Efficient Approximate Solution Research Articles

Deep learning for derivatives pricing: a comparative study of asymptotic and quasi-process corrections

AbstractIn this study, we compare two methods for using neural networks to efficiently learn the price of derivatives. The first method, proposed by Funahashi (Quant Financ 21(4):575–592, 2021a), involves training the neural networks to learn the difference between the derivative price and its asymptotic expansion, rather than learning the derivative price directly. The target derivative price is then obtained by adding the approximate solution with the predicted value of neural networks. This method reduces the required amount of learning data, often by a factor of one hundred to one thousand, compared to the case where the derivative price is directly learned through neural networks. In the second method, established in this paper, the neural networks learn the difference between the derivative price written on the underlying asset price that follows the target complex stochastic process and the derivative price written on the underlying asset price that has a relatively simple stochastic process that has a closed-form solution for the target derivative prices. This method provides an alternative valuation method when no efficient approximate solution for the derivative value is observed and if one can arbitrarily determine the model parameters of the quasi-process that approximates the original process. We also propose a unified method to determine the model parameters of quasi-processes from underlying asset processes. These methods prove valuable in cases where general analytic solutions are absent, as seen in widely used financial models such as the stochastic volatility models. These cases involve time-consuming numerical calculations to generate learning data, highlighting the value of the two methods, which significantly compress calculation times.

Computational analysis of a class of singular nonlinear fractional multi-order heat conduction model of the human head

The subject of the article is devoted to the development of a matrix collocation technique based upon the combination of the fractional-order shifted Vieta–Lucas functions (FSVLFs) and the quasilinearization method (QLM) for the numerical evaluation of the fractional multi-order heat conduction model related to the human head with singularity and nonlinearity. The fractional operators are adopted in accordance with the Liouville–Caputo derivative. The quasilinearization method (QLM) is first utilized in order to defeat the inherent nonlinearity of the problem, which is converted to a family of linearized subequations. Afterward, we use the FSVLFs along with a set of collocation nodes as the zeros of these functions to reach a linear algebraic system of equations at each iteration. In the weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document} norm, the convergence analysis of the FSVLFs series solution is established. We especially assert that the expansion series form of FSVLFs is convergent in the infinity norm with order O(1K3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}(\\frac{1}{K^3})}$$\\end{document}, where K represents the number of FSVLFs used in approximating the unknown solution. Diverse computational experiments by running the presented combined QLM-FSVLFs are conducted using various fractional orders and nonlinearity parameters. The outcomes indicate that the QLM-FSVLFs produces efficient approximate solutions to the underlying model with high-order accuracy, especially near the singular point. Furthermore, the methodology of residual error functions is employed to measure the accuracy of the proposed hybrid algorithm. Comparisons with existing numerical models show the superiority of QLM-FSVLFs, which also is straightforward in implementation.

Joint resource and trajectory optimization for secure UAV-based relay NOMA system

In this paper, we propose and analyze the maximal secrecy capability of an unmanned aerial vehicle (UAV)-based relay non-orthogonal multiple access system. In this system, a UAV acts as an aerial relay station to assist transmission from a source to two users in the presence of a terrestrial eavesdropper and a friendly jamming UAV. The minimum achievable secrecy rate (MASR) of two users is derived as a metric indicator to quantify the security of the system under the assumption that the location of the eavesdropper is imprecise. In order to maximize MASR, an efficient iterative algorithm is proposed to optimize system parameters, i.e., the trajectory of UAVs, transmit and jamming powers, as well as power allocation factors. Since the objective function and constraints of the total problem are non-convex, the successive convex approximation and block coordinate descent techniques are applied to find efficient approximate solutions. Simulation results demonstrate that the proposed algorithm significantly enhances the MASR of the system compared to benchmark schemes.

Approximate Solution of Schrodinger Equation to Diatomic Molecule for Harmonic Oscillator

This study has described the approximate solution of Schrodinger equation to diatomic molecule for harmonic oscillator. The solution procedure is developed by the Power series method and Newton’s second law. It consider an approximate solution of harmonic oscillator using Schrödinger equation in one dimension only because other analytical approaches are limited to the widely known method and consider two to five dimensions with various iteration method to obtain their results but here the solutions to be obtained and their efficiency will help other research to comprehend how the solution of this harmonic oscillator has been done over the years and also to use the most efficient approximate solution.

Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz’s function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems.

Approximate Subdifferential of the Difference of Two Vector Convex Mappings

This paper deals with the strong approximate subdifferential formula for the difference of two vector convex mappings in terms of the star difference. This formula is obtained via a scalarization process by using the approximate subdifferential of the difference of two real convex functions established by Martinez-Legaz and Seeger, and the concept of regular subdifferentiability. This formula allows us to establish approximate optimality conditions characterizing the approximate strong efficient solution for a general DC problem and for a multiobjective fractional programming problem.

A Self-Supervised Learning Approach for Accelerating Wireless Network Optimization

The prevailing issue in multi-hop wireless networking is interference management, which militates against the efficiency of traditional routing and scheduling algorithms. We develop a self-supervised learning approach to address the classic NP-hard problem of capacity optimization over a multi-hop wireless network, where the routing and scheduling decisions are deeply coupled. Our two-stage design leverages historical computation experiences to accelerate the optimization of new problem instances, where an instance represents the application-level input containing network topology, interference model, and other user-level traffic constraints. The first stage, Scheduling Structure Classification (SSC), distills the scheduling structure of the historical optimization instances into an appropriate number of classes, through a properly designed clustering algorithm without prior assumption or knowledge. The second stage uses the instances labelled with class information from the first stage to train an Application Identification (AID) neural network model capable of predicting a future problem instance's scheduling class given its application-level information. When solving the new instance, its predicted scheduling class and the associated scheduling structure are exploited to compute an efficient approximate solution, avoiding the time-consuming iterative search for such scheduling structure as practiced by the conventional approaches. We apply our method to different types of wireless multi-commodity flow problems across various network sizes and disparate flow requirements. The results demonstrate that our method significantly reduces the computation time robustly by at least 70% with only slight loss in the solution quality.

Rate-dependent JKR-type decohesion of a cylindrical punch from an elastic substrate

Recently published experimental data on non-quasistatic detachment of a flat-ended cylindrical punch from an adhesive rubber layer are analyzed in the framework of axisymmetric rate-dependent JKR-type model. The functional dependence of the work of adhesion on the velocity of the contour of contact area is assumed according to the known Gent–Schultz model. The evolution of the variable contact radius as a function of the punch displacement is described by a first-order ordinary differential equation, which possesses the localization property for its solutions, meaning that the detachment occurs at some nonzero contact radius. To facilitate the model fit to experimental force-displacement curve, a computationally efficient analytical approximate solution is suggested. A parametric analysis of the basic case (when the rubber layer is approximated by an elastic half-space) is presented.

A Fourier Galerkin method for ship waves

An efficient approximate solution to the problem of gravity–capillary ship-induced waves is derived using a Fourier Galerkin spectral approach. The proposed method solves linearised equations of finite-depth wave motion generated by a moving pressure distribution in a spectral framework. The method admits ship hulls of various shapes defined by their pressure signature and its Fourier transform in physical and spectral domains, respectively. For the present analysis, the ship hull of a given geometry is represented in a simplified form as a sum of two-dimensional Gaussian functions allowing analysis of interaction of wakes generated by bow-stern hull components or transverse multi-hull systems. An arbitrary movement of the pressure is realised through appropriate mathematical operations of translation and rotation of rigidly coupled component Gaussian forms directly in a spectral domain to preserve high model performance. Thus, the method is capable of giving fast predictions of wakes generated by an arbitrarily moving pressure distribution for different shapes of a simulated vessel.

Necessary optimality conditions for approximate Pareto efficient solutions of nonsmooth fractional interval-valued multiobjective optimization problems

This paper deals with approximate Pareto efficient solutions of a nonsmooth fractional interval-valued multiobjective optimization. We first introduce some types of approximate Pareto efficient solutions of the considered problem by considering the lower-upper interval order relation. Then we apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions of Karush–Kuhn–Tucker (KKT)-type for these solutions.

Modified Variational Iteration Method for Solving Sedimentary Ocean Basin Moving Boundary Value Problem

The motion of the shoreline in sedimentary ocean basin were investigated and solved using the modified variational iteration method since it provides an efficient approximate solution of the problem under consideration, which appears to be accurate when the consecutive error between iterative solutions of the problem is evaluated. A diffusion equation with specified beginning and boundary conditions, with a moving interface type reflecting sediment transportation, is used to simulate the physical phenomena of sand particle migration in sea water. This article also discusses the shifting boundary value problem that occurs from shoreline change in a sedimentary ocean basin.

Toward an efficient approximate analytical solution for 4-compartment COVID-19 fractional mathematical model

With the recent trend in the spread of coronavirus disease 2019 (Covid-19), there is a need for an accurate approximate analytical solution from which several intrinsic features of COVID-19 dynamics can be extracted. This study proposes a time-fractional model for the SEIR COVID-19 mathematical model to predict the trend of COVID-19 epidemic in China. The efficient approximate analytical solution of multistage optimal homotopy asymptotic method (MOHAM) is used to solve the model for a closed-form series solution and mathematical representation of COVID-19 model which is indeed a field where MOHAM has not been applied. The equilibrium points and basic reproduction number (R0) are obtained and the local stability analysis is carried out on the model. The behaviour of the pandemic is studied based on the data obtained from the World Health Organization. We show on tables and graphs the performance, behaviour, and mathematical representation of the various fractional-order of the model. The study aimed to expand the application areas of fractional-order analysis. The results indicate that the infected class decreases gradually until 14 October 2021, and it will still decrease slightly if people are being vaccinated. Lastly, we carried out the implementation using Maple software 2021a.

Numerical investigation of generalized perturbed Zakharov–Kuznetsov equation of fractional order in dusty plasma

In the present work, the new iterative method with a combination of the Laplace transform of the Caputo’s fractional derivative has been applied to the generalized (3 + 1) dimensional fractional perturbed Zakharov–Kuznetsov equation in a dusty plasma. The proposed method is applied without any discretization and linearization. The numerical and graphical results show the accuracy of the proposed method for nonlinear differential equations. Moreover, the methods are easy to implement and give the efficient approximate solutions.

On approximate vector variational inequalities and vector optimization problem using convexificator

<abstract><p>In the present article, we study a vector optimization problem involving convexificator-based locally Lipschitz approximately convex functions and give some ideas for approximate efficient solutions. In terms of the convexificator, we approximate Stampacchia-Minty type vector variational inequalities and use them to describe an approximately efficient solution to the nonsmooth vector optimization problem. Moreover, we give a numerical example that attests to the credibility of our results.</p></abstract>

The Properties of Quasiconvex Functions and Their Applications in Multiobjective Optimization Problems

A new type of approximate subdifferential was proposed for quasiconvex functions. Their properties were studied, and the approximate subdifferential was applied to the characterization of approximate solutions to quasiconvex multiobjective optimization problems. Firstly, the existing approximate subdifferentials were improved to get a new approximate subdifferential of the quasiconvex function, and their relationships and properties were given. Then, the optimality conditions for approximate efficient solutions and approximate properly efficient solutions to quasiconvex multiobjective optimization problems were obtained by means of the new approximate subdifferential.

Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

An influence game is a simple game represented over an influence graph (i.e., a labeled, weighted graph) on which the influence spread phenomenon is exerted. Influence games allow applying different properties and parameters coming from cooperative game theory to the contexts of social network analysis, decision-systems, voting systems, and collective behavior. The exact calculation of several of these properties and parameters is computationally hard, even for a small number of players. Two examples of these parameters are the length and the width of a game. The length of a game is the size of its smaller winning coalition, while the width of a game is the size of its larger losing coalition. Both parameters are relevant to know the levels of difficulty in reaching agreements in collective decision-making systems. Despite the above, new bio-inspired metaheuristic algorithms have recently been developed to solve the NP-hard influence maximization problem in an efficient and approximate way, being able to find small winning coalitions that maximize the influence spread within an influence graph. In this article, we apply some variations of this solution to find extreme winning and losing coalitions, and thus efficient approximate solutions for the length and the width of influence games. As a case study, we consider two real social networks, one formed by the 58 members of the European Union Council under nice voting rules, and the other formed by the 705 members of the European Parliament, connected by political affinity. Results are promising and show that it is feasible to generate approximate solutions for the length and width parameters of influence games, in reduced solving time.

Bicriteria scheduling of a material handling robot in an m-machine cell to minimize the energy consumption of the robot and the cycle time

This study considers a flowshop type production system consisting of m machines. A material handling robot transports the parts between the machines and loads and unloads the machines. We consider the sequencing of the robot moves and determining the speeds of these moves simultaneously. These decisions affect both the robot’s energy consumption and the production speed of the system. In this study, these two objectives are considered simultaneously. We propose a second order cone programming formulation to find Pareto efficient solutions. We also develop a heuristic algorithm that finds a set of approximate Pareto efficient solutions. The conic formulation can find robot schedules for small cells with less number of machines in reasonable computation times. Our heuristic algorithm can generate a large set of approximate Pareto efficient solutions in a very short computational time. Proposed solution approaches help the decision-maker to achieve the best trade-off between the throughput of a cell and the energy efficiency of a material handling robot.

Making the Most of Parallel Composition in Differential Privacy

Abstract We show that the ‘optimal’ use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that ‘overlap’ on the data domain, a quantity we call the maximum overlap of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of f-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for f-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to 1020000), and that its application can reduce the noise added to query answers by up to 60%.

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

A differential evolution algorithm for the customer order scheduling problem with sequence-dependent setup times

Although the customer order scheduling problem to minimize the total completion time has received a lot of attention from researchers, the literature has not considered so far the case where there are sequence-dependent setups between jobs belonging to different orders, a case that may occurs in real-life scenarios. For this NP-hard problem we develop a novel efficient approximate solution procedure. More specifically, we develop an innovative discrete differential evolution algorithm where differential mutations are performed directly in the permutation space and that uses a novel, parameter-free, restart procedure. The so-obtained solutions are improved by two proposed local search mechanisms that employ problem-specific, heuristic dominance relations. We carry out an extensive computational experience with randomly generated test instances to compare our proposal with existing algorithms from related problems. In these experiments, the proposed algorithm obtains the best results in terms of their average relative percentage deviation and success rate. Furthermore, an analysis of variance test, followed by a Tukey’s test, confirms the excellent performance of the algorithm proposed.