Large-scale Linear Programming Research Articles

Secure Outsourcing of Large-scale Linear Programming

Linear programming (LP) is often used in reality production and life, such as the use of resources, human resource management, production arrangement. The client should pay out huge amounts of overhead to dispose huge data sets with its resource-constraint devices. Fortunately, Cloud computing can finish off this deficiency. The client outsources the computation to cloud servers. Then the servers accomplish the task and return the result to the client. But in the process, the correctness, verifiability and privacy should be emphasized. In our paper, based on the studies and analysis of previous protocols for secure outsourcing of linear programming in cloud computing, we find the obvious deficiencies in efficiency and safety. On the basis of the existing protocols, we propose a new protocol that combines the advantages of existing protocols and improves the shortcomings of existing protocols. We also analyze the security and efficiency of the new protocol and design a suitable simulation to verify the efficiency of this protocol.

Biomass feedstock supply chain network design with biomass conversion incentives

Biomass has the potential to create sustainable energy systems, which is critical for societal welfare. A major issue regarding biomass resources is crop residues or leftover biomass that is burnt by farmers after harvesting; this happens due to high transportation costs which make burning the cheapest way to remove the residue. We develop a decision support system using a large-scale linear program with the goal of maximizing profit with and without the emission cost. This system helps identify farms that would benefit society were they to be incentivized under a biomass crop assistance program (BCAP). A case study of leftover corn stover in the state of North Dakota is analyzed to validate the model. Our results reveal that an incentive of $7.20 per ton of corn stover converted to ethanol when 20% of rail capacity is allocated is ideal, as it produces the lowest emissions of 16,784,953 metric tons with a $73,462,599 profit. Furthermore, penalizing emissions resulting from the transportation of corn stover also helps reduce emissions; a suitable value for the penalty could be $71.7 per metric ton of CO2 emitted. Such a policy would result in reducing dependency on petroleum, thus promoting a sustainable biomass supply chain.

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.

Organic Farming Suffices to Feed a Country: a Large-Scale Linear Programming Model to Develop an Organic Agriculture Plan for Turkey

A frequently voiced critique is that due to lower yields on organically managed farmlands, one cannot feed a country using organic agriculture. In this paper, we aim to mathematically disprove this claim by developing a linear programming model and produce a detailed agriculture plan for Turkey sufficient to feed her population with a 2400 kcal daily menu on average, solely comprising of organic foods. The model uses information about population sizes and food needs of 81 cities in Turkey, and yields of 120 food, feed, forage crops, and four animal products. Intensive and extensive livestock production methods as well as food transportation between cities has been incorporated into the model. The resulting problem with 950 thousand variables and 40 thousand constraints can be solved with an optimization package in under a minute. Results, prescribing how many acres of each crop should be grown in each city, indicate that to feed the country fully on organic produce, 63% of the arable land suffices, yielding 8.9 million hectares of unused land where further organic foods could be grown for export or aid. We also run the model under different scenarios: fully vegetarian diet, omnivore model, different transportation structures, drought conditions and a limit on fruit trees. With this work, we have shown that it is possible to feed the whole population of Turkey with an agricultural practice that is not harmful to human health, soil, water and air; respects biological cycles and reduces food miles and fossil fuel consumption, thus contributing to sustainability and fighting climate change. We tested preliminary scenarios to understand the robustness of organic agriculture in the face of extreme weather events. The proposed model can also be applied to other countries when appropriate data are used.

Penetration of Electric Vehicles toward 2050: Analysis Utilizing an Energy System Model Incorporating High-Temporal-Resolution Power Generation Sector

Abstract This paper examines possible future penetration of electric vehicles (EVs) toward decarbonization of energy system in Japan by utilizing a bottom-up energy technology mix model which incorporates a high-temporal-resolution power sector. The model is formulated as a large-scale linear programming model, which has capability of considering intermittent output of variable renewables, hourly-adjustment of electricity demand and supply using advanced technologies including large-scale batteries, EVs’ charging and energy conversion to hydrogen or heat. Simulation results reveal that EVs are highly likely to be deployed in the freight transport sector because the EVs’ annual utilization rate is generally higher than that in the passenger sector and it is relatively easy to recover higher investment cost through fuel savings. It is also found out that severe carbon reduction constraint does not necessarily bring about massive penetration of EVs because lowering carbon intensity in the power sector would play a primary role for decarbonization of whole energy system. However, little availability of decarbonized imported hydrogen and EVs’ cost reduction would lead to the brisk dissemination of the vehicles.

Low-Complexity Priority-Aware Interference-Avoidance Scheduling for Multi-user Coexisting Wireless Networks

In this paper, the priority-aware interference-avoidance scheduling for multi-user coexisting wireless networks with heterogeneous traffic demands is addressed. Both admission control and throughput maximization for admitted users are studied. These problems are addressed by a proposed sequential solution framework where at each step a large-scale linear program with a large number of variables is required to be solved. To efficiently solve the large-scale program, an accelerated column generation based method is proposed. In the proposed method, an efficient greedy initialization algorithm is first put forward by exploiting the proposed solution structure. After that, both upper and lower bounds on the optimal objective function of each optimization problem are derived, which are used to significantly alleviate the dependence of the whole solution procedure on deriving optimality of problems. Simulation results show that the proposed algorithm can effectively and efficiently handle the coexistence of multiple users with heterogeneous priorities and traffic demands.

An exact approach to extend network lifetime in a general class of wireless sensor networks

This paper provides a general framework to model and optimize lifetime maximization problems in wireless sensor networks with sensors having specialized capabilities like the ability to adjust their sensing range, change their directions, etc. In order to identify the set of tasks that a sensor carries out, the concept of role is introduced. These roles include sensor direction, sensing range, communication mode and combinations of these. The purpose is to identify schedules, represented as the allocation of roles to the sensors and a time interval for assuming such roles, while covering targets and transmitting signals to the base station. To do so, a large scale linear programming model is proposed and solved through an exact approach based on column generation, which is complemented with a branch-and-cut procedure used to address the pricing subproblem. The proposed approach is tested on an extensive set of randomly generated instances used to evaluate its performance. Computational results show the potential of the proposed approach for medium-large size instances for which it is possible to compute either the optimal or good quality solutions in short computational times.

Linear fractional approximations for master problems in column generation

In the context of large-scale linear programs solved by a column generation algorithm, we present a primal algorithm for handling the master problem. Successive approximations of the latter are created to converge to optimality. The main properties are that, for every approximation except the last one, the cost of the solution decreases whereas the sum of the variable values increases. Moreover, the minimum reduced cost of the variables also increases and converges to zero with a super-geometric growth rate.

Benders decomposition and column-and-row generation for solving large-scale linear programs with column-dependent-rows

In a recent work, Muter, Birbil, and Bülbül, (2013) identified and characterized a general class of linear programming (LP) problems – known as problems with column-dependent-rows (CDR-problems). These LPs feature two sets of constraints with mutually exclusive groups of variables in addition to a set of structural linking constraints, in which variables from both groups appear together. In a typical CDR-problem, the number of linking constraints grows very quickly with the number of variables, which motivates generating both columns and their associated linking constraints simultaneously on-the-fly. In this paper, we expose the decomposable structure of CDR-problems via Benders decomposition. However, this approach brings on its own theoretical challenges. One group of variables is generated in the Benders master problem, while the generation of the linking constraints is relegated to the Benders subproblem along with the second group of variables. A fallout of this separation is that only a partial description of the dual of the Benders subproblem is available over the course of the algorithm. We demonstrate how the pricing subproblem for the column generation applied to the Benders master problem does also update the dual polyhedron and the existing Benders cuts in the master problem to ensure convergence. Ultimately, a novel integration of Benders cut generation and the simultaneous generation of columns and constraints yields a brand-new algorithm for solving large-scale CDR-problems. We illustrate the application of the proposed method on a time-constrained routing problem. Our numerical experiments confirm the outstanding performance of the new decomposition method.

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.

VNS matheuristic for a bin packing problem with a color constraint

We study a new variant of the bin packing problem. Given a set of items, each item has a set of colors. Each bin has a color capacity, the total number of colors for a bin is the union of colors for its items and can not exceed the bin capacity. We want to pack all items into the minimal number of bins. For this NP-hard problem we apply the column generation technique based on the VNS matheuristic for the pricing problem. To get optimal or near optimal solutions we apply VNS matheuristic again using optimal solution for the large scale linear programming relaxation. Computational experiments are reported for the randomly generated test instances with large bin capacity and number of items up to 250.

Computer Oriented Interior Point Algorithm for Solving Linear Programming Problem with Application

In this paper, we study the interior point algorithm for solving linear programming (LP) problem developed by Narendra Karmarkar. As interior point algorithm for LP problem involves tremendous calculations, it is quite impossible to do so by hand calculation. To fulfill the requirement we develop computer code in MATLAB for LP which is based on this algorithm procedure. To illustrate the purpose, we formulate a real life sizeable large-scale linear program for diet problem and solve it using our computer code for interior point algorithm in MATLAB.
 Dhaka Univ. J. Sci. 65(1): 41-47, 2017 (January)

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...

Robust Co-Optimization to Energy and Ancillary Service Joint Dispatch Considering Wind Power Uncertainties in Real-Time Electricity Markets

This paper presents an adjustable robust co-optimization model for energy and ancillary service markets joint dispatch considering wind power uncertainties in real-time electricity markets. The participation factors are employed to generate adjustable generation outputs and ensure energy balance. The proposed robust optimization model can be equivalently transformed into a large-scale linear programming model with a huge number of dummy variables and additional constraints that challenge real-time markets. To further reduce the computation burden, the redundant constraint reduction strategy is proposed to fast eliminate redundant transmission security constraints before solving the model. The results from the IEEE 30-bus system with two wind farms show that the proposed robust joint optimization model provides a better solution compared to those cases using sequential dispatch or deterministic dispatch methodologies. Additional results from practical large-scale systems suggest that the number of redundant constraints is less than 6%, indicating the effectiveness of the proposed redundant constraint reduction strategy in improving the computation efficiency of the robust joint dispatch in real-time markets.

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.

Computationally efficient dynamic simulation of cellular kinetics via explicit solution of flux balance analysis: xDFBA modelling and its biochemical process applications

Recent developments in reconstruction of genome scale metabolism have seen their applications in predicting cell's dynamic responses by combining intracellular metabolism with macro-scale kinetic models. This, however, can be computationally prohibitive for genome-scale metabolic models because a large-scale linear program is solved repeatedly between integration time step. In addition, when this model is used to applications such as parameter estimation or optimal control, the resulting problem becomes bi-level optimization problem, which is difficult to solve real-time. This study proposes a novel dynamic flux balance analysis framework, which constructs intracellular fluxes parameterized as functions of boundary conditions a priori. This explicit solution does not require solving linear programs successively and can be readily exploited by macro-scale kinetic models. We refer to the proposed approach as xDFBA meaning dynamic flux balance analysis embedded with explicit solutions. The method is applied to the growth models of Escherichia coli and Saccharomyces cerevisiae and shows identical prediction capability but reduced computational load, compared to the result of existing DFBA schemes. Finally, by applying parameter estimation and optimal control to the kinetic model of E. coli with xDFBA approach, its biochemical applicability is demonstrated.

Optimization over structured subsets of positive semidefinite matrices via column generation

We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.

Optimal Integration of Variable Renewables in Electric Power Systems of Japan

AbstractAfter the Fukushima nuclear accident, nuclear policy has become uncertain in Japan, and alternatively, variable renewable (VR) energy, such as wind and photovoltaic (PV), is highlighted for addressing climate change and energy security issues. This paper attempts to assess the massive integration of VR into Japan’s long-term power-generation mix, employing the dynamic multiregional optimal power-generation mix model, which is a large-scale linear programming model including 200 million constraints. The feature of the model enables explicit consideration of the actual variability of wind and PV outputs in a 10-min temporal resolution and to analyze the impact of those variabilities on long-term expansion planning of power plants. Simulation results show that both nuclear decommission and carbon regulation policies make VR economically justified and encourage its expansion in Japan’s long-term power-generation mix. When nuclear is totally decommissioned and CO2 is mitigated by 50% from 2010 levels b...

Cutting planes for the multistage stochastic unit commitment problem

As renewable energy penetration rates continue to increase in power systems worldwide, new challenges arise for system operators in both regulated and deregulated electricity markets to solve the security-constrained coal-fired unit commitment problem with intermittent generation (due to renewables) and uncertain load, in order to ensure system reliability and maintain cost effectiveness. In this paper, we study a security-constrained coal-fired stochastic unit commitment model, which we use to enhance the reliability unit commitment process for day-ahead power system operations. In our approach, we first develop a deterministic equivalent formulation for the problem, which leads to a large-scale mixed-integer linear program. Then, we verify that the turn on/off inequalities provide a convex hull representation of the minimum-up/down time polytope under the stochastic setting. Next, we develop several families of strong valid inequalities mainly through lifting schemes. In particular, by exploring sequence independent lifting and subadditive approximation lifting properties for the lifting schemes, we obtain strong valid inequalities for the ramping and general load balance polytopes. Finally, branch-and-cut algorithms are developed to employ these valid inequalities as cutting planes to solve the problem. Our computational results verify the effectiveness of the proposed approach.