Showing papers on "Integer programming published in 2017"

Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks

Abstract: We present an approach for the verification of feed-forward neural networks in which all nodes have a piece-wise linear activation function. Such networks are often used in deep learning and have been shown to be hard to verify for modern satisfiability modulo theory (SMT) and integer linear programming (ILP) solvers.

On Dynamic Service Function Chain Deployment and Readjustment

Abstract: Network function virtualization (NFV) is a promising technology to decouple the network functions from dedicated hardware elements, leading to the significant cost reduction in network service provisioning. As more and more users are trying to access their services wherever and whenever, we expect the NFV-related service function chains (SFCs) to be dynamic and adaptive, i.e., they can be readjusted to adapt to the service requests’ dynamics for better user experience. In this paper, we study how to optimize SFC deployment and readjustment in the dynamic situation. Specifically, we try to jointly optimize the deployment of new users’ SFCs and the readjustment of in-service users’ SFCs while considering the trade-off between resource consumption and operational overhead. We first formulate an integer linear programming (ILP) model to solve the problem exactly. Then, to reduce the time complexity, we design a column generation (CG) model for the optimization. Simulation results show that the proposed CG-based algorithm can approximate the performance of the ILP and outperform an existing benchmark in terms of the profit from service provisioning.

Hermes: Latency Optimal Task Assignment for Resource-constrained Mobile Computing

Abstract: With mobile devices increasingly able to connect to cloud servers from anywhere, resource-constrained devices can potentially perform offloading of computational tasks to either save local resource usage or improve performance. It is of interest to find optimal assignments of tasks to local and remote devices that can take into account the application-specific profile, availability of computational resources, and link connectivity, and find a balance between energy consumption costs of mobile devices and latency for delay-sensitive applications. We formulate an NP-hard problem to minimize the application latency while meeting prescribed resource utilization constraints. Different from most of existing works that either rely on the integer programming solver, or on heuristics that offer no theoretical performance guarantees, we propose Hermes, a novel fully polynomial time approximation scheme (FPTAS). We identify for a subset of problem instances, where the application task graphs can be described as serial trees, Hermes provides a solution with latency no more than $(1+\epsilon)$ times of the minimum while incurring complexity that is polynomial in problem size and $\frac{1}{\epsilon}$ . We further propose an online algorithm to learn the unknown dynamic environment and guarantee that the performance gap compared to the optimal strategy is bounded by a logarithmic function with time. Evaluation is done by using real data set collected from several benchmarks, and is shown that Hermes improves the latency by $16$ percent compared to a previously published heuristic and increases CPU computing time by only $0.4$ percent of overall latency.

Dynamic passenger demand oriented metro train scheduling with energy-efficiency and waiting time minimization: Mixed-integer linear programming approaches

Abstract: In the daily operation of metro systems, the train scheduling problem aims to find a set of space-time paths for multiple trains that determine their departure and arrival times at metro stations, while train operations are in charge of selecting the best operational speed to satisfy the punctuality and operation costs. Different from the most existing researches that treat these two problems separately, this paper proposes an integrated approach for the train scheduling problem on a bi-direction urban metro line in order to minimize the operational costs (i.e., energy consumption) and passenger waiting time. More specifically, we simultaneously consider (1) the train operational velocity choices that correspond to the energy consumption of trains on each travelling arc, and (2) the dynamic passenger demands at each station for the calculation of total passenger waiting time in the planning horizon. By employing a space-time network representation in the formulations, this complex train scheduling and control problem with dynamic passenger demands is rigorously formulated into two optimization models with linear forms. The first model is an integer programming model that jointly minimizes train traction energy consumption and passenger waiting time. The second model, which is formulated as a mixed-integer programming model, further considers the utilization of regenerative braking energy on the basis of the first model. Due to the computational complexity of these two models, especially for large-scale real-world instances, we develop a Lagrangian relaxation (LR)-based heuristic algorithm that decomposes the primal problem into two sets of subproblems and thus enables to find a good solution in short computational time. Finally, two sets of numerical experiments, involving a relatively small-scale case and a real-world instance based on the operation data of Beijing metro are implemented to verify the effectiveness of the proposed approaches.

Mixed-Integer-Linear-Programming-Based Energy Management System for Hybrid PV-Wind-Battery Microgrids: Modeling, Design, and Experimental Verification

Abstract: Microgrids are energy systems that aggregate distributed energy resources, loads, and power electronics devices in a stable and balanced way. They rely on energy management systems to schedule optimally the distributed energy resources. Conventionally, many scheduling problems have been solved by using complex algorithms that, even so, do not consider the operation of the distributed energy resources. This paper presents the modeling and design of a modular energy management system and its integration to a grid-connected battery-based microgrid. The scheduling model is a power generation-side strategy, defined as a general mixed-integer linear programming by taking into account two stages for proper charging of the storage units. This model is considered as a deterministic problem that aims to minimize operating costs and promote self-consumption based on 24-hour ahead forecast data. The operation of the microgrid is complemented with a supervisory control stage that compensates any mismatch between the offline scheduling process and the real time microgrid operation. The proposal has been tested experimentally in a hybrid microgrid at the Microgrid Research Laboratory, Aalborg University.

An MILP-Based Optimal Power Flow in Multicarrier Energy Systems

Abstract: In this paper, an mixed integer linear programming (MILP) method is proposed for calculating the optimal power flow in a multicarrier energy system. A state variable-based linear energy hub model is developed, which avoids the introduction of dispatch factor variables applied traditionally to the optimal power flow problem. The multidimensional piecewise linear approximation method is proposed for representing nonconvex natural gas transmission constraints in which the approximation error is further analyzed. Accordingly, the optimal power flow is reformulated as an MILP problem. Compared with the nonlinear models, the proposed model can be solved by the existing optimization techniques, which can be easily implemented in the optimal power system planning problem. The proposed method is verified by case studies applied to the modified six-bus and the IEEE-118 systems. The test results show that the proposed method can provide a fast solution for the optimal power flow which can be applied to large scale hub systems with sufficient accuracy. The results also demonstrate that the proposed method outperforms the existing MILP methods in calculation time especially in large scale hub applications.

A Stochastic Multi-Objective Framework for Optimal Scheduling of Energy Storage Systems in Microgrids

Abstract: This paper presents a stochastic framework for day-ahead scheduling of microgrid energy storage systems in the context of multi-objective (MO) optimization. Operation cost of microgrid in normal conditions and load curtailment index in case of unscheduled islanding events (initiated by disturbances in the main grid) are chosen as main criteria of the proposed scheme. In practice, duration of disconnection from the upstream network is unknown in unscheduled islanding incidents and cannot be predicted with certainty. To properly handle the uncertainties associated with time and duration of such events as well as microgrid load and renewable power generation, stochastic models are involved in the MO scheduling framework and they are formulated as mixed integer linear programming problems. The non-dominated sorting genetic algorithm II is employed to effectively cope with the MO optimization problem and a fuzzy decision making approach is employed for appropriate representation of microgrid operator’s preferences in compromising between the two objectives. The proposed scheme is implemented on a test microgrid and the obtained results demonstrate the applicability and efficiency of this framework in dealing with conflicting requirements of microgrid security and economic operation.

An integrated optimization-simulation framework for vehicle and personnel relocations of electric carsharing systems with reservations

Abstract: One-way electric vehicle carsharing systems are receiving increasing attention due to their mobility, environmental, and societal benefits. One of the major issues faced by the operators of these systems is the optimization of the relocation operations of personnel and vehicles. These relocation operations are essential in order to ensure that vehicles are available for use at the right place at the right time. Vehicle availability is a key indicator expressing the level of service offered to customers. However, the relocation operations, that ensure this availability, constitute a major cost component for the provision of these services. Therefore, clearly there is a trade-off between the cost of vehicle and personnel relocation and the level of service offered. In this paper we are developing, solving, and applying, in a real world context, an integrated multi-objective mixed integer linear programming (MMILP) optimization and discrete event simulation framework to optimize operational decisions for vehicle and personnel relocation in a carsharing system with reservations. We are using a clustering procedure to cope with the dimensionality of the operational problem without compromising on the quality of the obtained results. The optimization framework involves three mathematical models: (i) station clustering, (ii) operations optimization and (iii) personnel flow. The output of the optimization is used by the simulation in order to test the feasibility of the optimization outcome in terms of vehicle recharging requirements. The optimization model is solved iteratively considering the new constraints restricting the vehicles that require further charging to stay in the station until the results of the simulation are feasible in terms of electric vehicles’ battery charging levels. The application of the proposed framework using data from a real world system operating in Nice, France sheds light to trade-offs existing between the level of service offered, resource utilization, and certainty of fulfilling a trip reservation.

Improved branch-cut-and-price for capacitated vehicle routing

Abstract: The best performing exact algorithms for the capacitated vehicle routing problem developed in the last 10 years are based in the combination of cut and column generation. Some authors only used cuts expressed over the variables of the original formulation, in order to keep the pricing subproblem relatively easy. Other authors could reduce the duality gaps by also using a restricted number of cuts over the master LP variables, stopping when the pricing becomes prohibitively hard. A particularly effective family of such cuts are the subset row cuts. This work introduces a technique for greatly reducing the impact on the pricing of these cuts, thus allowing much more cuts to be added. The newly proposed branch-cut-and-price algorithm also incorporates and combines for the first time (often in an improved way) several elements found in previous works, like route enumeration and strong branching. All the instances used for benchmarking exact algorithms, with up to 199 customers, were solved to optimality. Moreover, some larger instances with up to 360 customers, only considered before by heuristic methods, were solved too.

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Abstract: Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art metho...

Resilience Enhancement With Sequentially Proactive Operation Strategies

Abstract: Extreme weather events, many of which are climate change related, are occurring with increasing frequency and intensity and causing catastrophic outages, reminding the need to enhance the resilience of power systems This paper proposes a proactive operation strategy to enhance system resilience during an unfolding extreme event The uncertain sequential transition of system states driven by the evolution of extreme events is modeled as a Markov process At each decision epoch, the system topology is used to construct a Markov state Transition probabilities are evaluated according to failure rates caused by extreme events For each state, a recursive value function, including a current cost and a future cost, is established with operation constraints and intertemporal constraints An optimal strategy is established by optimizing the recursive model, which is transformed into a mixed integer linear programming by using the linear scalarization method, with the probability of each state as the weight of each objective The IEEE 30-bus system, the IEEE 118-bus system, and a realistic provincial power grid are used to validate the proposed method The results demonstrate that the proposed proactive operation strategies can reduce the loss of load due to the development of extreme events

Electric Vehicle Charging Station Placement for Urban Public Bus Systems

Abstract: Due to the low pollution and sustainable properties, using electric buses for public transportation systems has attracted considerable attention, whereas how to recharge the electric buses with long continuous service hours remains an open problem. In this paper, we consider the problem of placing electric vehicle (EV) charging stations at selected bus stops, to minimize the total installation cost of charging stations. Specifically, we study two EV charging station placement cases, with and without considering the limited battery size, which are called ECSP_LB and ECSP problems, respectively. The solution of the ECSP problem achieves the lower bound compared with the solution of the ECSP_LB problem, and the larger the battery size of the EV, the lower the overall cost of the charging station installation. For both cases, we prove that the placement problems under consideration are NP-hard and formulate them into integer linear programming. Specifically, for the ECSP problem we design a linear programming relaxation algorithm to get a suboptimal solution and derive an approximation ratio of the algorithm. Moreover, we derive the condition of the battery size when the ECSP problem can be applied. For the ECSP_LB problem, we show that, for a single bus route, the problem can be optimally solved with a backtracking algorithm, whereas for multiple bus routes we propose two heuristic algorithms, namely, multiple backtracking and greedy algorithms. Finally, simulation results show the effectiveness of the proposed schemes.

Verifying Neural Networks with Mixed Integer Programming

Abstract: Neural networks have demonstrated considerable success in a wide variety of real-world problems. However, the presence of adversarial examples - slightly perturbed inputs that are misclassified with high confidence - limits our ability to guarantee performance for these networks in safety-critical applications. We demonstrate that, for networks that are piecewise affine (for example, deep networks with ReLU and maxpool units), proving no adversarial example exists - or finding the closest example if one does exist - can be naturally formulated as solving a mixed integer program. Solves for a fully-connected MNIST classifier with three hidden layers can be completed an order of magnitude faster than those of the best existing approach. To address the concern that adversarial examples are irrelevant because pixel-wise attacks are unlikely to happen in natural images, we search for adversaries over a natural class of perturbations written as convolutions with an adversarial blurring kernel. When searching over blurred images, we find that as opposed to pixelwise attacks, some misclassifications are impossible. Even more interestingly, a small fraction of input images are provably robust to blurs: every blurred version of the input is classified with the same, correct label.

New Multistage and Stochastic Mathematical Model for Maximizing RES Hosting Capacity—Part I: Problem Formulation

Abstract: This two-part work presents a new multistage and stochastic mathematical model, developed to support the decision-making process of planning distribution network systems (DNS) for integrating large-scale “clean” energy sources. Part I is devoted to the theoretical aspects and mathematical formulations in a comprehensive manner. The proposed model, formulated from the system operator's viewpoint, determines the optimal sizing, timing, and placement of distributed energy technologies (particularly, renewables) in coordination with energy storage systems and reactive power sources. The ultimate goal of this optimization work is to maximize the size of renewable power absorbed by the system, while maintaining the required/standard levels of power quality and system stability at a minimum possible cost. From the methodological perspective, the entire problem is formulated as a mixed integer linear programming optimization, allowing one to obtain an exact solution within a finite simulation time. Moreover, it employs a linearized ac network model which captures the inherent characteristics of electric networks and balances well accuracy with computational burden. The IEEE 41-bus radial DNS is used to test validity and efficiency of the proposed model, and carry out the required analysis from the standpoint of the objectives set. Numerical results are presented and discussed in Part II of this paper to unequivocally demonstrate the merits of the model.

Smooth operator: Control using the smooth robustness of temporal logic

Abstract: Modern control systems, like controllers for swarms of quadrotors, must satisfy complex control objectives while withstanding a wide range of disturbances, from bugs in their software to attacks on their sensors and changes in their environments. These requirements go beyond stability and tracking, and involve temporal and sequencing constraints on system response to various events. This work formalizes the requirements as formulas in Metric Temporal Logic (MTL), and designs a controller that maximizes the robustness of the MTL formula. Formally, if the system satisfies the formula with robustness r, then any disturbance of size less than r cannot cause it to violate the formula. Because robustness is not differentiable, this work provides arbitrarily precise, infinitely differentiable, approximations of it, thus enabling the use of powerful gradient descent optimizers. Experiments on a temperature control example and a two-quadrotor system demonstrate that this approach to controller design outperforms existing approaches to robustness maximization based on Mixed Integer Linear Programming and stochastic heuristics. Moreover, it is not constrained to linear systems.

Output Range Analysis for Deep Neural Networks

Abstract: Deep neural networks (NN) are extensively used for machine learning tasks such as image classification, perception and control of autonomous systems. Increasingly, these deep NNs are also been deployed in high-assurance applications. Thus, there is a pressing need for developing techniques to verify neural networks to check whether certain user-expected properties are satisfied. In this paper, we study a specific verification problem of computing a guaranteed range for the output of a deep neural network given a set of inputs represented as a convex polyhedron. Range estimation is a key primitive for verifying deep NNs. We present an efficient range estimation algorithm that uses a combination of local search and linear programming problems to efficiently find the maximum and minimum values taken by the outputs of the NN over the given input set. In contrast to recently proposed "monolithic" optimization approaches, we use local gradient descent to repeatedly find and eliminate local minima of the function. The final global optimum is certified using a mixed integer programming instance. We implement our approach and compare it with Reluplex, a recently proposed solver for deep neural networks. We demonstrate the effectiveness of the proposed approach for verification of NNs used in automated control as well as those used in classification.

Multiperiod-based timetable optimization for metro transit networks

Abstract: This paper tackles the train timetable optimization problem for metro transit networks (MTN) in order to enhance the performance of transfer synchronization between different rail lines. Train timetables of connecting lines are adjusted in such a way that train arrivals at transfer stations can be well synchronized. This study particularly focuses on the timetable optimization problem in the transitional period (from peak to off-peak hours or vice versa) during which train headway changes and passenger travel demand varies significantly. A mixed integer nonlinear programming model is proposed to generate an optimal train timetable and maximize the transfer synchronization events. Secondly, an efficient hybrid optimization algorithm based on the Particle Swarm Optimization and Simulated Annealing (PSO-SA) is designed to obtain near-optimal solutions in an efficient way. Meanwhile, in order to demonstrate the effectiveness of the proposed method, the results of numerical example solved by PSO-SA are compared with a branch-and-bound method and other heuristic algorithms. Finally, a real-world case study based on the Beijing metro network and travel demand is conducted to validate the proposed timetabling model. Computational results demonstrate the effectiveness of adjusting train timetables and the applicability of the developed approach to real-world metro networks.

DARP: Divide Areas Algorithm for Optimal Multi-Robot Coverage Path Planning

Abstract: This paper deals with the path planning problem of a team of mobile robots, in order to cover an area of interest, with prior-defined obstacles. For the single robot case, also known as single robot coverage path planning (CPP), an źź(n) optimal methodology has already been proposed and evaluated in the literature, where n is the grid size. The majority of existing algorithms for the multi robot case (mCPP), utilize the aforementioned algorithm. Due to the complexity, however, of the mCPP, the best the existing mCPP algorithms can perform is at most 16 times the optimal solution, in terms of time needed for the robot team to accomplish the coverage task, while the time required for calculating the solution is polynomial. In the present paper, we propose a new algorithm which converges to the optimal solution, at least in cases where one exists. The proposed technique transforms the original integer programming problem (mCPP) into several single-robot problems (CPP), the solutions of which constitute the optimal mCPP solution, alleviating the original mCPP explosive combinatorial complexity. Although it is not possible to analytically derive bounds regarding the complexity of the proposed algorithm, extensive numerical analysis indicates that the complexity is bounded by polynomial curves for practical sized inputs. In the heart of the proposed approach lies the DARP algorithm, which divides the terrain into a number of equal areas each corresponding to a specific robot, so as to guarantee complete coverage, non-backtracking solution, minimum coverage path, while at the same time does not need any preparatory stage (video demonstration and standalone application are available on-line http://tinyurl.com/DARP-app).

Efficient and Autonomous Energy Management Techniques for the Future Smart Homes

Abstract: Smart grid with latest technologies provides solid foundation for the implementation of energy management systems at home premises. This paper proposes an autonomous energy management-based cost reduction solution for peak load times using a home energy management system (HEMS). Within a home environment, both the real time and the schedulable appliances are connected with smart meter through HEMS. We formulate an optimization problem under various practical constraints, which is shown to be a mixed integer programming problem that can be solved through a step-wise approach. A novel scheme based on Dijkstra algorithm is proposed, which results in the similar performance to that of the proposed optimal scheme while exhibiting much lower complexity. To further save the computational efforts, a low complexity scheme is also proposed, which produces considerably better results than the non-optimized scheme with the same complexity yet. Simulation results are presented at show the performance and complexity comparison of different proposed solutions and the existing methods.

Mixed-integer quadratic programming is in NP

Abstract: Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73---77 [17]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299---304 [1], von zur Gathen and Sieveking in Proc Am Math Soc 72:155---158 [18], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161---172. Springer [9], Papadimitriou in J Assoc Comput Mach 28:765---768 [15]).

Economic scheduling model of microgrid considering the lifetime of batteries

Abstract: Because of the uncertainty of renewable energy generation and load, batteries as energy storage devices play an important role in ensuring the safety and reliability of microgrid. To take full advantage of the batteries, the battery lifetime characteristics are analysed, and a weighted Wh throughput method is proposed to estimate the battery lifetime. To improve the economy of microgrid, an economic scheduling model of microgrid in grid-connected mode is established with the consideration of battery lifetime. For fast and efficiently solving the model, a technique is developed to convert the optimisation problem into a mixed integer linear programming problem and a mixed integer linear programming algorithm is applied to solve it. The proposed method has been validated on a microgrid, which consists of a wind turbine, a photovoltaic system, a micro turbine, a fuel cell, a diesel engine and a battery energy storage system (BESS). The simulation results show that the BESS is managed rationally and the total operation cost of microgrid over scheduling period is decreased by applying the proposed scheduling method.