Large-scale Linear Programming Problems Research Articles

A Reinforcement-Learning-Based Multiple-Column Selection Strategy for Column Generation

Column generation (CG) is one of the most successful approaches for solving large-scale linear programming (LP) problems. Given an LP with a prohibitively large number of variables (i.e., columns), the idea of CG is to explicitly consider only a subset of columns and iteratively add potential columns to improve the objective value. While adding the column with the most negative reduced cost can guarantee the convergence of CG, it has been shown that adding multiple columns per iteration rather than a single column can lead to faster convergence. However, it remains a challenge to design a multiple-column selection strategy to select the most promising columns from a large number of candidate columns. In this paper, we propose a novel reinforcement-learning-based (RL) multiple-column selection strategy. To the best of our knowledge, it is the first RL-based multiple-column selection strategy for CG. The effectiveness of our approach is evaluated on two sets of problems: the cutting stock problem and the graph coloring problem. Compared to several widely used single-column and multiple-column selection strategies, our RL-based multiple-column selection strategy leads to faster convergence and achieves remarkable reductions in the number of CG iterations and runtime.

Seismic vulnerability prediction of masonry aggregates: Iterative Finite element Upper Bound limit analysis approximating no tensile resistance

A novel 3D Finite Element Upper Bound limit analysis approach for the evaluation of the seismic vulnerability of masonry aggregates when the collapse may occur both locally and globally is presented.The model relies on a 3D discretization of the aggregate by means of infinitely resistant hexahedron elements and rigid plastic quadrilateral interfaces, where all plastic dissipation occurs.The interface behavior is assumed isotropic, with friction, cohesion, and tension cutoff. The collapse horizontal acceleration is obtained in the discretized model solving large-scale linear programming problems with standard numerical routines. According to a consolidated literature, the vulnerability prediction should be done with pre-assigned failure mechanisms formed by large macro-blocks mutually roto-translating, and assuming a material unable to withstand tensile stresses without mutual sliding between macro-blocks. However, such assumptions may be responsible for an overestimation of the load-carrying capacity and the determination of a realistic mechanism is neither obvious nor possible in any case. Discretizing the aggregate into 3D elements and interfaces, the failure mechanism is automatically found in the solution of the limit analysis problem, but enforcing the no-tension material hypothesis may be responsible for the activation of parasite sliding between contiguous elements or stalling issues of the numerical algorithm. In the first case, the failure mechanism changes considerably and the computational evaluation of the horizontal acceleration at collapse can be affected by unsafe inaccuracy.The present approach iteratively approximates the no-tension material model at low computational cost, progressively shrinking the failure surface and evaluating (i) collapse accelerations, (ii) failure mechanisms active, (iii) power dissipated on interfaces and (iv) power expended by loads not dependent on the collapse multiplier. By analyzing the failure mechanisms obtained, it is possible to select where spurious sliding is not dominant. Enforcing to zero the power dissipated on interfaces it is possible to estimate the collapse multiplier corresponding to the twofold assumption that (i) parasite sliding is negligible and (ii) the material approximates in a technically acceptable way the no-tension one.The model is first benchmarked on a masonry portal, where a reference solution is available, and then applied to two historical aggregates located in Italy.

An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography, and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving some large-scale CMOT problems on synthetic data. The preliminary experiments on the discrete tomography problem also highlight the potential modeling capability of our model.

Solution of Large-Scale Linear Programming Problem by Using Computer Technique

Linear programming (LP) is an important part of applied mathematics. This method has found its applications to important areas of product mix, blending, and diet problems. Steel, chemical, food processing industries and Oil refineries industry are also using LP with considerable success. But in practical LP can be very large. In this paper, our intent is to formulate an LP model of some large-scale real-life-oriented problems and to apply computer techniques for solving these problems. Starting with the graphical procedure which provides an ample amount of understanding into some fundamental concepts, the simple procedure of solving LP problems is developed. Finally, a special class of LP problem, namely Transportation is taken up and solved. We also solved the simplex system by using FORTRAN programming.

Transmission Network Dynamic Planning Based on a Double Deep-Q Network With Deep ResNet

Based on a Double Deep-Q Network with deep ResNet (DDQN-ResNet), this paper proposes a novel method for transmission network expansion planning (TNEP). Since TNEP is a large scale and mixed-integer linear programming (MILP) problem, as the transmission network scale and the optimal constraints increase, the numerical calculation and heuristic learning-based methods suffer from heavy computational complexities in calculation and training. Besides, due to the black box characteristic, the solution processes of the heuristic learning-based methods are inexplicable and usually require repeated training. By using DDQN-ResNet, this paper constructs a high-performance and flexible method to solve large-scale and complex-constrained TNEP problem. Firstly, we form a two-objective TNEP model, in which one objective is to minimize the comprehensive cost, and another objective is to maximize the transmission network reliability. The comprehensive cost takes into account the expansion cost, the network loss cost, and the maintenance cost. The transmission network reliability is evaluated by the expected energy not served (EENS) and the electrical betweenness. Secondly, from the TNEP model, the TNEP task is constructed based on the Markov decision process. By abstracting the task, the TNEP environment is obtained for DDQN-ResNet. In addition, to identify the construction value of lines, an agent is establish based on DDQN-ResNet. Finally, we perform the static planning and visualize the reinforcement learning process. The dynamic planning is realized by reusing the training experience. The validity and flexibility of DDQN-ResNet are verified on RTS 24-bus test system.

An interior point-proximal method of multipliers for convex quadratic programming

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

A Decomposition Algorithm for Solving Multi-Level Large-Scale Linear Programming Problems With Neutrosophic Parameters in the Constrains

Edge Federation: Towards an Integrated Service Provisioning Model

Edge computing is a promising computing paradigm for pushing the cloud service to the network edge. To this end, edge infrastructure providers (EIPs) need to bring computation and storage resources to the network edge and allow edge service providers (ESPs) to provision latency-critical services to users. Currently, EIPs prefer to establish a series of private edge-computing environments to serve specific requirements of users. This kind of resource provisioning mechanism severely limits the development and spread of edge computing for serving diverse user requirements. To this end, we propose an integrated resource provisioning model, named edge federation, to seamlessly realize the resource cooperation and service provisioning across standalone edge computing providers and clouds. To efficiently schedule and utilize the resources across multiple EIPs, we systematically characterize the provisioning process as a large-scale linear programming (LP) problem and transform it into an easily solved form. Accordingly, we design a dynamic algorithm to tackle the varying service demands from users. We conduct extensive experiments over the base station networks in Toronto city. Compared with the existing fixed contract model and multihoming model, edge federation can reduce the overall cost of EIPs by 23.3% to 24.5%, and 15.5% to 16.3%, respectively.

Switching preconditioners using a hybrid approach for linear systems arising from interior point methods for linear programming

In general, interior point methods are successful in solving large-scale linear programming problems. Their effectiveness is determined by how fast they calculate each solution of a linear system. When solving large-scale linear systems, iterative methods are useful options since they require slightly more computational memory in each iteration and preserve the sparsity pattern of the matrix system. In this context, a common choice is the conjugate gradient method using a preconditioning strategy. Choosing a single preconditioner that fits well during all iterations of the optimization method is not an easy task, because the distribution of the eigenvalues of the system matrix may vary significantly from the first to the last iterations. In order to simplify this task, one can use different preconditioners in different iterations, in a hybrid preconditioner strategy. In the case of symmetric positive definite systems, the Controlled Cholesky Factorization achieves excellent performance for the first interior point iterations, whereas the Splitting preconditioner is very useful in the last iterations. However, since we apply a hybrid approach combining both preconditioners, it is necessary to decide when using each preconditioner. This paper addresses the critical issue of choosing between the two preconditioners of the hybrid strategy by using the condition number of the matrix system. The Ritz values obtained from the conjugate gradient method provide an approximation to the eigenvalues, which offers an estimate of the condition number. The main contribution of this research is a new heuristic to switching the preconditioners based on the estimated condition number. Numerical results for large-scale problems show that our choice to change the preconditioners adds both speed and robustness to a hybrid approach that combines the Controlled Cholesky Factorization and the Splitting preconditioner.

An Asymptotically Superlinearly Convergent Semismooth Newton Augmented Lagrangian Method for Linear Programming

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of...

Self-Sustainable Dense Cellular M2M System With Hybrid Energy Harvesting and High Sensitivity Rectenna

The Internet of Things (IoT) is envisioned as an essential enabler of the smart city, which 5G small cell is going to cover. In this paper, we model the dense cellular M2M communication system using a Ginibre determinant point process and study the corresponding green communication metrics, including grid energy consumption and grid energy efficiency (EE). A dedicated millimeter-wave-based hybrid energy harvesting (EH) mechanism is proposed to maintain self-sustainable communication. Two EH modes are discussed, of which Mode 1 is suitable for long-term operation, while Mode 2 is preferred in dealing with a temporary energy shortage. We first minimize the average grid energy consumption by solving a large-scale linear programming problem. When the battery at machine device is charged with sufficient energy at the initial time, closed-form formula describing the average grid energy consumption of both the modes can be derived under moderate throughput constraints, which is proved to have clear relationship with 1) initial battery energy level at machine gateway and machine device; 2) specific hybrid EH time-division strategy; 3) system operation time; and 4) static power consumption at machine gateway and machine device. The numerical solutions are obtained using the interior-point method, and the results are illustrated and discussed. When dealing with the originally non-convex EE maximization problem, we first transform it into a series of sub-problems using the successive convex approximation method and then iteratively solve them with a trick combining Taylor expansion and square error minimization. Non-orthogonal multi-access (NOMA) transmission is investigated in our model to further improve the EE performance. The experiments show that NOMA achieves nearly three times throughput promotion and two times EE improvement.

Using groups in the splitting preconditioner computation for interior point methods

Interior point methods usually rely on iterative methods to solve the linear systems of large scale problems. The paper proposes a hybrid strategy using groups for the preconditioning of these iterative methods. The objective is to solve large scale linear programming problems more efficiently by a faster and robust computation of the preconditioner. In these problems, the coefficient matrix of the linear system becomes ill conditioned during the interior point iterations, causing numerical difficulties to find a solution, mainly with iterative methods. Therefore, the use of preconditioners is a mandatory requirement to achieve successful results. The paper proposes the use of a new columns ordering for the splitting preconditioner computation, exploring the sparsity of the original matrix and the concepts of groups. This new preconditioner is designed specially for the final interior point iterations; a hybrid approach with the controlled Cholesky factorization preconditioner is adopted. Case studies show that the proposed methodology reduces the computational times with the same quality of solutions when compared to previous reference approaches. Furthermore, the benefits are obtained while preserving the sparse structure of the systems. These results highlight the suitability of the proposed approach for large scale problems.

Core column prediction for protein multiple sequence alignments

BackgroundIn a computed protein multiple sequence alignment, the coreness of a column is the fraction of its substitutions that are in so-called core columns of the gold-standard reference alignment of its proteins. In benchmark suites of protein reference alignments, the core columns of the reference alignment are those that can be confidently labeled as correct, usually due to all residues in the column being sufficiently close in the spatial superposition of the known three-dimensional structures of the proteins. Typically the accuracy of a protein multiple sequence alignment that has been computed for a benchmark is only measured with respect to the core columns of the reference alignment. When computing an alignment in practice, however, a reference alignment is not known, so the coreness of its columns can only be predicted.ResultsWe develop for the first time a predictor of column coreness for protein multiple sequence alignments. This allows us to predict which columns of a computed alignment are core, and hence better estimate the alignment’s accuracy. Our approach to predicting coreness is similar to nearest-neighbor classification from machine learning, except we transform nearest-neighbor distances into a coreness prediction via a regression function, and we learn an appropriate distance function through a new optimization formulation that solves a large-scale linear programming problem. We apply our coreness predictor to parameter advising, the task of choosing parameter values for an aligner’s scoring function to obtain a more accurate alignment of a specific set of sequences. We show that for this task, our predictor strongly outperforms other column-confidence estimators from the literature, and affords a substantial boost in alignment accuracy.

USE OF CONTINUED ITERATION ON THE REDUCTION OF ITERATIONS OF THE INTERIOR POINT METHOD

ABSTRACT Interior point methods have been widely used to determine the solution of large-scale linear programming problems. The predictor-corrector method stands out among all variations of interior point methods due to its efficiency and fast convergence. In each iteration it is necessary to solve two linear systems to determine the predictor-corrector direction. Solving such systems corresponds to the step which requires more processing time, and therefore, it should be done efficiently. The most common approach to solve them is the Cholesky factorization. However, Cholesky factorization demands a high computational effort in each iteration. Thus, searching for effort reduction, the continued iteration is proposed. This technique consists in determining a new direction through projection of the search direction and it was inserted into PCx code. The computational results regarding medium and large-scale problems, have indicated a good performance of the proposed approach in comparison with the predictor-corrector method.

A new approach for finding a basis for the splitting preconditioner for linear systems from interior point methods

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra's predictor-corrector method for large scale linear programming problems needs to find a ba...

A Compromise Decision-making Model for Multi-objective Large-scale Programming Problems with a Block Angular Structure under Uncertainty

This paper proposes a compromise model, based on the technique for order preference through similarity ideal solution (TOPSIS) methodology, to solve the multi-objective large-scale linear programming (MOLSLP) problems with block angular structure involving fuzzy parameters. The problem involves fuzzy parameters in the objective functions and constraints. This compromise programming method is based on the assumption that the optimal alternative is closer to fuzzy positive ideal solution (FPIS) and at the same time, farther from fuzzy negative ideal solution (FNIS).An aggregating function that is developed from LP- metric is based on the particular measure of ‘‘closeness” to the ‘‘ideal” solution.An efficient distance measurement is utilized to calculate positive and negative ideal solutions. The solution process is as follows: first, the decomposition algorithm is used to divide the large-dimensional objective space into a two-dimensional space. A multi-objective identical crisp linear programming is derived from the fuzzy linear model for solving the problem. Then, a single-objective large-scale linear programming problem is solved to find the optimal solution. Finally, to illustrate the proposed method, an illustrative example is provided.

A column generation approach for evaluating delivery efficiencies of collimator technologies in IMRT treatment planning

Collimator systems used in Intensity Modulated Radiation Therapy can form different geometric aperture shapes depending on their physical capabilities. We compare the efficiency of using regular, rotating and dual multileaf collimator (MLC) systems under different combinations of consecutiveness, interdigitation and rectangular constraints. We also create a virtual freeform collimator, which can form any possible segment shape by opening or closing each bixel independently, to provide a basis for comparison. We formulate the problem of minimizing beam-on time as a large-scale linear programming problem. To deal with its dimensionality, we propose a column generation approach. We demonstrate the efficacy of our approach on a set of clinical problem instances. Our results indicate that the dual MLC under consecutiveness constraint yields very similar beam-on time to a virtual freeform collimator. Our approach also provides a ranking between other collimator technologies in terms of their delivery efficiencies.

Efficient GPU-based implementations of simplex type algorithms

Recent hardware advances have made it possible to solve large scale Linear Programming problems in a short amount of time. Graphical Processing Units (GPUs) have gained a lot of popularity and have been applied to linear programming algorithms. In this paper, we propose two efficient GPU-based implementations of the Revised Simplex Algorithm and a Primal–Dual Exterior Point Simplex Algorithm. Both parallel algorithms have been implemented in MATLAB using MATLAB’s Parallel Computing Toolbox. Computational results on randomly generated optimal sparse and dense linear programming problems and on a set of benchmark problems (netlib, kennington, Mészáros) are also presented. The results show that the proposed GPU implementations outperform MATLAB’s interior point method.

A New Compromise Decision-making Model Based on TOPSIS and VIKOR for Solving Multi-objective Large-scale Programming Problems with a Block Angular Structure under Uncertainty

This paper proposes a compromise model, based on a new method, to solve the multi-objective large-scale linear programming (MOLSLP) problems with block angular structure involving fuzzy parameters. The problem involves fuzzy parameters in the objective functions and constraints. In this compromise programming method, two concepts are considered simultaneously. First of them is that the optimal alternative is closer to fuzzy positive ideal solution (FPIS) and farther from fuzzy negative ideal solution (FNIS). Second of them is that the proposed method provides a maximum ‘‘group utility’’ for the ‘‘majority’’ and a minimum of an individual regret for the ‘‘opponent’’. In proposed method, the decomposition algorithm is utilized to reduce the large-dimensional objective space. A multi objective identical crisp linear programming derived from the fuzzy linear model for solving the problem. Then, a compromise solution method is applied to solve each sub problem based on TOPSIS and VIKOR simultaneously. Finally, to illustrate the proposed method, an illustrative example is provided.

A Decomposition Algorithm for Solving a Three Level Large Scale linear Programming Problem

This paper presents a three level large scale linear programming problem in which the objective functions at every level are to be maximized. A three level programming problem can be thought as a static version of the Stackelberg strategy. An algorithm for solving a three planner model and a solution method for treating this problem are suggested. At each level we attempt to optimize its problem separately as a large scale programming problem using Dantzig and Wolfe decomposition method. Therefore, we handle the optimization process through a series of sub problems that can be solved independently. Finally, a numerical example is given to clarify the main results developed in this paper.