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In this paper, we study a new generalization of the Lorentz cone $$\mathcal{L}^n_+$$ L + n , called the monotone extended second-order cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a nonlinear complementarity problem on a cylinder, which is equivalent to a suitable mixed complementarity problem, and provide a computational example illustrating applicability of MESOC.

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches., Comment: 30 pages, 14 figures

Coordinate descent algorithms are popular in machine learning and large-scale data analysis problems due to their low computational cost iterative schemes and their improved performances. In this work, we define a monotone accelerated coordinate gradient descent-type method for problems consisting of minimizing $$f+g$$ , where f is quadratic and g is nonsmooth and non-separable and has a low-complexity proximal mapping. The algorithm is enabled by employing the forward–backward envelope, a composite envelope that possess an exact smooth reformulation of $$f+g$$ . We prove the algorithm achieves a convergence rate of $$O(1/k^{1.5})$$ in terms of the original objective function, improving current coordinate descent-type algorithms. In addition, we describe an adaptive variant of the algorithm that backtracks the spectral information and coordinate Lipschitz constants of the problem. We numerically examine our algorithms on various settings, including two-dimensional total-variation-based image inpainting problems, showing a clear advantage in performance over current coordinate descent-type methods.

In this paper, we investigate the solvability for a coupled system involving nonlinear fractional q-difference stationary Schrodinger equation. The existence criterion of solutions is established by Schauder fixed point theorem, while the existence of iterative positive solutions is derived by monotone iteration method. As applications, an example is presented to illustrate the main results.

Team formation plays an essential role in the labor market. In this paper, we propose two bicriteria algorithms to construct a balance between gain and cost in a team formation problem under the streaming model, subject to a cardinality constraint. We formulate the problem as maximizing the difference of a non-negative normalized monotone submodular function and a non-negative linear function. As an extension, we also consider the case where the first function is $$\gamma $$ -weakly submodular. Combining the greedy technique with the threshold method, we present bicriteria streaming algorithms and give detailed analysis for both of these models. Our analysis is competitive with that in Ene’s work.

This paper is contributed to the structural stability of multi-wave configurations to Cauchy problem for the compressible non-isentropic Euler system with adiabatic exponent γ ∈ (1, 3]. Given some small BV perturbations of the initial state, the author employs a modified wave front tracking method, constructs a new Glimm functional, and proves its monotone decreasing based on the possible local wave interaction estimates, then establishes the global stability of the multi-wave configurations, consisting of a strong 1-shock wave, a strong 2-contact discontinuity, and a strong 3-shock wave, without restrictions on their strengths.

In this paper, we study the problem of maximizing a monotone non-decreasing submodular function $$f :2^\Omega \rightarrow {\mathbb {R}}_{+}$$ subject to a cardinality constraint, i.e., $$\max \{ f(A) : |A| \le k, A \subseteq \Omega \} $$ . We propose a deterministic algorithm based on a new greedy strategy for solving this problem. We prove that when the objective function f satisfies certain assumptions, the algorithm we propose can return a $$1 - \kappa _f (1 - \frac{1}{k})^k$$ approximate solution with O(kn) value oracle queries, where $$\kappa _f$$ is the curvature of the monotone submodular function f. We also show that our algorithm outperforms the traditional greedy algorithm in some cases. Furthermore, we demonstrate how to implement our algorithm in practice.

In this paper we introduce the notions of continuity, meet continuity and quasicontinuity of Δ-spaces, where a Δ-space is a monotone determined, weak monotone convergence space. We prove that: a Δ-space $(X, \mathcal {O}(X))$ is continuous iff the topology lattice $\mathcal {O}(X)$ is completely distributive; a Δ-space $(X, \mathcal {O}(X))$ is meet continuous iff $\mathcal {O}(X)$ is a complete Heyting algebra; a Δ-space $(X, \mathcal {O}(X))$ is continuous iff it is both meet continuous and quasicontinuous. The concepts of continuity, s2-continuity, 𝜃-continuity, strong continuity of posets are shown to be special cases of the continuity of Δ-spaces. We also introduce a type of directed completion of Δ-spaces, called Scott completion. For each Δ-space $(X, \mathcal {O}(X))$ , the topology lattice $\mathcal {O}(X)$ is isomorphic to the topology lattice of its Scott completion. The D-completion, D𝜃-completion and $D_{s_{2}}$ -completion of posets are included in the Scott completion.

This paper investigates dynamically consistent updates of weighted unanimity rules given by variational Bewley preferences (VBP), a class of incomplete preferences characterized by pairs of utility index over consequences and ambiguity index over priors. We show that these dynamically consistent updates are also VBP that preserves the same ranking over consequences and feature conditional ambiguity index obtained through the full Bayesian update of the unconditional one. Moreover, we study conditional forced-choice relations, which capture choices that a decision-maker must take after learning some event. Formally, they are monotone continuous weak orders. We show that any conditional forced-choice relation that preserves the ambiguity attitude of a given unconditional VBP has a variational representation with the same affine utility function over consequences and the ambiguity index of its corresponding dynamically consistent update. Thus, our result provides a novel foundation for full Bayesian updating of variational preferences.

We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.

In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119 , 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.

The paper concerns with a new iterative method for solving a monotone variational inclusion problem in a Hilbert space. The method is of the proximal contraction type incorporated with the regularization technique. Under the prediction stepsize conditions, we establish the strong convergence of the iterative sequences generated by the method to a particular solution of the problem satisfying a variational inequality problem. Finally, we give some numerical examples to illustrate the behavior of the new method in comparison with existing ones.

We construct a continuous nowhere monotone and nondifferentiable function depending on a single parameter q0 ϵ (0; 1). For functions from this continual class, we describe their structural, variational, fractal, and integrodifferential properties.

It is the purpose of this paper to introduce an iterative process which converges strongly to a common point of the set of solutions of a finite family of generalized equilibrium problems, the set of fixed points of a finite family of continuous asymptotically quasi- $$\phi$$ -nonexpansive mappings in the intermediate sense, and the set of zeros of a finite family of $$\gamma$$ -inverse strongly monotone mappings in uniformly convex and uniformly smooth real Banach space. Our results improve and unify most of the results that have been proved for this important class of nonlinear mappings.

In this paper, we propose a numerical approach for solving composite primal-dual monotone inclusions with a priori information. The underlying a priori information set is represented by the intersection of fixed point sets of a finite number of operators, and we propose and algorithm that activates the corresponding set by following a finite-valued random variable at each iteration. Our formulation is flexible and includes, for instance, deterministic and Bernoulli activations over cyclic schemes, and Kaczmarz-type random activations. The almost sure convergence of the algorithm is obtained by means of properties of stochastic Quasi-Fej\'er sequences. We also recover several primal-dual algorithms for monotone inclusions in the context without a priori information and classical algorithms for solving convex feasibility problems and linear systems. In the context of convex optimization with inequality constraints, any selection of the constraints defines the a priori information set, in which case the operators involved are simply projections onto half spaces. By incorporating random projections onto a selection of the constraints to classical primal-dual schemes, we obtain faster algorithms as we illustrate by means of a numerical application to a stochastic arc capacity expansion problem in a transport network., Comment: 23 Pages, 4 figures, 4 Tables

We are interested in the optimal control problem associated with certain quadratic cost functionals depending on the solution $X=X^\alpha$ of the stochastic mean-field type evolution equation in $\mathbb R^d$ $dX_t=b(t,X_t,\mathcal L(X_t),\alpha_t)dt+\sigma(t,X_t,\mathcal L(X_t),\alpha_t)dW_t,$ $X_0\sim \mu$ given, under assumptions that enclose a sytem of FitzHugh-Nagumo neuron networks, and where for practical purposes the control $\alpha_t$ is deterministic. To do so, we assume that we are given a drift coefficient that satisfies a one-sided Lipshitz condition, and that the dynamics is subject to a (convex) level set constraint of the form $\pi(X_t)\leq0$. The mathematical treatment we propose follows the lines of the recent monograph of Carmona and Delarue for similar control problems with Lipshitz coefficients. After addressing the existence of minimizers via a martingale approach, we show a maximum principle and then numerically investigate a gradient algorithm for the approximation of the optimal control., Comment: 32 pages; 11 figures

An edge-ordered graph is a graph with a total ordering of its edges. A path $$P=v_1v_2\ldots v_k$$ in an edge-ordered graph is called increasing if $$(v_iv_{i+1}) < (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ ; and it is called decreasing if $$(v_iv_{i+1}) > (v_{i+1}v_{i+2})$$ for all $$i = 1,\ldots,k-2$$ . We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing. Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $$\overline{\alpha}(G)$$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length $$\overline{\alpha}(G)$$ . Let $$\overline{\tau}(G)$$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of $$\overline{\tau}(G)$$ edges. In this paper we show that $$\overline \alpha(K_n) = \Omega(\log\log n)$$ , $$\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)$$ and $$\overline \tau(K_n) = O(\sqrt{n \log n})$$ .

In this paper, we propose an accelerated variant of the proximal point algorithm for computing a zero of an arbitrary maximally monotone operator A. The method incorporates an inertial term (based upon the acceleration techniques introduced by Nesterov), relaxation factors and a correction term. In a general Hilbert space setting, we obtain the weak convergence of the iterates $$(x_n)$$ to equilibria, with the fast rate $$\Vert x_{n+1}-x_{n}\Vert =o(n^{-1})$$ (for the discrete velocity). In particular, when using constant proximal indexes, we establish the accuracy of $$(x_n)$$ to a zero of A with the worst-case rate $$\Vert A_{\lambda }(x_n)\Vert =o(n^{-1})$$ ( $$A_{\lambda }$$ being the Yosida regularization of A with some index $$\lambda $$ ). Furthermore, given any positive integer p and using an appropriate adjustment of increasing proximal indexes, we obtain the worst-case convergence rate $$\Vert A_{\lambda }(x_n)\Vert =o(n^{-(p+1)})$$ . This acceleration process can be applied to various proximal-type algorithms such as the augmented Lagrangian method, the alternating direction method of multipliers, operator splitting methods and so on. Our methodology relies on a Lyapunov analysis combined with a suitable reformulation of the considered algorithm. Numerical experiments are also performed.

The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169, 2018), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including the Lov\'asz number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points). We show that every spectral point admits a probabilistic refinement and characterize the functions arising in this way. This reveals that the asymptotic spectrum can be parameterized with a convex set and the evaluation function at every graph is logarithmically convex. One consequence is that for any incomparable pair of spectral points $f$ and $g$ there exists a third one $h$ and a graph $G$ such that $h(G)

In this paper, we are concerned with the existence of positive solutions for nonlinear fourth-order equation with clamped beam boundary conditions $$\begin{aligned} \left\{ \begin{array}{ll} &{}u''''(x)=\lambda a(x)f(u(x))\ \ \ \text {for}\ \ x\in (0,1),\\ &{}u(0)=u(1)=u'(0)=u'(1)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$a\in L^{1}(0,1)$$ may change sign and $$f\in C([0,\infty ),[0,\infty ))$$ . We establish some new results of existence of positive solutions to this problem if the nonlinearity f is monotone on $$[0,\infty )$$ . The proofs of our main results are based upon a monotone iteration technique and the Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.

We consider a class of two-dimensional integral equations in ℝ2 with monotone nonlinear terms and prove the existence of a nonnegative bounded solution. We study the asymptotic behavior of this solution with respect to each variable. The result is illustrated by an example.

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. MKT is supported in part by ARC grant DE200100063.

We examine the problem of computing the highest density region (HDR) in a computational context where the user has access to a density function and quantile function for the distribution (e.g., in the statistical language R). We examine several common classes of continuous univariate distributions based on the shape of the density function; this includes monotone densities, quasi-concave and quasi-convex densities, and general multimodal densities. In each case we show how the user can compute the HDR from the quantile and density functions by framing the problem as a nonlinear optimisation problem. We implement these methods in R to obtain general functions to compute HDRs for classes of distributions, and for commonly used families of distributions. We compare our method to existing R packages for computing HDRs and we show that our method performs favourably in terms of both accuracy and average speed.

The present analysis provides insights and analytical findings for a cooperative system formulated with non-linear diffusion. This work discusses, from an analytical perspective, the interaction of two species when both cooperate to explore new territories. Firstly, the analysis focuses on regularity, existence and uniqueness of weak solutions considering slightly positivity in the non-linear operator together with a monotone argument in the independent terms. Afterwards, analytical solution profiles are provided making use of selfsimilar solutions. Non-linear diffusion induces the existence of a propagating front and determines the property of finite speed of propagation in both species when they cooperate to explore new territories.

Relevance and diversity are two desirable properties in data retrieval applications, an important field in data science and machine learning. In this paper, we consider three maximization problems to balance these two factors. The objective function in each problem is the sum of a monotone submodular function f and a supermodular function g, where f and g capture the relevance and diversity of any feasible solution, respectively. In the first problem, we consider a special supermodular diversity function g of a sum-sum format satisfying the relaxed triangle inequality, for which we propose a greedy-type approximation algorithm with an $$\left( 1-1/e,1/(2\alpha )\right) $$ -bifactor approximation ratio, improving the previous $$\left( 1/(2\alpha ),1/(2\alpha )\right) $$ -bifactor approximation ratio. In the second problem, we consider an arbitrary supermodular diversity function g, for which we propose a distorted greedy method to give a $$\min \left\{ 1-k_{f}e^{-1},1-k^{g}e^{-(1-k^{g})}\right\} $$ -approximation algorithm, improving the previous $$k_f^{-1}\left( 1-e^{-k_f(1-k^{g})}\right) $$ -approximation ratio, where $$k_f$$ and $$k^g$$ are the curvatures of the submodular function f and the supermodular funciton g, respectively. In the third problem, we generalize the uniform matroid constraint to the p matroid constraints, for which we present a local search algorithm to improve the previous $$\frac{1-k^g}{(1-k^g)k^f+p}$$ -approximation ratio to $$\min \left\{ \frac{p+1-k_f}{p(p+1)},\left( \frac{1-k^g}{p}+\frac{k^g(1-k^g)^2}{p+(1-k^g)^2}\right) \right\} $$ .

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are “best possible”. A computational evaluation shows the promise of this approach.

A projected extrapolated gradient method is designed for solving monotone variational inequality in Hilbert space. Requiring local Lipschitz continuity of the operator, our proposed method improves the value of the extrapolated parameter and admits larger step sizes, which are predicted based a local information of the involved operator and corrected by bounding the distance between each pair of successive iterates. The correction will be implemented when the distance is larger than a given constant and its main cost is to compute a projection onto the feasible set. In particular, when the operator is the gradient of a convex function, the correction step is not necessary. We establish the convergence and ergodic convergence rate in theory under the larger range of parameters. Related numerical experiments illustrate the improvements in efficiency from the larger step sizes.

This paper introduces a new subclass of convex functions called uniformly convex functions [this class is a slight extension of the class introduced earlier by Zalinescu (J Math Anal Appl 95:344–374, 1983; Convex analysis in general vector spaces, World Scientific, Singapore)] and studies some of its properties. It also defines a new subdifferentiability called $$(\phi ,c)$$ -subdifferentiability and investigates some of its properties. As a main result, we prove that a lower semicontinuous function defined on a Banach space is uniformly convex if and only if its Clarke subdifferential is uniformly monotone set-valued map. Then we move to present a result concerning optimization problems for uniformly convex functions. Finally, we apply our theory to conclude the Minty-Browder theorem for a subclass of monotone operators as well as the Lax–Milgram theorem without recourse to use the Riesz–Frechet theorem. Consequently, we establish the existence and uniqueness of weak solution for the p-Laplacian problem.

In this paper, we present some convergence results for various iterative algorithms built from Hardy-Rogers type generalized nonexpansive mappings and monotone operators in Hilbert spaces. We obtain some comparison results for the rates of convergence of these algorithms to the solution of variational inequality problem including Hardy–Rogers type generalized nonexpansive mappings and monotone operators. A numerical example is given to validate these results. We apply the iterative algorithms handled herein to solve convex minimization problem and illustrate this result by providing a non-trivial numerical example. The presented results considerably improve the corresponding results in Ali et al. (Comput Appl Math 39:74, 2020. https://doi.org/10.1007/s40314-020-1101-4 ).

Sample average approximation (SAA) approach for two-stage stochastic variational inequalities (SVIs) with continuous probability distributions, where the second-stage problems have multiple solutions, may not promise convergence assertions as the sample size tends to infinity. In this paper, a regularized SAA approach is proposed to numerically solve a class of two-stage SVIs with continuous probability distributions, where the second-stage problems are monotone and allowed to have multiple solutions. We first give some structural properties. After that, the convergence analysis of the regularized SAA approach for two-stage SVIs is investigated as the regularization parameter tends to zero and the sample size tends to infinity. Finally, we employ the progressive hedging algorithm to report some numerical results.

This paper investigates the exponential stability of traveling wave solutions for nonlinear delayed cellular neural networks. As a continuity of the past work (Wu and Niu, 2016; Yu, et al., 2011) on the existence and uniqueness of the traveling wave solutions, it is very reasonable and interesting to consider the exponential stability of the traveling wave solutions. By the weighted energy method, comparison principle and the first integral mean value theorem, this paper proves that, for all monotone traveling waves with the wave speed $$c < c_1^ \ast < 0$$ or $$c > c_2^ \ast > 0$$ , the solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations decay at some fields.

In this paper, the first-order forward–backward–half forward dynamical systems associated with the inclusion problem consisting of three monotone operators are analyzed. The framework modifies the forward–backward–forward dynamical system by adding a cocoercive operator to the inclusion. The existence, uniqueness, and weak asymptotic convergence of the generated trajectories are discussed. A variable metric forward–backward–half forward dynamical system with the essence of non-self-adjoint linear operators is introduced. The proposed notion, in turn, extends the forward–backward–forward dynamical system and forward–backward dynamical system in the framework of variable metric by relaxing some conditions on the metrics. The distributed dynamical system is further explored to compute a generalized Nash equilibrium in a monotone game as an application. A numerical example is provided to illustrate the convergence of trajectories.

In this work, we introduce two Bregman projection algorithms with self-adaptive stepsize for solving pseudo-monotone variational inequality problem in a Hilbert space. The weak and strong convergence theorems are established without the prior knowledge of Lipschitz constant of the cost operator. The convergence behavior of the proposed algorithms with various functions of the Bregman distance are presented. More so, the performance and efficiency of our methods are compared to other related methods in the literature.

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give a variational principle for limit shapes in this setting, and show that the resulting Euler-Lagrange equation can be integrated, giving explicit limit shapes parameterized by analytic functions., 37 pages, 21 figures

We consider the propagation dynamics of a general heterogeneous reaction-diffusion system under a shifting environment. By developing the fixed-point theory for second order non-autonomous differential system and constructing appropriate upper and lower solutions, we show there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed. We further show the uniqueness of forced waves by the sliding method and some analytical skills, and we obtain the global stability of forced waves by applying the dynamical systems approach. Moreover, we establish the spreading speed of the system by appealing to the abstract theory of monotone semiflows. Applications and numerical simulations are also given to illustrate the analytical results.

Assume that the function $f:[0,\infty )\rightarrow \mathbb {R}$ is operator monotone in $[0,\infty )$ . We can define the perspective $\mathcal {P}_{f}\left (B,A\right ) $ by setting $$ \mathcal{P}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, $$ where A, B > 0. In this paper, we show among others that, if σ ≥ C ≥ ρ > 0, D > 0, ς ≥ Q ≥ τ > 0 and 0 < n ≤ D − C ≤ N for some constants ρ, σ, ς, τ, n, N, then $$ \begin{array}{@{}rcl@{}} 0& \le& \frac{n}{N{\varsigma}^{2}}\left[ \mathcal{P}_{f}\left( {\varsigma} ,N+\sigma \right) -\mathcal{P}_{f}\left( {\varsigma} ,\sigma \right) \right] Q^{2} \\ & \leq& \mathcal{P}_{f}\left( Q,D\right) -\mathcal{P}_{f}\left( Q,C\right) \\ & \leq& \frac{N}{n\tau^{2}}\left[ \mathcal{P}_{f}\left( \tau ,n+\rho \right) -\mathcal{P}_{f}\left( \tau ,\rho \right) \right] Q^{2}. \end{array} $$ Applications for the weighted operator geometric mean and the perspective $$ \mathcal{P}_{\ln \left( \cdot +1\right) }\left( B,A\right) :=A^{1/2}\ln \left( A^{-1/2}BA^{-1/2}+1\right) A^{1/2},~ A,B>0 $$ are also provided.

We generalize the disutility of effort function in the linear-Constant Absolute Risk Aversion (CARA) pure moral hazard model. We assume that agents are heterogeneous in ability. Each agent’s ability is observable and treated as a parameter that indexes the disutility of effort associated with the task performed. In opposition to the literature (the “traditional” scenario), we find a new, “novel” scenario, in which a high-ability agent may be offered a weaker incentive contract than a low-ability one, but works harder. We characterize the conditions for the existence of these two scenarios: formally, the “traditional” (“novel”) scenario occurs if and only if the marginal rate of substitution of the marginal disutility of effort function is increasing (decreasing) in effort when evaluated at the second-best effort. If, further, this condition holds for all parameter values and matching is endogenous, less (more) talented agents work for principals with riskier projects in equilibrium. This implies that the indirect and total effects of risk on incentives are negative under monotone assortative matching.

The paper presents two inertial viscosity-type extragradient algorithms for finding a common solution of the variational inequality problem involving a monotone and Lipschitz continuous operator and of the fixed point problem with a demicontractive mapping in real Hilbert spaces. Our algorithms use a simple step size rule which is generated by some calculations at each iteration. Two strong convergence theorems are obtained without the prior knowledge of the Lipschitz constant of the operator. The numerical behaviors of the proposed algorithms in some numerical experiments are reported and compared with previously known ones.

This paper investigates the existence of a nondegenerate pure-strategy Bayesian-Nash equilibrium in a double auction between one seller and several heterogeneous buyers in the independent private value framework. It begins with three examples describing some of the model’s particular features. After studying a number of continuity-related properties of the interim payoff functions, we construct a sequence of strategy profiles that converges to a nondegenerate monotone Bayesian-Nash equilibrium of the game under some strict monotonicity conditions. The equilibrium existence result is applied to a double auction with risk-averse bidders.

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences $$\{f_{n}\}_{n=1}^{\infty}$$ and $$\{g_{n}\}_{n=1}^{\infty}$$ such that (1) $$f_{n}\leq f_{n+1}\leq f$$ and $$f\leq g_{n+1}\leq g_{n}$$ for all integers $$n \geq 1$$ ; (2) $$f_{n}$$ and $$g_{n}$$ are continuous and Gâteaux differentiable on $$X$$ ; (3) $$f_n \to f$$ and $${g_n \to f}$$ uniformly on $$X$$ ; (4) $$f_n$$ and $$g_n$$ are coercive and b-Lipschitzian. Moreover, we also prove that if $$X^{*}$$ is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists $$m>0$$ such that $$|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|$$ and $$|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|$$ whenever $$x, y \in X$$ and $$n\in N$$ .

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

The aim of this work is to study a stochastic obstacle problem governed by a T-monotone operator, a random force and a multiplicative stochastic reaction in the frame of Sobolev spaces. After proving a result of existence and uniqueness of the variational solution, by using an ad hoc perturbation of the stochastic reaction and a penalization of the constraint, we prove Lewy–Stampacchia’s inequalities associated with the problem finally.

Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

In this paper, we consider the existence of extremal solutions for a type of fractional compartment models. By use of a new comparison result, some new sufficient conditions for the existence of solutions are established by combining the monotone iterative technique and the methods of lower and upper solutions. Finally, an example is presented to illustrate the validity of our main results.