Linear Relaxation Research Articles

Optimal Design of Refinery Hydrogen Networks: Reduced Superstructure and Effective Algorithm

In refinery hydrogen networks, the cost related to hydrogen compressors is the second largest contributor to the total annualized cost (TAC). The optimal synthesis of hydrogen networks considering compressor arrangement is a computationally difficult optimization problem because of the highly nonlinear terms caused by hydrogen compression. In this study, a reduced superstructure for optimal design of hydrogen networks is first developed based on several heuristics. This reduced superstructure includes all possible connections among hydrogen sources, hydrogen sinks, and compressors. The maximum number of compressors is reduced dramatically, and the inlet and outlet pressures of all compressors can be preassigned, which notably reduces the bilinear terms and completely eliminates the linear fractional terms in the mathematical model. A mixed-integer nonlinear programming (MINLP) model is formulated, and a tailored algorithm that incorporates successive piecewise linear approximation of concave terms and linear relaxation of bilinear terms is proposed. The case studies show that the proposed model and algorithm can give better hydrogen network designs with only 2.9 and 4.9% of the computational time in the existing publication.

3DVerifier: efficient robustness verification for 3D point cloud models

3D point cloud models are widely applied in safety-critical scenes, which delivers an urgent need to obtain more solid proofs to verify the robustness of models. Existing verification method for point cloud model is time-expensive and computationally unattainable on large networks. Additionally, they cannot handle the complete PointNet model with joint alignment network that contains multiplication layers, which effectively boosts the performance of 3D models. This motivates us to design a more efficient and general framework to verify various architectures of point cloud models. The key challenges in verifying the large-scale complete PointNet models are addressed as dealing with the cross-non-linearity operations in the multiplication layers and the high computational complexity of high-dimensional point cloud inputs and added layers. Thus, we propose an efficient verification framework, 3DVerifier, to tackle both challenges by adopting a linear relaxation function to bound the multiplication layer and combining forward and backward propagation to compute the certified bounds of the outputs of the point cloud models. Our comprehensive experiments demonstrate that 3DVerifier outperforms existing verification algorithms for 3D models in terms of both efficiency and accuracy. Notably, our approach achieves an orders-of-magnitude improvement in verification efficiency for the large network, and the obtained certified bounds are also significantly tighter than the state-of-the-art verifiers. We release our tool 3DVerifier via https://github.com/TrustAI/3DVerifier for use by the community.

Group Delay-Aware Scalable Mobile Edge Computing Using Service Replication

The number of individuals and groups of users offloading independent and inter-related computational tasks to mobile edge computing (MEC) servers is rapidly increasing, thus overloading them and raising the risk of service interruptions. Hence, reactive service replication has been suggested to enable individuals and groups of users to access services from remote edge servers, thus guaranteeing system scalability. This paper proposes a task offloading and service replication scheme on local and remote MEC servers. The scheme minimizes the response time of all users while satisfying the delay requirements of user groups in traffic-heavy and multimedia-intense applications (e.g., online gaming, multimedia conferencing, augmenting reality). We formulate an integer linear problem that minimizes the average response time of all users while satisfying the time and time difference constraints of the user groups running the same applications. We then use linear relaxation programming using Lagrangian analysis and solve the problem using a numerical solver. In addition, we compare the optimal solution to distance-based and resource-based greedy approaches. The results demonstrate the merits of our proposed optimized decision scheme compared to these two greedy approaches.

Interval scheduling with economies of scale

Motivated by applications in cloud computing, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of jobs, each with start time, end time, and a function representing the cost of scheduling a subset of jobs on the same machine. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs that minimizes the total schedule cost, where overlapping jobs cannot be processed on the same machine. We propose a set cover formulation and a column generation algorithm to solve its linear relaxation. For the max-weight function, which is already NP-hard, we give a polynomial-time pricing algorithm; for the more general case of a function of the total weight, we have a pseudo-polynomial algorithm. To obtain integer solutions, we extend the column generation approach using branch-and-price. We computationally evaluate our methods on two different functions, using both random instances and instances derived from cloud computing data; our algorithm significantly outperforms known integer programming formulations (when these are available) and is able to provably optimize instances with hundreds of jobs.

Linear relaxation schemes for the Allen–Cahn-type and Cahn–Hilliard-type phase field models

This letter introduces novel linear relaxation schemes for solving the phase field models, particularly the Allen–Cahn (AC) type and Cahn–Hilliard (CH) type equations. The proposed schemes differ from existing schemes for the phase field models in the literature. The resulting semi-discrete schemes are linear by discretizing the AC and CH models on staggered time meshes. Only a linear algebra problem needs to be solved at each time marching step after the spatial discretization. Furthermore, our proposed schemes are shown to be unconditionally energy stable, i.e., the numerical solutions respect energy dissipation laws without restriction on the time steps. Several numerical examples are provided to illustrate the power of the proposed linear relaxation schemes for solving phase field models.

Effect of quasiperiodic and random noise on many-body dynamical decoupling protocols

Symmetries (and their spontaneous rupturing) can be used to protect and engender novel quantum phases and lead to interesting collective phenomena. In the work of K. Agarwal and I. Martin [Phys. Rev. Lett. 125, 080602 (2020)], the authors described a general dynamical decoupling (polyfractal) protocol that can be used to engineer multiple discrete symmetries in many-body systems. This work expands on the former by studying the effect of quasiperiodic and random noise on such a dynamical scheme. We find generally that relaxation of engineered symmetry generators proceeds by (i) an initial relaxation on microscopic timescales to a prethermal plateau whose height is independent of noise, (ii) a linear relaxation regime with a noise-dependent rate, followed by (iii) a slow logarithmic relaxation regime that is only present for quasiperiodic noise. We glean the essential features of these regimes via scaling collapses and show that they can be generally explained by the spectral properties of the various noise waveforms considered. In particular, the quasiperiodic noise is characterized by highly time-dependent spectrum with a noise floor that mimics white noise, and peaks that grow sharper with time. We argue that both the noise floor and peaks contribute to the initial linear-in-time relaxation while the logarithmic regime is initiated when the peaks become sufficiently well resolved and cease to contribute to further relaxation. We provide numerical evidence to justify these findings.

Machine-Learning–Based Arc Selection for Constrained Shortest Path Problems in Column Generation

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

Solving min–max linear fractional programs based on image space branch-and-bound scheme

Based on the image space branch-and-bound scheme, this paper presents a novel algorithm for globally solving the min–max linear fractional programs (MMLFP), which has many applications in management optimization, engineering optimization, economic investment and so on. For finding the global optimal solution of the MMLFP, by leading into new parameter, we firstly transform the MMLFP into an equivalent fractional problem (EFP). Next, by using convex hull and concave hull approximation of bilinear function, we construct the linear relaxation problem (LRP) for computing the lower bound of the global minimum value of the EFP in the image space branch-and-bound algorithm. By subsequently solving a series of linear relaxation problems and refining the initial image space rectangle, the proposed algorithm is globally convergent to the optimum solution of the EFP. In addition, we give the computational complexity analysis of the algorithm based on the exhaustive branching rule. Finally, computational comparisons show better computational performance of the algorithm.

Fluctuating Demand-Oriented Optimization of Train Line Planning Considering Carriage Resources Transfer under Flexible Compositions

The intercity railway is subject to variation and fluctuation in demand both in time and space over a day to a large extent. In that case, more advanced line planning techniques are practically needed to match the non-equilibrium passenger demand. We propose an integer linear programming model for adapting to the fluctuating demand and improving rail line profit, in which the multi-period planning approach and flexible train composition mode are taken into consideration. In particular, we also consider the limitations of the carriage and the dynamic transfer of resources during a finite period to ensure the better implementation of the line planning and subsequent operation plans. Our purpose is to improve on previous line planning models by integrating the multi-period strategic-level line planning decision with resource constraints. Since the problem is computationally intractable for realistic size instances, an improved round heuristic algorithm that is based on linear relaxation is proposed and we compare the round heuristic performance with the commercial solver Gurobi on artificial instances. The numerical experiments that are based on an intercity railway in China certify the effectiveness and applicability of the proposed model and algorithm. We evaluate the impact of different optimization parameters and reserved carriages and the computation results show that in comparison to the fixed composition mode, the proposed approach can improve the utilization efficiency of carriage resources and increase the line profit by 1.9% under the same carriage resource conditions.

Stochastic Dual Dynamic Programming for Multiechelon Lot Sizing with Component Substitution

This work investigates lot sizing with component substitution under demand uncertainty. The integration of component substitution with lot sizing in an uncertain demand context is important because the consolidation of the demand for components naturally allows risk-pooling and reduces operating costs. The considered problem is relevant not only in a production context, but also in the context of distribution planning. We propose a stochastic programming formulation for the static–dynamic type of uncertainty, in which the setup decisions are frozen but the production and consumption quantities are decided dynamically. To tackle the scalability issues commonly encountered in multistage stochastic optimization, this paper investigates the use of stochastic dual dynamic programming (SDDP). In addition, we consider various improvements of SDDP, including the use of strong cuts, the fast generation of cuts by solving the linear relaxation of the problem, and retaining the average demand scenarios. Finally, we propose two heuristics, namely, a hybrid of progressive hedging with SDDP and a heuristic version of SDDP. Computational experiments conducted on well-known instances from the literature show that the heuristic version of SDDP outperforms other methods. The proposed method can plan with up to 10 decision stages and 20 scenarios per stage, which results in 2010 scenario paths in total. Moreover, as the heuristic version of SDDP can replan to account for new information in less than a second, it is convenient in a dynamic context. Summary of Contribution: We believe our paper is suitable for the mission and scope of IJOC because we design efficient algorithms to solve an operations research problem. More precisely, we investigate the use of stochastic dual dynamic programming (SDDP) for lot sizing with component substitution under demand uncertainty. In this work, we consider the static–dynamic decision framework, and a good approximation of the expected costs in this context requires us to solve the problem with a large number of scenarios of future demand. As solving the considered problem is computationally intensive, we investigate the use of SDDP, which decomposes the problem per decision stage. We study several enhancements of SDDP, such as the use of strong cuts, the incorporation of a lower bound computed with the average demand scenario, the multicut version of SDDP, and scenario sampling with randomized quasi–Monte Carlo. Despite these improvements, the convergence of SDDP remains slow. Consequently, we propose a heuristic version of SDDP and a hybrid of progressive hedging and SDDP. We present the results of an extensive computational study performed on well-known instances from the literature. The results show that the heuristic SDDP outperforms the hybrid of progressive hedging with SDDP and state-of-the-art methods from the literature. Besides, our analysis shows that component substitution can pool the risk, and it allows maintaining the same service level with less inventory. The presented methodology can be used by practitioners to size their production lots, and subsequent researchers can build upon our results to consider uncertainty in other parameters, such as lead times, yields, and production capacities. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by Mitacs and the Institut de Valorisation des Données (IVADO). Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2022.1215 .

The edge-labeled survivable network design problem: Formulations and branch-and-cut

In this paper, we study a variant of the survivable network design problem, that is the survivable network design problem with labels (colors) on the edges. In particular, we address the Generalized Labeled Two Edge Connected Subgraph Problem (GLTECSP) that has many applications in telecommunication and transportation. Given a connected undirected graph G such that with each edge is associated a set of labels (colors), the GLTECSP consists in finding a two-edge connected spanning subgraph of G with a minimum number of distinct labels. We propose two Integer Programming (IP) formulations for the problem, a natural formulation using cuts on the edges, and a compact formulation using color-cuts. We devise Branch-and-Cut algorithms to solve both formulations and compare them on sets of randomly generated instances. Computational results show that the compact formulation outperforms the natural one regarding the linear relaxation and the computational time. Moreover, the compact formulation is able to solve to optimality several instances left unsolved within the time limit by the natural formulation.

On the design and analysis of near-term quantum network protocols using Markov decision processes

The quantum internet is one of the frontiers of quantum information science. It will revolutionize the way we communicate and do other tasks, and it will allow for tasks that are not possible using the current, classical internet. The backbone of a quantum internet is entanglement distributed globally in order to allow for such novel applications to be performed over long distances. Experimental progress is currently being made to realize quantum networks on a small scale, but much theoretical work is still needed in order to understand how best to distribute entanglement, especially with the limitations of near-term quantum technologies taken into account. This work provides an initial step toward this goal. In this work, we lay out a theory of near-term quantum networks based on Markov decision processes (MDPs), and we show that MDPs provide a precise and systematic mathematical framework to model protocols for near-term quantum networks that is agnostic to the specific implementation platform. We start by simplifying the MDP for elementary links introduced in prior work and by providing new results on policies for elementary links in the steady-state (infinite-time) limit. Then, we show how the elementary link MDP can be used to analyze a complete quantum network protocol. We then provide an extension of the MDP formalism to two elementary links. Here, as new results, we derive linear programing relaxations that allow us to obtain optimal steady-state policies with respect to the expected fidelity and waiting time of the end-to-end link.

Optimizing Information Freshness in RF-Powered Multi-Hop Wireless Networks

Many applications operating in the Internet of Things (IoT) require timely and fair data collection from devices. This has motivated research into a new metric called Age of Information (AoI). This paper contributes to this effort by proposing to minimize the maximum average AoI (min-max AoI) in a multi-hop IoT network comprising of solar-powered Power Beacons (PBs). It outlines a Mixed Integer Linear Program (MILP) that jointly optimizes: (i) the beamforming vector used by PBs to charge devices, and (ii) routing, which determines how samples from devices are forwarded to a sink node, and (iii) the sampling time of sources. It also presents two protocols: Centralized Linear Relaxation (CLR) and Distributed Path Selection (DPS), respectively. CLR is run by the sink to determine the transmit power of PBs and the path of each source using two Linear Programs (LPs). On the other hand, DPS is a distributed approach whereby PBs and sources make their own decisions using local information. Our simulation results show that min-max AoI increases with the number of sources, but reduces with increasing number of PBs. The number of paths available to a source, the number of frames, and solar panel size have limited impact on performance. The min-max AoI of CLR and DPS is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.60\times $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.95\times $ </tex-math></inline-formula> higher than that of MILP.

A nonlinear constitutive model for entangled symmetric dendrimers

An orientation-stretch-coupled constitutive equation is suggested for symmetric Cayley tree-like dendrimers, which can self-consistently describe the linear relaxation spectrum and nonlinear viscoelastic behavior. The molecular stress is determined by orientation-stretch-coupled conformation tensors of all segments. The linear relaxation spectrum is determined according to hierarchical arm retraction with branch point hopping and dynamic tube dilation. Under strong flows, the orientation relaxation time and the stretch relaxation time are affected by the convective constraint release effect and branch point withdrawal. The coupling between segmental orientation and stretch in each generation is represented in the evolution of the coupled conformation tensor, while the possible stretch coupling among different generations is ignored. The predictions on both linear and nonlinear rheological behaviors are consistent with experiments, proving the rationality in treating coupling problems.

Constant modulus shaped beam synthesis via linear fractional semidefinite relaxation

Excitation amplitude control is a critical issue regarding the shaped beam synthesis of array antennas. Ideally, excitations can have the same amplitude, and the desired beam can be achieved only by changing the phases. This method is called constant modulus synthesis, which can reduce the complexity of array antennas and is more conducive to the feeding, but it is really difficult to realize. In this paper, the authors derive a novel constant modulus beam synthesis algorithm, which uses the linear fractional semidefinite relaxation to relax the constant modulus constraints. Simulation results show that this method is stable and can maintain the excitation modulus while realizing the shaped beam.

Image Space Accelerating Algorithm for Solving a Class of Multiplicative Programming Problems

This paper interprets an image space accelerating branch and bound algorithm for globally solving a class of multiplicative programming problems (MP). In this algorithm, in order to obtain the global optimal solution, the problem (MP) is transformed into an equivalent problem (P2) by introducing new variables. By utilizing new linearizing relaxation technique, the problem (P2) can be converted into a series of linear relaxation programming problems, which provide the reliable lower bound in the branch and bound search. Meanwhile, an image space accelerating method is constructed to improve the computational performance of the algorithm by deleting the subintervals which have no global optimal solution. Furthermore, the global convergence of the algorithm is proved. The computational complexity of the algorithm is analyzed, and the maximum iterations of the algorithm are estimated. Finally, numerical experimental results show that the algorithm is robust and efficient.

Microscopic dynamics underlying the stress relaxation of arrested soft materials

Arrested soft materials such as gels and glasses exhibit a slow stress relaxation with a broad distribution of relaxation times in response to linear mechanical perturbations. Although this macroscopic stress relaxation is an essential feature in the application of arrested systems as structural materials, consumer products, foods, and biological materials, the microscopic origins of this relaxation remain poorly understood. Here, we elucidate the microscopic dynamics underlying the stress relaxation of such arrested soft materials under both quiescent and mechanically perturbed conditions through X-ray photon correlation spectroscopy. By studying the dynamics of a model associative gel system that undergoes dynamical arrest in the absence of aging effects, we show that the mean stress relaxation time measured from linear rheometry is directly correlated to the quiescent superdiffusive dynamics of the microscopic clusters, which are governed by a buildup of internal stresses during arrest. We also show that perturbing the system via small mechanical deformations can result in large intermittent fluctuations in the form of avalanches, which give rise to a broad non-Gaussian spectrum of relaxation modes at short times that is observed in stress relaxation measurements. These findings suggest that the linear viscoelastic stress relaxation in arrested soft materials may be governed by nonlinear phenomena involving an interplay of internal stress relaxations and perturbation-induced intermittent avalanches.

Loading Cost-Aware Model Caching And Request Routing In Edge-enabled Wireless Sensor Networks

Abstract Existing works on caching in multi-access edge computing focus on service caching and request routing. However, loading cost and execution time influenced by resource sharing have not been well exploited. To fill this gap, we investigate the joint optimization problem over deep neural network (DNN) model caching and DNN request routing with edge collaboration in edge-enabled wireless sensor networks. A problem is formulated, with the objective of maximizing throughput, under constraints of budget, accuracy and latency etc. The proof of NP-hardness for the formulated problem is provided. To solve the problem, an approximation algorithm based on randomized rounding is presented. In addition, the approximation ratio for the presented algorithm is proved to be $1/(1-\sqrt{4\ln S/\xi^\dagger})$, where $S$ is the number of edge servers and $\xi^\dagger$ is the objective value from linear relaxation. Extensive experiments demonstrate that the system throughput for the presented algorithm can be improved by 58.8% on average, compared with that of the baseline algorithm.

A Priori Direct Numerical Simulation Analysis of the Closure of Cross-Scalar Dissipation Rate of Reaction Progress Variable and Mixture Fraction in Turbulent Stratified Flames

The cross-scalar dissipation rate of reaction progress variable and mixture fraction widetilde{{varepsilon_{cxi } }} plays an important role in the modelling of stratified combustion. The evolution and statistical behaviour of widetilde{{varepsilon_{cxi } }} have been analysed using a direct numerical simulation (DNS) database of statistically planar turbulent stratified flames with a globally stochiometric mixture. A parametric analysis has been conducted by considering a number of DNS cases with a varying initial root-mean-square velocity fluctuation u^{prime } and initial scalar integral length scale ell_{phi }. The explicitly Reynolds averaged DNS data suggests that the linear relaxation model for widetilde{{varepsilon_{cxi } }} is inadequate for most cases, but its performance appears to improve with increasing initial ell_{phi } and u^{prime } values. An exact transport equation for widetilde{{varepsilon_{cxi } }} has been derived from the first principle, and the budget of the unclosed terms of the widetilde{{varepsilon_{cxi } }} transport equation has been analysed in detail. It has been found that the terms arising from the density variation, scalar-turbulence interaction, chemical reaction rate and molecular dissipation rate play leading order roles in the widetilde{{varepsilon_{cxi } }} transport. These observations have been justified by a scaling analysis, which has been utilised to identify the dominant components of the leading order terms to aid model development for the unclosed terms of the widetilde{{varepsilon_{cxi } }} transport equation. The performances of newly proposed models for the unclosed terms have been assessed with respect to the corresponding terms extracted from DNS data, and the newly proposed closures yield satisfactory predictions of the unclosed terms in the widetilde{{varepsilon_{cxi } }} transport equation.

Minimizing electricity consumption in a personal rapid transit transportation system

ABSTRACT In this study, we address the minimisation of the consumed electric energy in a personal rapid transit transportation system (PRT). The PRT system can play a significant role in the supply chain, especially in the last mile delivery of goods in urban areas. Instead of transporting passengers and goods on different vehicles, the capacities offered by PRT system can be consolidated and utilised in a shared mode. Two compact formulations of polynomial size are developed for sequencing the vehicle trips. Since the existing model from the literature performed poorly, the first and the second models are developed by applying the lifting technique and Reformulation-Linearisation Technique, respectively. These techniques are well known for strengthening existing models whose linear relaxation is weak. In addition, we embed strengthened cuts within a constraint generation solution procedure. The performance of the proposed formulations is verified through computational experiments conducted on some instances from the literature. Furthermore, optimal solutions are provided for all the unsolved instances.