Class Of Problems Research Articles

On a class of obstacle problem via Young measure in generalized Sobolev space

On a class of obstacle problem via Young measure in generalized Sobolev space

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper investigates the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.

A Bregman proximal subgradient algorithm for nonconvex and nonsmooth fractional optimization problems

In this paper, we study a class of nonconvex and nonsmooth fractional optimization problem, where the numerator of which is the sum of a nonsmooth and nonconvex function and a relative smooth nonconvex function, while the denominator is relative weakly convex nonsmooth function. We propose a Bregman proximal subgradient algorithm for solving this type of fractional optimization problems. Under moderate conditions, we prove that the subsequence generated by the proposed algorithm converges to a critical point, and the generated sequence globally converges to a critical point when the objective function satisfies the Kurdyka-Łojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, two numerical experiments illustrate the effectiveness and superiority of the algorithm. Our results give a positive answer to an open problem proposed by Bot et al. [14].

Fragmentation of colliding liquid rims

We present direct numerical simulations of the splashing process between two cylindrical liquid rims. This belongs to a class of impact and collision problems with a wide range of applications in science and engineering, and motivated here by splashing of breaking ocean waves. Interfacial perturbations with a truncated white noise frequency profile are introduced to the rims before their collision, whose subsequent morphological development is simulated by solving the two-phase incompressible Navier–Stokes equation with the adaptive mesh refinement technique, within the Basilisk software environment. We first derive analytical solutions predicting the unsteady interfacial and velocity profiles of the expanding sheet forming between the two rims, and develop scaling laws for the evolution of the lamella rim under capillary deceleration. We then analyse the formation and growth of transverse ligaments ejected from the lamella rims, which we find to originate from the initial corrugated geometry of the perturbed rim surface. Novel scaling models are proposed for predicting the decay of the ligament number density due to the ongoing ligament merging phenomenon, and found to agree well with the numerical results presented here. The role of the mechanism in breaking waves is discussed further and necessary next steps in the problem are identified.

R-linear and strong convergence of Tseng methods for solving certain split and non-split problems

In the recent past, authors have fondly assumed the co-coercive condition for solving certain class of split inverse problems. Recently, some attempts have been made to relax this very strict condition, however, only few results exists in literature in such case. In this paper, we present some new Mann-type Tseng methods for solving split monotone variational inclusion and common fixed point problems for a finite collection of quasi-pseudocontractive operators. We prove strong convergence and R − linear convergence rate results of our methods, while the co-coerciveness property is dispensed with. Our methods incorporate the relaxation, inertial extrapolation and self-adaptive step size techniques to achieve better performance and faster convergence. With the aid of relevant numerical experiments, we showcase the performance profile of our methods in comparison with recent methods in literature.

Parameter-setting heuristic for the quantum alternating operator ansatz

The quantum alternating operator ansatz (QAOA) is a generalized approach for solving challenging optimization problems that builds on the alternating structure of the quantum approximate optimization algorithm. Finding high-quality parameters efficiently for QAOA remains a major challenge in practice. In this work, we introduce a classical strategy for parameter setting, suitable for cases in which the number of distinct cost values grows only polynomially with the problem size, such as is common for constraint-satisfaction problems. The crux of our strategy is that we replace the cost function expectation value step of QAOA with a classical model that can be efficiently evaluated classically and has parameters which play an analogous role to the QAOA parameters. This model is based on empirical observations that, in some QAOA states, variable configurations with the same cost have the same amplitudes from step to step. We define this class of states as homogeneous states. For problems with particular symmetries, QAOA states are guaranteed to be homogeneous. More generally, high overlaps between QAOA states and homogeneous states have been empirically observed in a number of settings. Building on this idea, we define a classical homogeneous proxy for QAOA in which this property holds exactly and which yields information describing both states and expectation values. We then classically determine high-quality parameters for this proxy and then use these parameters in QAOA, an approach we label the homogeneous heuristic for parameter setting. We numerically examine this heuristic for MaxCut on unweighted Erdős-Rényi random graphs. For up to three QAOA levels we demonstrate that the heuristic is easily able to find parameters that match approximation ratios corresponding to previously found globally optimized approaches. For levels up to 20 we obtain parameters using our heuristic with approximation ratios monotonically increasing with depth, while a strategy that uses parameter transfer instead fails to converge with comparable classical resources. These results suggest that our heuristic may find good parameters in regimes that are intractable with noisy intermediate-scale quantum devices. Finally, we outline how our heuristic may be applied to wider classes of problems. Published by the American Physical Society 2024

Sufficiency of -cyclical monotonicity in a class of multi-marginal optimal transport problems

Abstract $c$ -cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach, and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

Interatomic Interaction Models for Magnetic Materials: Recent Advances

Abstract Atomistic modeling is a widely employed theoretical method of computational materials science. It has found particular utility in the study of magnetic materials. Initially, magnetic empirical interatomic potentials or spin-polarized density functional theory (DFT) served as the primary models for describing interatomic interactions in atomistic simulations of magnetic systems. Furthermore, in recent years, a new class of interatomic potentials known as magnetic machine-learning interatomic potentials (magnetic MLIPs) has emerged. These MLIPs combine the computational efficiency, in terms of CPU time, of empirical potentials with the accuracy of DFT calculations. In this review, our focus lies on providing a comprehensive summary of the interatomic interaction models developed specifically for investigating magnetic materials. We also delve into the various problem classes to which these models can be applied. Finally, we offer insights into the future prospects of interatomic interaction model development for the exploration of magnetic materials.

A line search technique for a class of multi-objective optimization problems using subgradient

A line search technique for a class of multi-objective optimization problems using subgradient

Axisymmetric virtual elements for problems of elasticity and plasticity

AbstractThe virtual element method (VEM) allows discretization of elasticity and plasticity problems with polygons in 2D and polyhedrals in 3D. The polygons (and polyhedrals) can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. However, to the author's knowledge axisymmetric virtual elements have not appeared before in the literature. Hence, in this work a novel first order consistent axisymmetric VEM is applied to problems of elasticity and plasticity. The VEM specific implementation details and adjustments needed to solve axisymmetric simulations are presented. Representative benchmark problems including pressure vessels and circular plates are illustrated. Examples also show that problems of near incompressibility are solved successfully. Consequently, this research demonstrates that the axisymmetric VEM formulation successfully solves certain classes of solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

Shape optimization for interface identification in nonlocal models

Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface–dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques. The code for obtaining the results in this paper is published at (https://github.com/schustermatthias/nlshape).

Solving Euclidean Max-Sum problems exactly with cutting planes

This paper studies binary quadratic programs in which the objective is defined by the maximisation of a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems, which we refer to as the Euclidean Max-Sum problem, includes the capacitated, generalised and max-sum diversity problems as special cases. Due to the nonconcave objective, traditional cutting plane algorithms are not guaranteed to converge globally. In this paper, we introduce two exact cutting plane algorithms to address this limitation. The new algorithms remove the need for a concave reformulation, which is known to significantly slow down convergence. We establish exactness of the new algorithms by examining the concavity of the quadratic objective in a given direction, a concept we refer to as directional concavity. Numerical results show that the algorithms outperform other exact methods for benchmark diversity problems (capacitated, generalised and max-sum), and can easily solve problems of up to three thousand variables.

Logical Platforms for Mobile Application in Decision Support Systems Based on Color Information Processing

Logical Platforms for Mobile Application in Decision Support Systems Based on Color Information Processing

Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems

Existence, sign and asymptotic behaviour for a class of integro-differential elliptic type problems

Existence results for boundary value problems of Hadamard fractional differential equations on unbounded domain

Abstract In this paper we investigate the existence and uniqueness of solutions for a class of boundary value problem for fractional differential equations involving the Hadamard’s fractional derivative, by applying a nonlinear alternative of Leray–Schauder due to Frigon and Granas for contraction maps in Fréchet spaces.

Various duality models associated with some constrained robust nonlinear optimal control problems

This paper deals with various duality results for a new class of constrained robust nonlinear control problems. Concretely, we introduce a minimisation problem associated with a vector of functionals of multiple integral type. In addition, the considered multi-objective controlled optimisation problem is governed by several mixed constraints generated by partial derivatives of second order and uncertain data. For this new class of control problems, we establish and characterise Wolfe, Mond-Weir and mixed type robust dual models. To this aim, by using the notion of convex multiple integral functional, generated by a controlled second-order Lagrangian with uncertain data, and the concept of robust weak efficient solution to the problem under study, we build a new mathematical context to state and prove the duality theorems.

The Global existence and blow-up of solutions for a class of Steklov parabolic problems

In this paper, we study weak solutions to the following Steklov parabolic problem: We prove theorems of existence of weak solutions, via Galerkin approximation. Moreover, we show the existence of solutions which blow up in a finite time.

On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

A solution to the only one object problem with dissolution rules

In the framework of membrane computing, (non-)uniform families of recognizer membrane systems are usually defined to solve abstract decision problems. In this sense, the use of finite resources for each member of the family makes the difference with respect to Turing machines solving these problems. While keeping the finite nature of these systems, it is interesting to know which type of problems can be solved by means of a single membrane system. For this purpose, the complexity class PMCR1p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{PMC}^{1p}_{\\mathcal {R}}$$\\end{document} was defined as the class of problems that can be solved by means of a single membrane system in polynomial time. Due to the polynomial-time encoding of the input, at least all the problems from P can be solved with a trivial system. To go below P, the class PMCR1f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{PMC}^{1f}_{\\mathcal {R}}$$\\end{document} restricts the definition of this encoding. In this work, we study the capability of different types of membrane systems to solve the ONLY-ONE-OBJECT problem, while having the encoding restriction.

Efficient mini-batch stochastic gradient descent with Centroidal Voronoi Tessellation for PDE-constrained optimization under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is frequently utilized in engineering, biology and finance. Although stochastic gradient descent is a well-known stochastic optimization technique for solving the approximate optimal control problem, it often exhibits a slow convergence rate due to the inherent variance in gradient approximation. To address this issue, we propose a more efficient stochastic optimization algorithm based on Centroidal Voronoi Tessellation sampling. This strategy reduces the error in gradient estimation compared to the conventional mini-batch stochastic gradient descent method. Our approach involves dividing the whole snapshot set into several Voronoi cells with low variance and extracting samples with good uniformity from each region to construct an unbiased estimation of the full gradient. Our method can economically and stably approximate numerical optimal control functions even with a fixed step size. Numerical results demonstrate that the proposed method is a reliable algorithm with great potential for applications in the field of optimization.