Monotone Assumption Research Articles

Distributed mirror descent method with operator extrapolation for stochastic aggregative games

In this work, we investigate a class of stochastic aggregative games, where each player has an expectation-valued objective function depending on its local strategy and the aggregate of all players’ strategies. We propose a distributed algorithm with operator extrapolation to search for the Nash equilibrium, in which each player maintains an estimate of this aggregate by exchanging this information with its neighbors over a time-varying network, and updates its equilibrium estimate through the mirror descent method. Specially, we embed an operator extrapolation utilizing two step historical gradient samples at the search direction so as to accelerate the convergence. Under the generalized strongly monotone assumption characterized by the Bregman’s distance on the pseudo-gradient mapping, we prove that the proposed algorithm can achieve the optimal convergence rate O(1/k) in terms of the expected Bregman distance to the Nash equilibrium. Finally, the algorithm performance is demonstrated via numerical simulations.

Ground state solutions and periodic solutions with minimal periods to second-order Hamiltonian systems

In this paper, we study the existence of periodic solutions to second-order Hamiltonian systems. The nonlinear term has a special form, which satisfies a nondecreasing monotone assumption. Making use of a generalized Nehari manifold, we built the homomorphism between the Nehari manifold and the unit sphere of some space. Some sufficient conditions are obtained to guarantee the existence of periodic solutions with the prescribed minimal period to Hamiltonian systems. Our results extend the results in references.

A Modified Scaled Spectral-Conjugate Gradient-Based Algorithm for Solving Monotone Operator Equations

This paper proposes a modified scaled spectral-conjugate-based algorithm for finding solutions to monotone operator equations. The algorithm is a modification of the work of Li and Zheng in the sense that the uniformly monotone assumption on the operator is relaxed to just monotone. Furthermore, unlike the work of Li and Zheng, the search directions of the proposed algorithm are shown to be descent and bounded independent of the monotonicity assumption. Moreover, the global convergence is established under some appropriate assumptions. Finally, numerical examples on some test problems are provided to show the efficiency of the proposed algorithm compared to that of Li and Zheng.

Design considerations for phase I/II dose finding clinical trials in Immuno-oncology and cell therapy

Immuno-oncology (IO) and cell therapy, the frontier of cancer treatment, is a rapidly developing area that brings new opportunities to patients. In IO and cell therapy clinical trial development, it is critical to identify the right dose level in early phase of trials thus improving the probability of success in confirmatory trials to test the superiority over other therapies. Given the complex mechanism interacting with immune system for IO drugs especially cell therapy, the traditional oncology dose finding trial designs may not serve the purpose. Specifically, it is questionable to believe the monotone relationship between dose level and safety/efficacy, which will likely result in inappropriate dose selection using designs with monotone assumption. Additionally, considering the immune system pathway, designs ignoring the heterogeneity of the patient populations may provide misleading dose decisions, which could be either unsafe or lead to selection mistakes for targeted population. Therefore, in our paper, we review and present the limitations of the traditional dose finding designs. Then we discuss improved dose finding designs that consider both safety and efficacy outcomes simultaneously. Furthermore, we propose novel dose finding designs for multiple populations: BNP-mTPI and fBNP-mTPI, which extend the modified toxicity probability interval designs by utilizing Bayesian non-parametric priors. The proposed designs consider patient population differences meanwhile flexibly borrowing information across populations with similar profiles to improve the efficiency of dose search and accuracy of estimation of optimal dose level. Simulations are provided to demonstrate the model performance. Finally, we conclude the recommendations for IO and cell therapy dose finding designs in the discussion and offer insights for future research direction.

Existence result for fractional Schrödinger–Poisson systems involving a Bessel operator without Ambrosetti–Rabinowitz condition

The present study is concerned with the nontrivial solutions for fractional Schrödinger–Poisson system with the Bessel operator. Under certain assumptions on the nonlinearity f, a nontrivial nonnegative solution is obtained by perturbation method for the given problem. In particular, the Ambrosetti–Rabinowitz type condition or the monotone assumption on the nonlinearity is unnecessary.

Identification of Fully Measurable Grand Lebesgue Spaces

We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm ρ(f)=ess supx∈X⁡δ(x)ρp(x)(f), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x), and p(·) and δ(·) are measurable functions over a measure space (X,ν), p(x)∈[1,∞], and δ(x)∈(0,1] almost everywhere. We prove that every such space can be expressed equivalently replacing p(·) and δ(·) with functions defined everywhere on the interval (0,1), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded p(·), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.

Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with large perturbation and gradient terms

In this paper, we investigate boundary blow-up solutions of the problem \begin{equation*} \quad \left\{ \begin{array}{l} -\Delta _{p(x)}u+f(x,u)=\rho (x,u)+K(x)|\nabla u|^{m(x)}\mbox{ in }\Omega , \\[2mm] u(x)\rightarrow +\infty \mbox{ as }d(x,\partial \Omega )\rightarrow 0, \end{array} \right. \end{equation*} where $\Delta _{p(x)}u=\mathrm{div}\,(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where $p(\cdot )\equiv 2 $, without gradient term, to the case where $p(\cdot )$ is a function, with gradient term. It also extends the previous work of Y. Liang, Q.H. Zhang and C.S. Zhao [38] from the radial case in the problem to the non-radial case. The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $, which means that $\rho (x,u)$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,\partial \Omega )$. Hence, the results of this paper are new even in the canonical case $p(\cdot )\equiv 2$. In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term.

Morse theory for the Hofer length functional

Following [16], we develop here a connection between Morse theory for the (positive) Hofer length functional L : Ω Ham (M, ω) → ℝ, with Gromov–Witten/Floer theory, for monotone symplectic manifolds (M, ω). This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.

Asymptotic pattern of a migratory and nonmonotone population model

In this paper, we consider atime-delayed and nonlocal population model with migration and relaxthe monotone assumption for the birth function. We study the globaldynamics of the model system when the spatial domain is bounded. Ifthe spatial domain is unbounded, we investigate the spreading speed$c^*$, the non-existence of traveling wave solutions with speed$c\in(0,c^*)$, the existence of traveling wave solutions with $c\geqc^*$, and the uniqueness of traveling wave solutions with $c>c^*$.It is shown that the spreading speed coincides with the minimal wavespeed of traveling waves.

A new approach to measure socioeconomic inequality in health

This paper introduces a new approach to rank and measure socioeconomic inequality in health. A novel feature of the approach is the use of an income-health matrix that relates socioeconomic class with health status; each row of the matrix corresponds to a socioeconomic class and contains the respective probability distribution of health. By invoking a monotone assumption on the income-health matrix, which is a direct translation of the well-known hypothesis of social gradients in health outcomes, we derive a set of welfare-dominance and inequality-dominance conditions for ranking health distributions. Unlike other existing health inequality measures, our conditions require no cardinal specification of ordinal heath data and, hence, are robust. We then apply the dominance conditions to compare socioeconomic health inequality in the US and Canada using the newly released JCUSH data.

Benefit-Risk Assessments in Dose-Finding Trials

The objective of dose-finding trials is to identify the doses that are sufficiently effective and safe for either future late-phase trials or the ultimate patients. Most methodologies for analyzing data from Phase II dose-finding trials mainly focus on finding the effective doses or minimum effective dose through hypothesis testing, while Phase I studies focus on finding the maximum tolerated dose. In this article, we focus on benefit–risk assessment through both hypothesis testing and point estimation. For purposes of efficiency, monotone dose–response relationships will be assumed for the safety parameter. However, the monotone assumption cannot be made directly for the benefit–risk measure. This consideration can limit the use of downward or upward sequential procedures for multiplicity adjustment for benefit-risk assessment. Several different approaches will be discussed and evaluated. Numerical and data examples will be used to illustrate the properties of these approaches.

Existence of solutions of generalized vector variational-type inequalities with set-valued mappings

In this paper, we introduce a new class of generalized vector variational-type inequalities with set-valued mappings (GVVTI). First, the solvability of generalized vector variational-type inequalities without monotone assumption is considered in Banach spaces by using Brouwer’s fixed point theorem. Second, by the introduction of the concepts of pseudomonotone, v -coercive and η -hemicontinuous, the solvability of (GVVTI) is obtained with the assumption of η -hemicontinuous and pseudomonotone.

Generalized vector variational-type inequalities

Solvabilities of generalized vector variational-type inequalities (GVVTI) are discussed in this work. First, the solvability of GVVTI without monotone assumption for mappings is considered in (reflexive) Banach spaces by using Brouwer fixed point theorem (Browder fixed point theorem). Second, the solvability of GVVTI with monotone assumption for mappings is considered in reflexive Banach spaces by using Ky Fan lemma.

Monotone iterative technique for delay differential equations

In proving existence of extremal solutions for delay differential equations, one usually assumes nondecreasing property on the function involved without which the proof breaks down. Our main purpose, in this paper, is to remove the monotone assumption by proving a new comparison result which is required in the analysis and which avoids the standard line of argument. Furthermore, our comparison theorem shows that the choice of initial functions cannot be arbitrary for the results to hold. Using this result we then develop monotone technique to obtain extremal solutions.