Split Problem Research Articles

Cloud‐fog assisted human tracing and disaster evacuation framework

AbstractMany kinds of natural hazards and man‐made calamities have occurred in recent years, each of which has left a huge number of victims and has been challenging to deal with. The solution is an effective evacuation planning system, which is gaining the interest of governments, academic institutions, and industries all across the world. This study focuses on quick and organized evacuation from densely populated areas during disasters to decrease damage successfully. This research proposes a fog‐cloud‐assisted effective disaster evacuation framework. The data accumulation layer uses IoT, mobile networks, and GPS technologies to collect environmental and location‐based data to track the inhabitants. The fog layer is utilized for (i) disaster event detection and (ii) human tracing and crowd density analysis. A cloud layer makes it easier for an evacuation algorithm to produce an evacuation map that will point evacuees to the exit while computing a quick and safe path utilizing environmental and inhabitant information gathered from the fog. The findings reveal that the performance of Model A surpasses the split delivery vehicle routing problem model with the minimum demand of vehicles and with improvement of 38.1% average. Also, A* takes less time (18.3 s) than other state‐of‐the‐art techniques to find the best optimal route for evacuation. Implementation and performance analysis demonstrate the efficiency and effectiveness of the proposed framework.

Solving a Continent-Scale Inventory Routing Problem at Renault

This paper is the fruit of a partnership with Renault. Their reverse logistic requires solving a continent-scale multiattribute inventory routing problem (IRP). With an average of 30 commodities, 16 depots, and 600 customers spread across a continent, our instances are orders of magnitude larger than those in the literature. Existing algorithms do not scale, so we propose a large neighborhood search (LNS). To make it work, (1) we generalize existing split delivery vehicle routing problems and IRP neighborhoods to this context, (2) we turn a state-of-the-art matheuristic for medium-scale IRP into a large neighborhood, and (3) we introduce two novel perturbations: the reinsertion of a customer and that of a commodity into the IRP solution. We also derive a new lower bound based on a flow relaxation. In order to stimulate the research on large-scale IRP, we introduce a library of industrial instances. We benchmark our algorithms on these instances and make our code open source. Extensive numerical experiments highlight the relevance of each component of our LNS. Funding: This work was supported by Renault Group. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0342 .

New iterative algorithms for solving a class of split common solution problems and their applications

The purpose of this paper is to introduce new iterative algorithms for approximating a solution to a class of monotone operator equations. More precisely, we study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. In order to solve this problem, we propose three new algorithms and establish strong convergence theorems for them.

New inertial self-adaptive algorithms for the split common null-point problem: application to data classifications

In this paper, we propose two inertial algorithms with a new self-adaptive step size for approximating a solution of the split common null-point problem in the framework of Banach spaces. The step sizes are adaptively updated over each iteration by a simple process without the prior knowledge of the operator norm of the bounded linear operator. Under suitable conditions, we prove the weak-convergence results for the proposed algorithms in p-uniformly convex and uniformly smooth Banach spaces. Finally, we give several numerical results in both finite- and infinite-dimensional spaces to illustrate the efficiency and advantage of the proposed methods over some existing methods. Also, data classifications of heart diseases and diabetes mellitus are presented as the applications of our methods.

An algorithm for approximating solutions of the split variational inclusion problem

We introduce and study an iterative algorithm to approximate a solution to the split variational inclusion problem in real Hilbert spaces. The strong convergence of the proposed algorithm is proved via some suitable conditions. As applications, we derive consequences for several special cases such as the split variational inequality problem, the split minimum point problem, and the split common fixed point problem. Finally, we present some numerical experiments to demonstrate the performance of the proposed algorithm.

Inertial method for solving split equality Hammerstein-type equation problems in reflexive Banach spaces

The purpose of this paper is to introduce an inertial scheme for solving Hammerstein-type equation problems in general reflexive real Banach spaces, where the underlying mappings are generally continuous monotone, not just uniformly continuous monotone. A strong convergence theorem is proved under some mild conditions, and finally numerical examples are provided to demonstrate the applicability of the algorithm.

A Modified Viscosity-Type Self-Adaptive Iterative Algorithm for Common Solution of Split Problems with Multiple Output Sets in Hilbert Spaces

A modified viscosity-type self-adaptive iterative algorithm is presented in this study, having a strong convergence theorem for estimating the common solution to the split generalized equilibrium problem along with the split common null point problem with multiple output sets, subject to some reasonable control sequence restrictions. The suggested algorithm and its immediate consequences are also discussed. The effectiveness of the proposed algorithm is finally demonstrated through analytical examples. The findings presented in this paper will help to consolidate, extend, and improve upon a number of recent findings in the literature.

An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

A parallel algorithm for solving a pair of two variational inequalities in Hilbert spaces

The purpose of this paper is to introduce a new parallel algorithm for finding solutions to pairs of variational inequalities over the solution sets of systems of split equality null-point problems in Hilbert spaces. More precisely, we propose a new self-adaptive algorithm based on the hybrid steepest descent method and establish a strong convergence theorem for it. Moreover, we also give some applications of our proposed algorithm to solving some related problems, such as systems of split equality minimum point problems and systems of split equality feasibility problems.

An approximation algorithm for solving a split problem of fixed point and variational inclusion

In this paper, the split fixed point and variational inclusion problems are considered. With the help of fixed point technique, Tseng-type splitting method and self-adaptive rule, an iterative algorithm is proposed for solving this split problem in which the involved operators f and g are demicontractive operators and ϕ is plain monotone. Strong convergence theorem is proved under some mild conditions.

Inertial proximal point algorithm for the split common solution problem of monotone operator equations

Inertial proximal point algorithm for the split common solution problem of monotone operator equations

New inertial modification of regularized algorithms for solving split variational inclusion problem

In this paper, we introduce two new self-adaptive iterative algorithms for finding the solution of a split variational inclusion problem in real Hilbert spaces. The strong convergence theorems are proved under suitable conditions. In applications, the main results are applied for solving the split feasibility problem, the split minimization problem, and the split common fixed point problem. In order to test the numerical performances, the proposed algorithms are compared with several existing algorithms in the literature. Finally, the computational experiments show that the proposed algorithms are fast and more efficient than other approaches.

Perturbation Resilience of Self-Adaptive Step-Size Algorithms for Solving Split Variational Inclusion Problems and their Applications

In this paper, two viscosity proximal type algorithms involving the superiorization method are presented for solving the split variational inclusion problems in real Hilbert spaces. Strong convergence theorems and bounded perturbation resilience analysis of the proposed algorithms are obtained under mild conditions. The split feasibility problems, the split minimization problems, and the variational inequality problems are concerned as the applications, and several numerical experiments are performed to show the efficiency and implementation of the proposed algorithms.

A New Family of Route Formulations for Split Delivery Vehicle Routing Problems

We propose a new family of formulations with route-based variables for the split delivery vehicle routing problem with and without time windows. Each formulation in this family is characterized by the maximum number of different demand quantities that can be delivered to a customer during a vehicle visit. As opposed to previous formulations in the literature, the exact delivery quantities are not always explicitly known in this new family. The validity of these formulations is ensured by an exponential set of nonrobust constraints. Additionally, we explore a property of optimal solutions that enables us to determine a minimum delivery quantity based on customer demand and vehicle capacity, and this number is often greater than one. We use this property to reduce the number of possible delivery quantities in our formulations, improving the solution times of the computationally strongest formulation in the family. Furthermore, we propose new variants of nonrobust cutting planes that strengthen the formulations, namely limited-memory subset-row covering inequalities and limited-memory strong k-path inequalities. Finally, we develop a branch-cut-and-price (BCP) algorithm to solve our formulations enriched with the proposed valid inequalities, which resorts to state-of-the-art algorithmic enhancements. We show how to effectively manage the nonrobust cuts when solving the pricing problem that dynamically generates route variables. Numerical results indicate that our formulations and BCP algorithm establish new state-of-the-art results for the variant with time windows, as many benchmark instances with 50 and 100 customers are solved to optimality for the first time. Several instances of the variant without time windows are solved to proven optimality for the first time. Funding: This work was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 306033/2019-4, 313220/2020-4, and 314088/2021-0], the Région Nouvelle Aquitaine, France [Grant AAPR2020A-2020-8601810], the Agence Nationale de la Recherche [Grant ANR-20-CE40-0021-01], the Fundação de Amparo à Pesquisa do Estado de São Paulo [Grants 13/07375-0, 16/01860-1, and 19/23596-2], and the Paraíba State Research Foundation [Grants 261/2020 and 041/2023]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0085 .

Beamforming Analysis and Design for Wideband THz Reconfigurable Intelligent Surface Communications

Reconfigurable intelligent surface (RIS)-aided terahertz (THz) communications have been regarded as a promising candidate for future 6G networks because of its ultra-wide bandwidth and ultra-low power consumption. However, there exists the beam split problem, especially when the base station (BS) or RIS owns the large-scale antennas, which may lead to serious array gain loss. Therefore, in this paper, we investigate the beam split and beamforming design problems in the THz RIS communications. Specifically, we first analyze the beam split effect caused by different RIS sizes, shapes and deployments. On this basis, we apply the fully connected time delayer phase shifter hybrid beamforming (FC-TD-PS-HB) architecture at the BS and deploy distributed RISs to cooperatively mitigate the beam split effect. We aim to maximize the achievable sum rate by jointly optimizing the hybrid analog/digital beamforming, time delays at the BS and reflection coefficients at the RISs. To solve the formulated problem, we first design the analog beamforming and time delays based on different RISs’ physical directions, and then it is transformed into an optimization problem by jointly optimizing the digital beamforming and reflection coefficients. Next, we propose an alternatively iterative optimization algorithm to deal with it. Specifically, for given the reflection coefficients, we propose an iterative algorithm based on the minimum mean square error technique to obtain the digital beamforming. After, we apply Lagrangian dual reformulation (LDR) and multidimensional complex quadratic transform (MCQT) methods to transform the original problem to a quadratically constrained quadratic program, which can be solved by alternating direction method of multipliers (ADMM) technique to obtain the reflection coefficients. Finally, the digital beamforming and reflection coefficients are obtained via repeating the above processes until convergence. Simulation results verify that the proposed scheme can effectively alleviate the beam split effect and improve the system capacity.

Preferences on financing mechanisms for thermal retrofit measures in multi-owner buildings: A discrete choice experiment with landlords and owner-occupiers in France

Thermal retrofit of existing buildings is a major challenge for the energy transition. Retrofitting multi-owner buildings is particularly challenging because it involves multiple co-owners with heterogeneous preferences and incentives to renovate. We conduct a discrete choice experiment on thermal retrofit measures with landlords and owner-occupiers of condominiums in multi-owner buildings in France. Attributes include financing mechanisms (equity, private loan, and collective loan), transferability of loans (i.e. loans, which may be transferred to the next owner in case the condominium is sold), heating cost savings (absolute and relative to other co-owners in the same building), and investment costs. Results from estimating mixed logit models suggest that participants prefer on average equity financing over loan financing and collective loans over private loans. Aversion to loan financing appears to be partly due to debt aversion. The results also suggest that for loan financing, co-owners prefer transferable loans. Further, co-owners do not like retrofit measures that result in higher heating cost savings for other co-owners in the same building than for themselves. Thus, asymmetric distribution of benefits of thermal retrofit measures may impede acceptability of such measures in multi-owner buildings. Finally, the findings provide no evidence for a landlord-tenant split incentives problem in our sample.

Inertial Viscosity Approximation Methods for General Split Variational Inclusion and Fixed Point Problems in Hilbert Spaces

The purpose of this paper is to find a common element of the fixed point set of a nonexpansive mapping and the set of solutions of the general split variational inclusion problem in symmetric Hilbert spaces by using the inertial viscosity iterative method. Some strong convergence theorems of the proposed algorithm are demonstrated. As applications, we use our results to study the split feasibility problem and the split minimization problem. Finally, the numerical experiments are presented to illustrate the feasibility and effectiveness of our theoretical findings, and our results extend and improve many recent ones.

A branch-and-cut embedded matheuristic for the inventory routing problem

This paper presents an improved version of the solution method that won the inventory routing problem track of the 12th DIMACS Implementation Challenge. The solution method is a branch-and-cut embedded matheuristic where a matheuristic is called every time a new primal solution is found in a branch-and-cut method. The matheuristic consists of a construction heuristic and an improvement heuristic. The construction heuristic uses a giant tour method and a shifting assignments method to generate a set of promising routes which, in turn, are combined into a feasible solution to the problem by solving a route-based mathematical program. The improvement heuristic then solves a series of extended route-based mathematical programs where clusters of customers may be inserted and/or removed from the routes of the initial feasible solution. We have, to the best of our knowledge, gathered all detailed results from previously published methods for the inventory routing problem and made this overview available online. Compared with these results, the proposed method found the best-known solution for 741 out of 878 multi-vehicle inventory routing instances, where 247 of them are strictly better than the previously best-known solutions. Furthermore, we prove optimality for 458 of these solutions. The proposed method is also able to find the best-known solution for 116 out of 226 benchmark instances for the split delivery vehicle routing problem, and improve three best-known solutions from the CVRPlib for the capacitated vehicle routing problem.

Mann-type approximation scheme for solving a new class of split inverse problems in Hilbert spaces

ABSTRACT In this paper, we introduce and study a new class of split inverse problems, named split hierarchical monotone variational inclusion problem with multiple output sets in real Hilbert spaces. By using the inertial technique and self-adaptive step size strategy, we propose and analyze a new Mann-type iterative method for solving the problem. The convergence analysis of the proposed iterative method under some suitable conditions is studied. Also, we show that the sequence of iterates generated by this method converges strongly to a minimum-norm solution of the problem. As theoretical applications, we apply our results to approximate the solutions of other classes of split inverse problems. Finally, we present some numerical experiments to illustrate the practical potential and advantages of our proposed method.

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.