partitional signal structures in all selling mechanisms with independent private values and quasi-linear preferences. We also provide sharp characterization of optimal signal structures for second-price auctions and posted-price sales.

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern’s iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

We study joint numerical and spectral radii defined for d-tuples of bounded operators on a Hilbert space and related to noncommutative notions of independence. The definitions are in analogy with the ones of Popescu, where his formulations turned out to be related with free creation operators, and in this way related to the free independence of Voiculescu. In our study the definitions are related with either weakly monotone creation operators, and thus associated with the monotone independence of Muraki, or with boolean creation operators, and hence related with the boolean independence.

We establish strong and weak Hardy–Littlewood–Sobolev inequalities for the subadditive operators majorized by operators from a certain class of integral convolutions of the Riesz-potential type with almost monotone kernels generated both by operators of ordinary shift and by operators of generalized shift associated with the differential Laplace–Bessel operator.

In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the e-subdifferential. We obtain a strong convergence theorem for the new algorithm.

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set.

We consider a class of “filtered” schemes for first order time dependent Hamilton–Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter $$\varepsilon =\varepsilon ({\Delta t,\Delta x})>0$$ which goes to 0 as the time and space steps $$(\Delta t,\Delta x)$$ are going to 0 and does not depend on the time $$t_n$$, for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of $$\varepsilon =\varepsilon ^n (\Delta t, \Delta x)$$ at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold $$\varepsilon ^n$$. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

We consider collective evaluation problems, where individual grades given to candidates are combined to obtain a collective grade for each of these candidates. In this paper, we prove the following two results: (1) a collective evaluation rule is update monotone and continuous if and only if it is a min-max rule, and (2) a collective evaluation rule is update monotone and consistent if and only if it is an extreme min-max rule.

This work aims to prove the large deviation principle for a class of stochastic partial differential equations with locally monotone coefficients under the extended variational framework, which generalizes many previous works. Using stochastic control and the weak convergence approach, we prove the Laplace principle, which is equivalent to the large deviation principle in our framework. Instead of assuming compactness of the embedding in the corresponding Gelfand triple or finite dimensional approximation of the diffusion coefficient in some existing works, we only assume some temporal regularity in the diffusion coefficient.

In some applications, it may be assumed or known that the survival function has a nondecreasing hazard function. The maximum likelihood estimator under this assumption is called the monotone nonparametric maximum likelihood estimator (MoNPMLE). In this article, we establish the conditional consistency of the MoNPMLE under the mixed case interval-censored model with left truncation. We also investigate the unconditional consistency of the MoNPMLE through simulation study. The Gradient Projection Method algorithm is used to obtain the MoNPMLE. Simulation results indicate that the MoNPMLE is consistent and outperforms the NPMLE.

Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

The main purpose of this paper is to prove the mean ergodic theorem for nonexpansive mappings and semigroups in locally compact Hadamard spaces, including finite dimensional Hadamard manifolds. The main tool for proving ergodic convergence is the almost periodicity of orbits of a nonexpansive mapping. Therefore, in the first part of the paper, we study almost periodicity (and as a special case, periodicity) in metric and Hadamard spaces. Then, we prove a mean ergodic theorem for nonexpansive mappings and continuous semigroups of contractions in locally compact Hadamard spaces. Finally, an application to the asymptotic behavior of the first order evolution equation associated to the monotone vector field on Hadamard manifolds is presented., Comment: 23 pages, 3 figures

The purpose of this paper is to study and analyze an iterative method for split feasibility problem and variational inclusion problem (also known as the problem of finding a zero of the sum of two monotone operators) in the framework of real Banach spaces. By combining Mann’s and Halpern’s approximation methods, we propose an iterative algorithm for approximating a common solution of the aforementioned problems. Furthermore, we derive the strong convergence of the proposed algorithm under appropriate conditions. In all our results, we use the new way introduced by Suanti et al. to select the step-size which ensures the convergence of the sequences generated by our scheme. We also gave an application of our results and a numerical example of the proposed algorithm in comparison with the algorithm of Suanti et al. to show the efficiency and advantage of our algorithm. Our results extend and complement many known related results in the literature.

We study and solve one class of discrete matrix equations with cubic nonlinearity. The existence of a two-parameter family of monotone and bounded solutions is proved. Under certain additional conditions, we determine the asymptotic behavior of the constructed solutions. The obtained results are extended to the corresponding inhomogeneous discrete matrix equations and to some more general cases of nonlinearity.

In this paper, we introduce a new algorithm with self adaptive step-size for finding a common solution of a split feasibility problem and a fixed point problem in real Hilbert spaces. Motivated by the self adaptive step-size method, we incorporate the self adaptive step-size to overcome the difficulty of having to compute the operator norm in the proposed method. Under standard and mild assumption on the control sequences, we establish the strong convergence of the algorithm, obtain a common element in the solution set of a split feasibility problem for sum of two monotone operators and fixed point problem of a demimetric mapping. Numerical examples are presented to illustrate the performance and the behavior of our method. Our result extends, improves and unifies other results in the literature.

In this paper, we introduce and study strong convergence of a proximal point algorithm for approximating common solution of a finite family of monotone inclusion problems and fixed point problem of a generalized demimetric mapping in complete CAT(0) spaces. An illustrative example is given to validate theoretical result obtained herein. Our results improve and generalize some well-known results in the literature.