Number Of Integer Variables Research Articles

Optimal and efficient planning of charging stations for electric vehicles in urban areas: formulation, complexity and solutions

The deployment of charging infrastructure for electric vehicles is crucial in extending their range. Many studies on the charging infrastructure deployment adopt the Mixed Integer Linear Programming (MILP) method to optimize various objectives. However, as the number of integer variables and constraints increases, the computational time and memory requirements of MILP models increase exponentially. This makes it impractical to use MILP models to solve large-scale optimization problems. In this paper, we formulate and prove that the Planning of Electric Vehicle Charging Stations (PEVCS) is an NP-complete combinatorial optimization problem. We also prove that PEVCS has a significant effect, that is, submodularity. Additionally, we propose two efficient methods that use submodularity to improve the conventional methodology for PEVCS. Furthermore, we provide a provable guarantee for the performance of our proposed methods. Our experimental results demonstrate the efficiency and effectiveness of these methods on small-scale and large-scale datasets, especially in realistic large-scale situations.

An approximation algorithm for indefinite mixed integer quadratic programming

In this paper, we give an algorithm that finds an \(\epsilon \)-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.

Improved heuristics for finding balanced teams

This research addresses the problem of dividing a group of people into a collection of teams that need to be “balanced” across a variety of different attributes. This type of problem arises, for example, in an academic setting where it is necessary to partition students into a number of balanced study teams and also in a youth camp in which children need to be formed into sports teams that are competitive with each other. Recent work has resulted in both linear and nonlinear integer programing models for solving this problem. In the research here, improvements to the models are made together with a linear approximation to the nonlinear objective function that significantly reduce the number of integer variables and constraints. Computational experiments are performed on random instances of the problem, as well as on instances for which there are almost perfectly balanced teams, the latter providing a way to determine the quality of the optimal solution obtained by the heuristics. These tests show that the approach developed here almost always obtain better balanced teams than those from prior research.

Disjunctive Programming for Multiobjective Discrete Optimisation

In this paper, I view and present the multiobjective discrete optimisation problem as a particular case of disjunctive programming where one seeks to identify efficient solutions from within a disjunction formed by a set of systems. The proposed approach lends itself to a simple yet effective iterative algorithm that is able to yield the set of all nondominated points, both supported and nonsupported, for a multiobjective discrete optimisation problem. Each iteration of the algorithm is a series of feasibility checks and requires only one formulation to be solved to optimality that has the same number of integer variables as that of the single objective formulation of the problem. The application of the algorithm shows that it is particularly effective when solving constrained multiobjective discrete optimisation problem instances. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0804 .

Distances between optimal solutions of mixed-integer programs

A classic result of Cook et al. (Math. Program. 34:251–264, 1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is given in terms of the number of variables and a parameter $$ \varDelta $$, which quantifies sub-determinants of the underlying linear inequalities. We show that this distance can be bounded in terms of $$ \varDelta $$ and the number of integer variables rather than the total number of variables. To this end, we make use of a result by Olson (J. Number Theory 1:8–10, 1969) in additive combinatorics and demonstrate how it implies feasibility of certain mixed-integer linear programs. We conjecture that our bound can be improved to a function that only depends on $$ \varDelta $$, in general.

A holarchic approach for multi-scale distributed energy system optimisation

The benefits of decentralised energy systems can be realised through the optimal siting of distributed energy systems and the design of highly interlinked district heating networks within existing electrical and gas networks. The problem is often formulated as a Mixed Integer Linear Programming (MILP) problem. MILP formulations are efficient and reliable, however the computational burden increases drastically with the number of integer variables, making detailed optimisation infeasible at large urban scales. To tackle complex problems at large scale the development of an efficient and robust simplification method is required. This paper presents an aggregation schema to facilitate the optimisation of urban energy systems at city scale.Currently, spatial and/or temporal aggregation are commonly employed when modelling energy systems at spatio-temporal resolutions from plant scheduling up to national scenarios. This paper argues for solving different scales separately using a bottom-up approach, while keeping track of the error made by reducing the resolution when moving from building to urban scale. Novel modelling formulations and optimisation techniques are presented. They enable drastic reduction of the computational time (by up to a factor of 100) required to find an optimal solution in reasonable time without sacrificing the quality of the results (no more than 1% loss in accuracy).A density-based clustering algorithm enables intelligent division of a large city-scale problem into sub-optimisation problems by creating clusters of different density. In each cluster, the trade-off between centralised and decentralised energy systems and the associated district heating network design is evaluated. A solution is selected based on a local optimisation of the network costs. Demand profiles of each building are assigned appropriately, then at an upper level the energy optimisation problem is solved considering the network losses at lower levels. This method enables large-scale modelling of urban energy systems while taking into account building-scale levels of detail. The clustering method enables assessment of the potential of district heating networks on city scale based on building characteristics and available urban energy systems.

Efficient virtual network embedding algorithm based on restrictive selection and optimization theory approach

Network virtualization (NV) is widely considered as a key component of the future network and promises to allow multiple virtual networks (VNs) with different protocols to coexist on a shared substrate network (SN). One main challenge in NV is virtual network embedding (VNE). VNE is a NP-hard problem. Previous VNE algorithms in the literature are mostly heuristic, while the remaining algorithms are exact. Heuristic algorithms aim to find a feasible embedding of each VN, not optimal or sub-optimal, in polynomial time. Though presenting the optimal or sub-optimal embedding per VN, exact algorithms are too time-consuming in small-scaled networks, not to mention moderately sized networks. To make a trade-off between the heuristic and the exact, this paper presents an effective algorithm, labeled as VNE-RSOT (Restrictive Selection and Optimization Theory), to solve the VNE problem. The VNE-RSOT can embed virtual nodes and links per VN simultaneously. The restrictive selection contributes to selecting candidate substrate nodes and paths and largely cuts down on the number of integer variables, used in the following optimization theory approach. The VNE-RSOT fights to minimize substrate resource consumption and accommodates more VNs. To highlight the efficiency of VNE-RSOT, a simulation against typical and state-of-art heuristic algorithms and a pure exact algorithm is made. Numerical results reveal that virtual network request (VNR) acceptance ratio of VNE-RSOT is, at least, 10% higher than the best-behaved heuristic. Other metrics, such as the execution time, are also plotted to emphasize and highlight the efficiency of VNE-RSOT.

On approximation algorithms for concave mixed-integer quadratic programming

Concave mixed-integer quadratic programming is the problem of minimizing a concave quadratic polynomial over the mixed-integer points in a polyhedral region. In this work we describe an algorithm that finds an $$\epsilon $$ -approximate solution to a concave mixed-integer quadratic programming problem. The running time of the proposed algorithm is polynomial in the size of the problem and in $$1/\epsilon $$ , provided that the number of integer variables and the number of negative eigenvalues of the objective function are fixed. The running time of the proposed algorithm is expected unless $$\mathcal {P}=\mathcal {NP}$$ .

Mixed integer reformulations of integer programs and the affine TU-dimension of a matrix

We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables. To this end, we introduce a generalization of total unimodularity called the affine TU-dimension of a matrix and study related theory and algorithms for determining the affine TU-dimension of a matrix. We also present bounds on the number of integer variables needed to represent certain integer hulls.

Resource allocation in shared spectrum access communications for operators with diverse service requirements

In this paper, we study inter-operator spectrum sharing and intra-operator resource allocation in shared spectrum access communication systems and propose efficient dynamic solutions to address both inter-operator and intra-operator resource allocation optimization problems. For inter-operator spectrum sharing, we present two competent approaches, namely the subcarrier gain based sharing and fragmentation based sharing, which carry out fair and flexible allocation of the available shareable spectrum among the operators subject to certain well-defined sharing rules, traffic demands and channel propagation characteristics. Subcarrier gain based spectrum sharing scheme has been found to be more efficient in terms of achieved throughput. However, fragmentation based sharing is more attractive in terms of computational complexity. For intra-operator resource allocation, we consider resource allocation problem with users' dissimilar service requirements, where the operator supports users with delay-constraint and non-delay constraint service requirements, simultaneously. This optimization problem is a mixed integer nonlinear programming problem and nonconvex, which is computationally very expensive, and the complexity grows exponentially with the number of integer variables. We propose less-complex and efficient suboptimal solution based on formulating exact linearization, linear approximation and convexification techniques for the nonlinear and/or non-convex objective functions and constraints. Extensive simulation performance analysis has been carried out that validates the efficiency of the proposed solution.

Robust Optimization of Schedules Affected by Uncertain Events

In this paper, we present a new method for finding robust solutions to mixed-integer linear programs subject to uncertain events. We present a new modeling framework for such events that result in uncertainty sets that depend parametrically on the decision taken. We also develop results that can be used to compute corresponding robust solutions. The usefulness of our proposed approach is illustrated by applying it in the context of a scheduling problem. For instance, we address uncertainty on the start times chosen for the tasks or on which unit they are to be executed. Delays and unit outages are possible causes for such events and can be very common in practice. Through our approach, we can accommodate them without altering the remainder of the schedule. We also allow for the inclusion of recourse on the continuous part of the problem, that is, we allow for the revision of some of the decisions once uncertainty is observed. This allows one to increase the performance of the robust solutions. The proposed scheme is also computationally favorable since the robust optimization problem to be solved remains a mixed-integer linear program, and the number of integer variables is not increased with respect to the nominal formulation. We finally apply the method to a concrete batch scheduling problem and discuss the effects of robustification in this case.

Reducing Computation Time with a Rolling Horizon Approach Applied to a MILP Formulation of Multiple Urban Energy Hub System

Abstract Energy hub model is a powerful concept allowing the interactions of many energy conversion and storage systems to be optimized. Solving the optimal configuration and operating strategy of an energy hub combining multiple energy sources for a whole year can become computationally demanding. Indeed the effort to solve a mixed-integer linear programming (MILP) problem grows dramatically with the number of integer variables. This paper presents a rolling horizon approach applied to the optimisation of the operating strategy of an energy hub. The focus is on the computational time saving realized by applying a rolling horizon methodology to solve problems over many time-periods. The choice of rolling horizon parameters is addressed, and the approach is applied to a model consisting of a multiple energy hubs. This work highlights the potential to reduce the computational burden for the simulation of detailed optimal operating strategies without using typical-periods representations. Results demonstrate the possibility to improve by 15 to 100 times the computational time required to solve energy optimisation problems without affecting the quality of the results.

Local search heuristics for the mobile facility location problem

In the mobile facility location problem (MFLP), one seeks to relocate (or move) a set of existing facilities and assign clients to these facilities so that the sum of facility movement costs and the client travel costs (each to its assigned facility) is minimized. This paper studies formulations and develops local search heuristics for the MFLP. First, we develop an integer programming (IP) formulation for the MFLP by observing that for a given set of facility destinations the problem may be decomposed into two polynomially solvable subproblems. This IP formulation is quite compact in terms of the number of nonzero coefficients in the constraint matrix and the number of integer variables; and allows for the solution of large-scale MFLP instances. Using the decomposition observation, we propose two local search neighborhoods for the MFLP. We report on extensive computational tests of the new IP formulation and local search heuristics on a large range of instances. These tests demonstrate that the proposed formulation and local search heuristics significantly outperform the existing formulation and a previously developed local search heuristic for the problem.

Integer programming-based method for designing synthetic metabolic networks by Minimum Reaction Insertion in a Boolean model.

In this paper, we consider the Minimum Reaction Insertion (MRI) problem for finding the minimum number of additional reactions from a reference metabolic network to a host metabolic network so that a target compound becomes producible in the revised host metabolic network in a Boolean model. Although a similar problem for larger networks is solvable in a flux balance analysis (FBA)-based model, the solution of the FBA-based model tends to include more reactions than that of the Boolean model. However, solving MRI using the Boolean model is computationally more expensive than using the FBA-based model since the Boolean model needs more integer variables. Therefore, in this study, to solve MRI for larger networks in the Boolean model, we have developed an efficient Integer Programming formalization method in which the number of integer variables is reduced by the notion of feedback vertex set and minimal valid assignment. As a result of computer experiments conducted using the data of metabolic networks of E. coli and reference networks downloaded from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, we have found that the developed method can appropriately solve MRI in the Boolean model and is applicable to large scale-networks for which an exhaustive search does not work. We have also compared the developed method with the existing connectivity-based methods and FBA-based methods, and show the difference between the solutions of our method and the existing methods. A theoretical analysis of MRI is also conducted, and the NP-completeness of MRI is proved in the Boolean model. Our developed software is available at “http://sunflower.kuicr.kyoto-u.ac.jp/~rogi/minRect/minRect.html.”

Threshold Boolean form for joint probabilistic constraints with random technology matrix

We develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations for the studied stochastic problem, and we derive a set of strengthening valid inequalities. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the mixed-integer linear programming formulations are orders of magnitude faster to solve than the mixed-integer nonlinear programming formulation. The method integrating the valid inequalities in a branch-and-bound algorithm has the best performance.

Minimization of economical losses due to utility disturbances in the process industry

A process industrial site may consist of several production areas, some producing intermediate products for further refinement in other areas, and some producing end products. The areas may share the same utilities, such as steam and cooling water, which means that the areas could be connected both by the flow of products through the site and by the use of the same utilities. Management of utility disturbances thus becomes an interesting topic. In this paper, a simple approach for modeling utilities is suggested and used to formulate a mixed-integer quadratic program (MIQP) that aims at minimizing the total economic loss at the site, due to utility disturbances. The optimization problem is reformulated as an ordinary quadratic program (QP), where auxiliary variables are utilized to avoid the use of integer variables. For suitable choices of the optimization weights, the solutions to the MIQP and the QP are in many cases equal. Two examples are given, where one is a small example inspired by a real site at the specialty chemicals company Perstorp, and the second is a larger problem that aims to show the advantage of the QP formulation when the number of areas, and thus the number of integer variables, becomes large.

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233---256, 2010) and Averkov (Discret Optimiz 9(4):209---215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $$m=2$$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuejols and Margot (Math Program 120:429---456, 2009) and obtain polynomial complexity results about the mixed integer hull.

Threshold Boolean Form for Joint Probabilistic Constraints with Random Technology Matrix

We develop a new modeling and exact solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations equivalent to the studied stochastic problem. We derive a set of strengthening valid inequalities for the reformulated problems. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the MIP reformulations are orders of magnitude faster to solve than the MINLP formulation. The method integrating the derived valid inequalities in a branch-and-bound algorithm has the best performance.

Optimal regenerator assignment and resource allocation strategies for translucent optical networks

Physical layer impairments in wavelength-routed networks limit the maximum distance, a signal can travel in the optical domain, without significant distortion. Therefore, signal regeneration is required at some intermediate nodes for long-haul lightpaths. In translucent WDM networks, sparsely located regenerators at certain nodes can be used to offset the impact of physical layer impairments. The routing and wavelength assignment (RWA) techniques in such translucent networks need to take into consideration the availability of regenerators and the maximum optical reach of the transparent lightpaths (without any regeneration). Although there has been significant research interest in RWA algorithms for translucent networks, much of the research has focused on dynamic RWA techniques. Only a handful of recent papers have considered the static (offline) case, and they typically propose heuristic algorithms to solve this complex design problem for practical networks. In this paper, we propose a generalized integer linear program (ILP) based formulation for static regenerator assignment and RWA in translucent WDM optical networks, with sparse regenerator placement. To the best of our knowledge, such a formulation that optimally allocates resources for a set of lightpaths for translucent networks, given the physical network, the locations of the regenerators, and the maximum optical reach has not been considered before. The proposed formulation is important for two reasons. First, it can serve as a benchmark for evaluating different heuristic approaches that may be developedin the future. Second, we show that using a novel node representation technique, it is possible to drastically reduce the number of integer variables. This means that unlike existing ILP formulations, our approach can actually be used to generate optimal solutions for practical networks, with hundreds of lightpath demands.

Selection of typical demand days for CHP optimization

Optimizing the configuration and operation of a CHP system for a whole year becomes a computationally demanding task when, for example, integer variables are used to model the status (on/off) of different pieces of equipment. The reason is that a discrete optimization problem is fundamentally an enumerative problem, featuring that the number of possible solutions grows exponentially with the number of integer variables. This computational difficulty is known as the curse of dimensionality, and severely limits the chances to use mixed integer programming methods to design CHP systems. To work out this problem, this paper presents a new and unambiguous method to reduce a full year of demand data to a few representative days that adequately preserve significant characteristics such as the peak demands, the demand duration curves, and the temporal inter-relationship between the different types of demands (power, heating, and cooling). Days are selected using a partitional clustering method known as the k-medoids method, and their ability to resemble the original data is tested by means of two quality indexes and a calendar visual inspection. Two case studies are discussed for the completeness of the paper, showing how the method and the quality indexes can be used in practice.