Linear Programming Algorithm Research Articles

A step-truncated method in a wide neighborhood interior-point algorithm for linear programming

A step-truncated method in a wide neighborhood interior-point algorithm for linear programming

Solution of the real and complex eigenvalue problems in the ABS class

Abstract The class of ABS methods was originally developed for solving systems of linear equations in a finite number of iterations. Later it was shown that methods for solving nonlinear systems of equations, linear programming algorithms, quadratic programming, etc. are also members of the ABS class. In this paper, we show that QR factorization can also be derived from three ABS subclasses, which is an essential step in methods solving eigenvalue problems. Another possibility to use the ABS class in the eigenvalue problems is to transform a matrix to Hessenberg form which is an important initial step when solving eigenvalue problems by QR-type methods. Here we study versions of the plain QR-method which are based on the QR factorization computed by the ABS methods and compare them with the MATLAB eig () function. We also compare the ABS Hessenberg method to the MATLAB hess() function. The preliminary results show the competitiveness of 3 ABS eigenvalue solving methods with the MATLAB eig () function.

Scheduling Scheme Optimization for Emergency Resources on a Deep-sea Considering Environmental Impacts

An efficient and timely emergency response to a major accident is more challenging for the exploration and utilization of deep-sea oil and gas fields due to harsh external environment factors. An optimal solution is needed to tackle the external environment changes in the demand for emergency resources. To effectively balance emergency response time and the satisfaction of resource scheduling quantity, this paper proposes a multiple rescue points and multi-objective optimization model to address such challenges. Such model takes into account the environment factors involving wind speed and wave height caused by the variability of the deep-sea monsoon climate. Additionally, to handle the uncertainty of the resource demand at the accident sites, a linear programming and heuristic hybrid algorithm is proposed to help decision makers select the ideal routes and minimal scheduling time of deep-sea emergency resources, as well as the optimal operation season. A case study of a blowout accident is conducted to demonstrate the application of the proposed model and the real-world implications.

An exact reduction technique for the k-Colour Shortest Path Problem

The k-Colour Shortest Path Problem is a variant of the classic Shortest Path Problem. This problem consists of finding a shortest path on a weighted edge-coloured graph, where the maximum number of different colours used in a feasible solution is fixed to be k. The k-CSPP has several real-world applications, particularly in network reliability. It addresses the problem of reducing the connection cost while improving the reliability of the network. In this work, we propose a heuristic approach, namely Colour-Constrained Dijkstra Algorithm (CCDA), which is able to produce effective solutions. We propose a graph reduction technique, namely the Graph Reduction Algorithm (GRA), which removes more than 90% of the nodes and edges from the input graph. Finally, using a Mixed-Integer Linear Programming (MILP) model, we present an exact approach, namely Reduced Integer Linear Programming Algorithm (RILP), that takes advantage of the heuristic CCDA and the GRA. Several tests were performed to verify the effectiveness of the proposed approaches. The computational results indicate that the produced approaches perform well, in terms of both the solution’s quality and computation times.

Optimization and Performance Analysis of a Distributed Energy System Considering the Coordination of the Operational Strategy and the Fluctuation of Annual Hourly Load

The operation strategies of a distributed energy system (DES) are usually proposed according to the electrical load (FEL) and the thermal load (FTL), which take the cooling/heating load or electric load as unique constraint conditions that result in a too high or too low equipment load rate. This paper proposes a new hybrid operation strategy (HOS) that takes the full utilization of natural gas and the minimization of power consumption from the power grid as constraints and coordinates the cooling/electricity ratio and heating/electricity ratio of buildings and equipment. In the optimization phase of a DES, an optimization method based on the discretization of the load is proposed to investigate the influence of the uncertainty of the load on the DES, which helps to avoid repeated load simulations and provides stronger adjustability by quoting the normal distribution function to obtain multiple sets of load data with different fluctuations. Further, a multi-objective optimization model combining the genetic algorithm (GA) and mixed integer linear programming algorithm (MILP) was established to find the optimal configuration of equipment capacities by comprehensively considering the annual total cost, carbon emissions, and energy efficiency of the DES. Finally, an office building example was used to validate the feasibility of the above theories and methods. Compared with the FEL and FTL, the HOS reduced the energy waste of the DES by 19.7% and 15.5%, respectively. Compared with using a typical daily load, using an annual hourly load to optimize the DES-HOS produced a better comprehensive performance and lower adverse impacts derived from load fluctuations.

Simple and fast algorithm for binary integer and online linear programming

In this paper, we develop a simple and fast online algorithm for solving a class of binary integer linear programs (LPs) arisen in general resource allocation problem. The algorithm requires only one single pass through the input data and is free of matrix inversion. It can be viewed as both an approximate algorithm for solving binary integer LPs and a fast algorithm for solving online LP problems. The algorithm is inspired by an equivalent form of the dual problem of the relaxed LP and it essentially performs (one-pass) projected stochastic subgradient descent in the dual space. We analyze the algorithm in two different models, stochastic input and random permutation, with minimal technical assumptions on the input data. The algorithm achieves \(O\left( m \sqrt{n}\right) \) expected regret under the stochastic input model and \(O\left( (m+\log n)\sqrt{n}\right) \) expected regret under the random permutation model, and it achieves \(O(m \sqrt{n})\) expected constraint violation under both models, where n is the number of decision variables and m is the number of constraints. The algorithm enjoys the same performance guarantee when generalized to a multi-dimensional LP setting which covers a wider range of applications. In addition, we employ the notion of the permutational Rademacher complexity and derive regret bounds for two earlier online LP algorithms for comparison. Both algorithms improve the regret bound by a factor of \(\sqrt{m}\) by paying more computational cost. Furthermore, we demonstrate how to convert the possibly infeasible solution to a feasible one through a randomized procedure. Numerical experiments illustrate the general applicability and effectiveness of the algorithms.

A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.

Stable Torsion Length

Abstract The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the 1st exact computations of stable torsion length for nontrivial examples.

Optimizing Gradient Methods for IoT Applications

Solving linear programming (LP) and nonlinear programming (NLP) problems is momentous because of their wide applications in real-life problems. There is no unified way to find the global optimum for NLPs. But on the other hand, the simplex algorithm, as the dominating methodology for LPs for several decades moves only on the boundary (vertices) and ignores the vast majority of the feasible region in the process of searching. In this article, we study two gradient-based methodologies that explore the whole feasible region, which guarantee faster convergence rates for both LP and NLP optimization problems including IoT problems such as Software-Defined Internet of Vehicles (SDIoV) and vehicular ad hoc networks (VANETs). The gradient-simplex algorithm (GSA) for LPs, which moves inside the feasible region in the gradient direction at first to reduce the search space and then explores the reduced boundary to find an optimal solution. The evolutionary-gradient algorithm (EGA), on the other hand, is for NLPs and uses an evolutionary population to estimate gradients by evolving to find better solutions in steps. Based on extensive simulations, the obtained numerical results show that both approaches provide efficient solutions and outperform the state-of-the-art methods on optimization problems with large feasible spaces. Comparative results of applying the GSA on SDIoV and VANETs with different sizes are included.

Design and field measurement of dipole magnets for a 300-MeV proton and heavy ion synchrotron in SESRI

Space Environment Simulation and Research Infrastructure (SESRI) is a new national research infrastructure under construction at Harbin Institute of Technology (HIT) in China. Its 300-MeV proton and heavy ion accelerator is a major radiation source, and a compact synchrotron has been designed with a circumference of less than 44 m. The synchrotron dipoles are essential components of the accelerator, and they have been designed and tested carefully to meet the stringent physical requirements. The measured transverse field homogeneity is less than 2 × 10-4 from 0.2 T to 1.4 T and roughly agrees with the simulated results. With a modified linear programming algorithm, the integral magnetic field uniformity within 0.2–1.1 T is optimised from 6 × 10-4 to 1.5 × 10-4, and the corresponding multi-pole field-component errors are greatly reduced. By shortening the pole ends, the reproducibility of integral field among six magnets at 0.2 T is less than 3 × 10-4. In addition, the transient effect of the magnetic field is also evaluated and tested. Through analysis of the transient effect and structural strength, low-carbon steel and stainless steel are selected for the side plates and end plates, respectively. According to the transient measurement results, the feedforward compensation can be implemented to produce a more accurate current waveform. On 1 January 2022, the beam was successfully accumulated with the preset current waveform, and the beam intensity met the design requirements.

Day-ahead local flexibility market for active and reactive power with linearized network constraints

This paper presents a Day-Ahead Local Flexibility Market in the Distribution Network cleared using a novel Successive Linear Programming algorithm for Optimal Power Flow in the Distribution Network. The proposed algorithm is shown to be robust and produce high quality, AC feasible solutions with satisfactory solution speed. The efficiency of the proposed algorithm is validated via its application in various distribution networks and under different loading scenarios. The proposed Flexibility Market encompasses flexibility products for upward and downward active energy as well as service products for the provision of reactive power. The efficiency of the proposed Flexibility Market in solving operational issues of the distribution network is demonstrated via a case-study.

A Novel Solution for Optimized Energy Management Systems Comprising an AC/DC Hybrid Microgrid System for Industries

A novel solution for optimized energy management comprising a microgrid system for industries in Pakistan is proposed. The proposed study considered microgrids based on photovoltaics, wind turbines, power storage systems, and dual-fuel (DF) generators as backup. A heuristic methodology with a cuckoo search algorithm (CSA) is presented for efficient power trading by scheduling machines. The study was conducted to prove that CSA is adaptable and flexible for self-governing choices for the efficient management and scheduling of machines and power trade between the microgrid and commercial grid. A mixed integer linear programming algorithm is introduced to optimize the system design problems that control decision making for the ideal operation management. A real-time pricing scheme is utilized for electricity price figures. The simulation results show the efficient performance of the proposed scheme to maximize profitability, reduction in electricity cost, and peak to average ratio. Furthermore, the proposed optimization technique was compared with a highly in-use strawberry algorithm to prove the supremacy of the proposed technique. The proposed efficient and robust energy management system was implemented in Shafi Dyeing Industry, Faisalabad, to validate the simulated model.

THE FUZZY OPTIMISTIC-REASONABLE-PESSIMISTIC INVENTORY MODEL

In inventory and production decision problems, decision makers are interested to identify the optimal inventory and production level. In a certain decision environment, the optimal inventory level could be determined through traditional inventory methods and the optimal ptoduction level could be determined through linear programming algorithms. In an uncertain decision environment, the traditional methods and algorithms can not provide efficient and relevant solutions for these levels, due to the vague and changing parameters. In this case it is neccesary to develop new methods and models that can deal with vague variables and provide optimal levels. In this paper, the optimal inventory and production levels are determined through a single model that uses fuzzy linear programming. This new model is Fuzzy Optimistic-Reasonable-Pessimistic Inventory Model. It has three scenario: optimistic, reasonable and pessimistic, that are defined through triangular fuzzy numbers. In this way, decision makers can deal with vague parameters. These scenarios help managers to divide the Fuzzy ORP Model into three sub-models, that can be easily solved through traditional Simplex Algorithms. Each sub-model provides a crisp solution for each scenario. The solutions forms the final fuzzy optimal solution. The Fuzzy PRO Inventory Model helps managers to identify three optimal levels and to rank them according to their evaluations. This is useful, also, in predictions, where the decision makers should predict different scenarios for the production process. The limit of this model is the definition of the variables and scenarios. This model consider that all values for all variables and coefficients have the same definition: the inferior limit is related to the optimistic sceanrio, the peak is represents the reasonable limit and the superior limit is related to the pessimistic scenario. In real problem, the decision variables could have different definition than coefficients. The inferior limit of the cost is related to the optimistic scenario, but the superior limit of the production level can be related to the optimistic scenario. There are different representations for the scenarios.

Greenhouse gas emission reduction in residential buildings: A lightweight model to be deployed on edge devices

Electricity produced and used in the residential sector is responsible for approximately 30% of the greenhouse gas emissions (GHGE). Insulating houses and integrating renewable energy and storage resources are key for reducing such emissions. However, it is not only a matter of installing renewable energy technologies but also of optimizing the charging/discharging of the storage units. A number of optimization models have been proposed lately to address this problem. However, they are often limited in several respects: (i) they often focus only on electricity bill reduction, placing GHGE reduction on the backburner; (ii) they rarely propose hybrid-energy storage optimization strategies considering thermal and storage heater units; (iii) they are often designed using Linear Programming (LP) or metaheuristic techniques that are computational intensive, hampering their deployment on edge devices; and (iv) they rarely evaluate how the model impacts on the battery lifespan. Given this state-of-affairs, the present article compares two approaches, the first one proposing an innovative sliding grid carbon intensity threshold algorithm developed as part of a European project named RED WoLF, the second one proposing an algorithm designed based on LP. The comparison analysis is carried out based on two distinct real-life scenarios in France and UK. Results show that both algorithms contribute to reduce GHGE compared to a solution without optimization logic (between 10 to 25%), with a slight advantage for the LP algorithm. However, RED WoLF makes it possible to reduce significantly the computational time (≈25 min for LP against ≈1ms for RED WoLF) and to extend the battery lifespan (4 years for LP against 12 years for RED WoLF).

Projected orthogonal vectors in two-dimensional search interior point algorithms for linear programming

Projected orthogonal vectors in two-dimensional search interior point algorithms for linear programming

SIR Epidemics with State-Dependent Costs and ICU Constraints: A Hamilton–Jacobi Verification Argument and Dual LP Algorithms

The aim of this paper is twofold. On one hand, we strive to give a simpler proof of the optimality of greedy controls when the cost of interventions is control-affine and the dynamics follow a state-constrained controlled SIR model. This is achieved using the Hamilton–Jacobi characterization of the value function, via the verification argument and explicit trajectory-based computations. Aside from providing an alternative to the Pontryagin complex arguments in Avram et al. (Appl Math Comput 418:126816, 2022) (see also Avram et al. in Appl Math Comput 423:127012, 2022), this method allows one to consider more general classes of costs; in particular state-dependent ones. On the other hand, the paper is completed by linear programming methods allowing one to deal with possibly discontinuous costs. In particular, we propose a brief exposition of classes of linearized dynamic programming principles based on our previous work and ensuing dual linear programming algorithms. We emphasize the particularities of our state space and possible generations of forward scenarios using the description of reachable sets.

Hierarchical Scheduling Control Method for Cascade Hydro-PV-Pumped Storage Generation System

High photovoltaic penetration in a power system has significantly challenged its safety and economic operation. To use the complementary characteristics of various renewable energy sources (RESs) fully, a novel hierarchical scheduling control (HSC) method is presented to accommodate the variability and uncertainty of a cascade hydro-PV-pumped storage (CH-PV-PS) generation system. Considering the optimization functions and execution requirements of the CH-PV-PS system, the HSC method is divided into two layers: the dynamic optimization layer and the static optimization layer. The static optimization layer focuses on the economy of the CH-PV-PS system, and the dynamic optimization layer focuses on the safety of the CH-PV-PS system. In the first layer, that is, the static optimization layer, the objectives of the day-ahead and hour-ahead schedules are established, and a heuristic algorithm is combined with a linear programming algorithm to optimize the energy allocation. Considering the uncertainty of the PV power output and hour-ahead load, a real-time schedule is established in the second layer; that is, in the second layer, the dynamic optimization layer, real-time scheduling and prediction of active output are established. Model predictive control methods are introduced to correct for prediction bias at different time scales in order to fully utilize the control capability of hydropower generation. A CH-PV-PS real-world system in Southwest China is chosen as a case study. In the three scenarios, where only PV fluctuations are considered, the simulation results reveal that, compared with the traditional open-loop optimized and hierarchical open-loop optimization methods, the HSC method reduces the average relative deviation of PV and increases the system economics. After a large amount of RESs are connected to the power grid, the HSC method provides a solution for improving the consumption of RESs.

Optimization of aircraft remote power distribution unit placement using linear programming to decrease the wire weight

The modern large civil aircraft has the trend to utilize the remote distribution network. This paper presents an optimization concept for this distribution network to optimize remote power distribution unit placement in aircraft to decrease the wire weight. A model formulation has been created to minimize the total wire weight. The length of wire was calculated approximately by the different longitinal axis of center of gravity of wire ends. To solve the wiring weight reduction problem, the loads are selected to connect to specific remote power distribution unit which position is close, then model formulation has been transformed to the linear programming algorithm. In order to validate the concept a simulation model and different calculation case have been built. The calculation results show the method can decrease wires weight in remote power distribution system and has shown high benefits.

Optimization on Ready-Mixed Concrete Dispatching Problem via Sliding Time Window Searching

The dispatching of ready-mixed concrete (RMC) is usually scheduled manually and difficult to meet the demand of construction sites. The inefficient dispatching plan causes concrete defects of discontinuous casting (cold joint formation) or long on-site queuing (concrete early hardening inside RMC truck). The RMC dispatching literature focus on optimizing the dispatching intervals to minimize RMC truck on-site queuing, system operation costs of RMC plant and construction sites, or tradeoff. The previous research methods include mixed-integer linear programming (MILP) and evolutionary algorithms. They have the limitations of providing optimal solutions or computation efficiency. This paper develops a Binary Linear programming with Sliding Time-Window (BLSTW) to overcome these limitations. The BLSTW model subtly determines the dispatch of RMC trucks through binary variables incorporated into sliding time window search approach. Two perspectives, from the RMC plant supplier and construction site demanders, are discussed respectively. Three case studies are conducted to verify the efficiency improvement of the developed model including one largescale example of 25 sites. The comparisons made with two cited results shows the superior of the developed model.

A Hybrid Prioritization Model for Funding School Maintenance and Rehabilitation Projects Using Data Envelopment Analysis and Linear Programming Algorithms

A Hybrid Prioritization Model for Funding School Maintenance and Rehabilitation Projects Using Data Envelopment Analysis and Linear Programming Algorithms