Large-Number Optimization: Exact-Arithmetic Mathematical Programming with Integers and Fractions Beyond Any Bit Limits

With the emergence of increasingly computing-intensive and delay-sensitive tasks, the processing of computing tasks on cloud servers cannot meet the current needs any longer. The emergence of mobile edge computing (MEC) technology and the popularity of 5G applications can solve these demands. Offloading tasks to the MEC server reduces the energy consumption of local devices, and also has a lower latency than offloading to the cloud server. In this paper, an MEC–edge cloud server collaborative system model with energy harvesting technology is designed to minimize the processing delay of computing tasks by allocating computing resources. We propose an optimal integer linear programming (OILP) algorithm with two steps. Firstly, we propose a Lyapunov stability optimization algorithm based on task priority. With the constraints of local mobile device power stability, the divide-and-conquer idea is used for solving the target values of the processing tasks locally, and the MEC and edge cloud servers separately. Therefore, the objective problem is transformed into an integer linear programming problem, and then an integer linear programming algorithm based on CPU utilization optimization is proposed to obtain a resource allocation scheme. Simulation results show that the proposed OILP algorithm can further reduce the delay, improve the CPU’s utilization of the MEC server, and reduce the number of the tasks that cannot be processed under the condition of the energy stability of the local device.

A suggested algorithm to solve triangular fuzzy rough integer linear programming (TFRILP) problems with α-level is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are triangular fuzzy rough numbers. In real-life situations, the parameters of a linear programming problem model may not be defined precisely, because of the current market globalization and some other uncontrollable factors. In order to solve this problem, a proper methodology is adopted to solve the TFRILP problems by the slice-sum method with the branch-and-bound technique, through which two fuzzy integers linear programming (FILP) problems with triangular fuzzy interval coefficients and variables were constructed. One of these problems is an FILP problem, where all of its coefficients are the upper approximation interval and represent rather satisfactory solutions; the other is an FILP problem, where all of its coefficients are the lower approximation interval and represent completely satisfactory solutions. Moreover, α-level at α = 0.5 is adopted to find some other rough value optimal solutions and decision rough integer variables. Integer programming is used, since a lot of the linear programming problems require that the decision variables be integers. In addition, the motivation behind this study is to enable the decision makers to make the right decision considering the proposed solutions, while dealing with the uncertain and imprecise data. A flowchart is also provided to illustrate the problem-solving steps. Finally, two numerical examples are given to clarify the obtained results.

Ranking in quadratic integer programming problems

Data-parallel applications generate a mix of coflows with and without deadlines. Deadline coflows are mission-critical and must be completed within deadlines, while the non-deadline coflows desire to be completed as soon as possible. Scheduling such mix-coflows is an important problem in modern datacenters. However, existing solutions only focus on one of the two types of coflows: they either solely concentrate on meeting the deadlines of deadline-aware coflows or reducing the coflow completion times (CCTs) of non-deadline coflows. In this article, we study the problem of optimizing deadline and non-deadline coflows simultaneously. To this end, we present a new optimization framework, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixCoflow</i> , to schedule deadline coflows to minimize and balance their bandwidth footprint, such that non-deadline coflows can be scheduled as early as possible. Specifically, we develop the mathematical model and formulate the scheduling problem for deadline coflows as a lexicographical min-max integer linear programming (ILP) problem. Through rigorous theoretical analysis, this ILP problem has been proved to be equivalent to a linear programming (LP) problem that can be solved with standard LP solvers. By solving this LP, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixCoflow</i> is able to balance the bandwidth footprint of deadline coflows while guaranteeing their deadlines. As a result, non-deadline coflows can be scheduled as soon as possible whenever they arrive. To demonstrate the effectiveness of our work, we have conducted extensive simulations based on a widely used Facebook data trace. The simulation results verify that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixCoflow</i> can achieve significant improvement on the average CCT of non-deadline coflows, at no expense of increasing the deadline miss rates of deadline coflows, when compared to the state-of-art solutions.

This paper addresses the fault diagnosis problem in discrete event systems using backward conflict-free Petri nets, where two different transitions have no common output places. The notion of generalized markings is employed, where the number of tokens in a place can be negative. Based on the special structures of backward conflict-free Petri nets and the generalized marking corresponding to an observed transition sequence, an integer linear programming (ILP) problem is constructed where the numbers of unknown integer variables and constraints are independent of the length of the observed transition sequence. A diagnosis algorithm is proposed by associating two different objective functions with the ILP problem and solving it. The computational efficiency of the proposed algorithm depends on the net structure only, regardless of the length of an observed transition sequence.

In this paper we deal with multiobjective linear and quadratic programming problem with uncertain information. So far in the field of statistical analysis and data mining, e.g., mean-variance portfolio problem, support vector machine and their varieties, we have encountered various kinds of quadratic and linear programming problems with multiple criteria. Moreover coefficients in such problems have uncertainty that is expressed by interval, probabilistic distribution or possibilistic (fuzzy) distribution. In this paper, we define a robust basis for all possible perturbation of coefficients within intervals in objective functions and constraints that is regarded as secure and conservative solution under uncertainty. According to the conventional multi-objective programming literature, it is required to solve test subproblem for each basis. Therefore, in case of our interval problem excessive computational demand is estimated. In this paper investigating the properties of robust basis by means of combination of interval extreme points we obtained the result that the robust basis can be examined by working with only a finite subset of possible perturbations of the coefficients

We propose a framework designed to tackle a multi-objective optimization challenge related to the placement of applications in fog computing, employing a deep reinforcement learning (DRL) approach. Unlike other optimization techniques, such as integer linear programming or genetic algorithms, DRL models are applied in real time to solve similar problem situations after training. Our model comprises a learning process featuring a graph neural network and two actor-critics, providing a holistic perspective on the priorities concerning interconnected services that constitute an application. The learning model incorporates the relationships between services as a crucial factor in placement decisions: Services with higher dependencies take precedence in location selection. Our experimental investigation involves illustrative cases where we compare our results with baseline strategies and genetic algorithms. We observed a comparable Pareto set with negligible execution times, measured in the order of milliseconds, in contrast to the hours required by alternative approaches.

This paper presents a distributed server allocation model with preventive start-time optimization against a single server failure. The presented model preventively determines the assignment of servers to users under each failure pattern to minimize the largest maximum delay among all failure patterns. We formulate the proposed model as an integer linear programming (ILP) problem. We prove the NP-completeness of the considered problem. As the number of users and that of servers increase, the size of ILP problem increases; the computation time to solve the ILP problem becomes excessively large. We develop a heuristic approach that applies simulated annealing and the ILP approach in a hybrid manner to obtain the solution. Numerical results reveal that the developed heuristic approach reduces the computation time by 26% compared to the ILP approach while increasing the largest maximum delay by just 3.4% in average. It reduces the largest maximum delay compared with the start-time optimization model; it avoids the instability caused by the unnecessary disconnection permitted by the run-time optimization model.

In this paper, we define a new problem of provisioning lightpaths and computing resources for a set of location-transparent scheduled grid demands in optical grid networks. A location-transparent scheduled grid demand specifies only an amount of computing resources needed in a specified time interval to process input data. The network node generating a demand is called a client node. There are several network nodes which have sufficient resources for a demand. These nodes are called resource nodes. An algorithm is used to choose a resource node to reserve a specified amount of computing resources and provision a lightpath between the resource node and the client node. Given a set of location-transparent scheduled grid demands, it is required to provision the best lightpath (i.e. wavelength resources) as well as computing resources available during the specified time interval for each demand so as to optimize a certain objective function. In our work, we develop integer linear programming (ILP) formulations for 2 objective functions: 1) Given a network capacity, maximize the number of demands accepted; 2) Minimize the total number of wavelength-links to honor a given set of demands. Because the ILP algorithms are computationally expensive, we also develop heuristics to deal with large networks. The simulation results show that our heuristics achieve good performance.

Nodes in the wireless sensor networks are battery operated, and thus, the efficient use of the node's energy during wireless communication is pivotal for the long battery life. This paper addresses the challenges in energy-efficient data aggregation and transmission scheduling for each node by using time division multiple access (TDMA). We introduce multi-channel TDMA scheduling algorithms with the objective of minimizing the total energy consumption in the network. The proposed algorithms utilize multiple radio channels to bestow efficient scheduling while eliminating collisions and overhearing. First, we formulate an integer linear programming (ILP) algorithm that finds the minimum bound for the network's energy consumption. Then, we propose a near-optimal heuristic algorithm based on backtracking, which uses memoization to spurn the suboptimal schedules. Subsequently, we propose a computationally efficient heuristic algorithm by using Langford subset generation. The algorithm reduces the energy consumption in each timeslot while avoiding revisiting the same timeslot. We conducted extensive simulations to evaluate three proposed algorithms. The simulation results demonstrate that the proposed heuristic algorithms provide performance comparable to the optimum results of the ILP algorithm and reduce the magnitude of computation time.

As millimeter wave (mmWave) technology becomes more integrated into today's communication infrastructure, its application to mobile scenarios has attracted considerable interests. In particular, mmWave technology is one of the leading candidates in Vehicle-to-Everything (V2X) community to address the ever-rising data exchange demand from vehicles. However, the inherent directionality of the technology poses challenges to peer discovery and transmission scheduling in typical V2X scenarios, causing significant loss of network capacity. To mitigate this issue, we propose a novel sub-6 GHz V2X-assisted synchronous mmWave communication scheduler in this paper. The sub-6 GHz channel serves as control channel for mmWave scheduling. The scheduler carefully considers data importance and freshness in determining communication schedule, such that network utility across mmWave links is maximized. We formulate the scheduling problem into an integer linear programming (ILP) problem. However, ILP problem is NP-hard and has exponential computational complexity for the worst cases. We then develop a scheduling algorithm, maximum weight matching based iterative algorithm (MWMIA), that operates in polynomial time with performance guarantees for the NP-hard ILP, by leveraging structural properties of link conflicts. Comparative simulation studies verify the effectiveness of our scheduler against other solutions.

A Survey of Lagrangean Techniques for Discrete Optimization

The electric energy consumption in passenger vehicles is rapidly increasing. To limit the associated increase in fuel consumption, an energy management system has been developed. This system exploits the fact that the losses in the internal combustion engine vary with the operating point, and uses the possibility to temporarily store electric energy in a battery, such that electric energy is only produced at moments when it is cheap to generate. To come to a practically applicable solution, a vehicle model is derived, containing only the component characteristics relevant for this application. The energy management problem is formulated as an optimization problem. The fuel consumption over a driving cycle is minimized, while respecting physical limitations of the components and maintaining an acceptable energy level of the battery. Several optimization methods are studied to come to a solution. The dynamic optimization problem is solved using Dynamic Programming. After rewriting it as a static optimization problem and approximating the cost function by a quadratic function, the problem is solved using Quadratic Programming, which requires less computation time. A real-time implementable strategy has been derived from the Quadratic Programming problem, that does not require a prediction of the future driving cycle. This strategy compares the current cost of producing electric energy with the estimated average cost. By adapting the average cost based on the energy level of the battery, it is ensured that the battery energy level will remain around the desired value. Simulations show that a fuel reduction up till 2% can be obtained on a conventional vehicle without major hardware changes. Higher reductions are possible on the exhaust emissions. To predict and explain the amount of fuel reduction that can be obtained with a given vehicle configuration, a set of engineering rules is derived based on typical component characteristics. Their results correspond reasonably well with the simulations. An advanced power net topology is studied which contains both a battery and an ultracapacitor that are connected by a DC-DC converter and a switch. Because of the increased complexity, this system is modeled using linear and piecewise linear approximations of the component characteristics, such that the energy management problem can be casted as a Linear Programming problem. The discrete switch makes it a Mixed Integer Linear Programming Problem. A realtime strategy, similar to the strategy for the conventional power net, has been derived. The addition fuel reduction that is obtained with the dual storage power net is small, because the maximum profit that can be obtained with an ideal lossless battery is not much higher than with a normal battery. Subsequently, hybrid electric vehicles are studied that use an Integrated Starter Generator which can be used for generating electric power and for vehicle propulsion. Several conSummary figurations are studied with respect to their potential fuel reduction. Configurations that enable start-stop operation of the engine obtain a much higher fuel reduction, up to 40%. The controllers are tested in real-time on a Hardware-in-the-Loop environment, where a vehicle simulation model is combined with existing electric components. It is shown that the electric power setpoints provided by an energy management strategy can be realized in practice.

Integer Programming for Optimal Control of Geostationary Station Keeping of Low-Thrust Satellites

We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.