Quadratic Integer Research Articles

Resilient planning strategies to support disruption-tolerant production operations

The challenge being addressed in this paper is how best to prepare production for the potential disruptive events in order to minimise the impact of these unexpected disturbances. In particular, the paper is focusing on two particular approaches to disruption preparation. The first examines the optimal location and quantity of inventory or “work-in-progress”. The second explores the role of inspection stations in providing timely warning of quality issues. A mathematical model of the operations of a production system is suggested, which covers several types of disruptions: resource breakdowns with unknown start time, product quality losses and random demand fluctuations. The optimal solution minimises a total cost objective function, which is made up of production costs, inventory holding costs, inspection costs, as well as penalty costs for the lost or uncompleted orders. As an output, the model gives the best location and quantity of inventory and the best location of inspection points for any arbitrary production system. The model is first formulated as an optimal control problem, and it is shown how to transform it to a quadratic integer programming problem. The approach was applied to study an industrial strength laboratory production system that reflects the actual industrial disruption scenarios experienced by an industrial collaborator in a key production facility. The optimal solution for this laboratory system is presented and its evaluation is shown under several scenarios with disruptions. The model was also tested on large scale production systems with up to 500 resources.

A new algorithm for quadratic integer programming problems with cardinality constraint

Quadratic integer programming problems with cardinality constraint have many applications in real life. Portfolio selection is an important application in financial optimization. In this paper we develop an exact and efficient algorithm for quadratic integer programming problems with cardinality constraint. This iterative algorithm is actually a branch and bound method, which adopts a domain cut and partition scheme. Removing cardinality constraint and integrality constraints on variables, the relaxation problem is a quadratic programming problem. By solving the quadratic programming problem, we find a lower bound for the optimal objective function value of the primal problem. The domain cut technique is used to cut off some regions that do not contain any feasible solution better than the incumbent solution. Thus the optimality gap is reduced greatly. The finite number of iterations is clear from the fact that there are only a finite number of feasible integer solutions in the feasible region. Encouraging computational results are also reported in the paper.

Study on Structure Optimization Design for Outrigger of In-situ Slewing Device

To solve the problem of insufficient turning radius for heavy-duty wreckers in narrow and congested road environment, this paper designed a new type of in-situ slewing device, and conducted a finite element simulation analysis on it. On this basis, by combining the optimization theory, this paper further optimized the structure of in-situ slewing device by using three different optimization design methods, including screening method, nonlinear second-order Lagrange method and quadratic integer sequence algorithm. The total weight loss of outrigger structure was 69.96 kilogram, the weight-loss ratio is equal to 7.34%. These results prove that the goal of structural optimization design has been achieved.

Influence Maximization From Cascade Information Traces in Complex Networks in the Absence of Network Structure

Influence maximization refers to the problem of selecting the most influential source set of a given size in a diffusion network. Most of the existing literature assumes the knowledge of the underlying network and the knowledge of the diffusion model of propagation to solve the influence maximization problem. However, both the real-world information networks and their diffusion models are not easy to determine in practice. In this article, an influence maximization algorithm based on the observed cascades has been proposed. A novel idea that a good subset of nodes for influence maximization should be composed of nodes that are not only active and strong or influential by themselves but also independent of each other has been proposed. Based on this premise, the problem has been cast as a quadratic integer programming problem with constraints. Furthermore, this problem has been shown to be equivalent to a particular class of Max-GP problem, namely, max-not-cut with size ${k}$ (MNC), for which the state-of-the-art semidefinite programming (SDP) solution techniques exist with guaranteed performance ratios, although the original problem is NP-hard. The new SDP-based method presented in this article is tested on a number of synthetic as well as real-world data sets and has been shown to perform better than the state-of-the-art influence maximization algorithms which work with the apriori knowledge of the network and assume a particular diffusion model. The results demonstrate its practical applicability in applications using Twitter, blogosphere, and other social networks which are being increasingly used to target influential customers for viral marketing.

Branching with hyperplanes in the criterion space: The frontier partitioner algorithm for biobjective integer programming

We present an algorithm for finding the complete Pareto frontier of biobjective integer programming problems. The method is based on the solution of a finite number of integer programs. The feasible sets of the integer programs are built from the original feasible set, by adding cuts that separate efficient solutions. Providing the existence of an oracle to solve suitably defined single objective integer subproblems, the algorithm can handle biobjective nonlinear integer problems, in particular biobjective convex quadratic integer optimization problems. Our numerical experience on a benchmark of biobjective integer linear programming instances shows the efficiency of the approach in comparison with existing state-of-the-art methods. Further experiments on biobjective integer quadratic programming instances are reported.

The railway rapid transit network construction scheduling problem

We consider the problem of scheduling the construction of a railway rapid transit transportation network. We assume that the network topology is already known. The problem consists of sequencing the construction tasks in order to maximize the long term profit of the project. The problem can be viewed as a resource-constrained project scheduling problem, where both the budget and available construction equipment act as resources influencing the schedule. Since lines segments can be put into operation as soon as they are finished, both the costs and the revenues are dependent on the schedule. We propose a quadratic integer programming model which is solved to optimality by branch-and-cut. To illustrate the methodology we apply the model to the construction of the Metro network of the city of Seville, and we perform sensitivity analyses on several model parameters.

A new approximation algorithm for unrelated parallel machine scheduling with release dates

In the current study, an unrelated parallel machine scheduling problem with release dates is considered, which is to obtain a job assignment with minimal sum of weighted completion times. Although this problem is NP-hard in the strong sense, which renders the optimality seeking a formidable task within polynomial time, a 4-approximation algorithm based on the constant worst-case bound is devised and proved in comparison with the existing 16/3-approximation (Hall et al. in Math Oper Res 22(3):513–544, 1997). In the newly proposed algorithm, the original scheduling problem is divided into several sub-problems based on release dates. For each sub-problem, a convex quadratic integer programming model is constructed in accordance with the specific problem structure. Then a semi-definite programming approach is implemented to produce a lower bound via the semi-definite relaxation of each sub-problem. Furthermore, by considering the binary constraint, a branch and bound based method and a local search strategy are applied separately to locate the optimal solution of each sub-problem. Then the solution of the original scheduling problem is derived by integrating the outcomes of the sub-problems via the proposed approximation algorithm. In the numerical analysis, it is discovered that the proposed methods could always obtain a scheduling result within $$1\%$$ of the optimal solution. And an asymptotic trend could be observed where the quality of solutions improves even further as the scale of the scheduling problem increases.

Global optimum economic designing of grid-connected photovoltaic systems with multiple inverters using binary linear programming

Nowadays, grid–connected photovoltaic (GCPV) system is known as a top leading technology among all resources. However, it still suffers from drastic investment costs. Detailed economic studies should be conducted in this regard to make this technology as gainful as possible. A practical approach is the “optimum economic design”, trying to find an electrically possible layout, i.e. number of series modules and parallel strings as well as the inverter number with the highest profit. This problem is inherently an quadratic integer programming owing to the multiplication of two integer variables in its objective function and some technical constraints. This nonlinear problem should be solved by exhaustive search methods, including comparative and evolutionary algorithms (EAs) such as particle swarm optimization (PSO) and genetic algorithm (GA). In this paper, a new formulation based on the definition of new binary variables has been proposed to convert this problem to the binary linear programming (BLP). The provided method finds the global optimum solution in a scale of seconds while EAs have to be run numerous times in a scale of hours to reach a sufficiently good answer. Moreover, although current methodologies are rarely covered GCPV systems with multiple inverter types, this formulation can be easily developed for systems with several inverter types. The simulations of a 1.1 MW power plant system endorse that the output design provided by the proposed method assures 95,000 $ (1.94%) higher profit compared with those presented by GA. The sensitivity analysis, provided for the prototype system by the efficient new algorithm, also unveils it is economically viable for even 52% of the current feed–in tariff, 40% energy generation lower than the estimated value and 1.1 $/W price rise for the initial investment.

A multilevel analysis of the Lasserre hierarchy

This paper analyzes the relation between different orders of the Lasserre hierarchy for polynomial optimization (POP). Although for some cases solving the semidefinite programming relaxation corresponding to the first order of the hierarchy is enough to solve the underlying POP, other problems require sequentially solving the second or higher orders until a solution is found. For these cases, and assuming that the lower order semidefinite programming relaxation has been solved, we develop prolongation operators that exploit the solutions already calculated to find initial approximations for the solution of the higher order relaxation. We can prove feasibility in the higher order of the hierarchy of the points obtained using the operators, as well as convergence to the optimal as the relaxation order increases. Furthermore, the operators are simple and inexpensive for problems where the projection over the feasible set is “easy” to calculate (for example integer {0, 1} and {−1,1} POPs). Our numerical experiments show that it is possible to extract useful information for real applications using the prolongation operators. In particular, we illustrate how the operators can be used to increase the efficiency of an infeasible interior point method by using them as an initial point. We use this technique to solve quadratic integer {0, 1} problems, as well as MAX-CUT and integer partition problems.

Topology optimization via sequential integer programming and Canonical relaxation algorithm

The mathematical essence of structural topology optimization is large-scale nonlinear integer programming. To overcome its huge computational burden, a popular way is to relax the 0–1 variable constraints and transform the integer programming problem into a continuous variable programming problem which can be solved by using gradient based mathematical programming methods. To cope with the variable transformation, the well-known SIMP (Solid Isotropic Material with Penalty) method introduces interpolation schemes for the material properties versus design variables with penalty and achieves great success and popularity. However, there is no doubt that directly tackling the large-scale nonlinear integer programming is very important. This paper solves the structural topology optimization problems with single or multiple constraints by applying the Canonical Dual Theory (CDT) by Gao and Ruan (2010) together with Sequential Approximate Programming approach under the classic structural topology optimization formulations.Following the Sequential Approximate Programming (SAP) frame from the structural optimization, the present paper firstly utilizes sensitivity information to construct the explicit and separable approximate Sequential Quadratic Integer Programming (SQIP) or Sequential Linear Integer Programming (SLIP) subproblems for the topology optimization. And then, the subproblems are solved by applying the Canonical relaxation algorithm based on CDT theory. Their special mathematical structures are exploited to develop analytic solution of Kuhn–Tucker condition of the dual programming. Numerical experiments of two linear and quadratic integer programming problems with random coefficients assert that the Canonical relaxation algorithm can obtain approximate solutions with good properties very efficiently and the dual gap is negligible when the number of design variables increases.Because move limit strategies play a key role in many search algorithms of structural optimization, this paper combines one of two different move limit strategies within the new method. The new method first solves a set of classic topology optimization problems with only material usage constraint, including minimum structural compliance design under constant load, maximum heat transfer efficiency for the heat conduction problem. And then we apply the method to the topology optimization problems with multiple constraints, including minimum structural compliance design under an additional local displacement constraint and minimum structural compliance design under infill constraints. The results of these problems demonstrate that the new method can efficiently solve the discrete variable structural topology optimization problems with multiple nonlinear constraints or many local linear constraints in a unified and systematic way. Beyond that, the new method can achieve integer solutions when combined with the move limit strategy of controlling volume fraction parameter. It can deal with much more design variables than the general branch-and-bound method and makes no use of any sensitivity threshold or heuristic stabilization scheme during the iterative process in comparison with BESO method. Finally, the new method in this paper can be further developed as a general solver for these large-scale discrete variable structural topology optimization problems.

A result on the Slope conjectures for 3-string Montesinos knots

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.

SDP-based branch-and-bound for non-convex quadratic integer optimization

Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed by Buchheim and Wiegele (Math Program 141(1--2):435--452, 2013), which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST uses an off-the-shelf interior point algorithm. In this paper, we present a tailored coordinate ascent algorithm for solving the dual problems of these SDPs. Building on related ideas of Dong (SIAM J Optim 26(3):1962--1985, 2016), it exploits the particular structure of the SDPs, most importantly a small rank of the constraint matrices. The latter allows both an exact line search and a fast incremental update of the inverse matrices involved, so that the entire algorithm can be implemented to run in quadratic time per iteration. Moreover, we describe how to extend this approach to a certain two-dimensional coordinate update. Finally, we explain how to include arbitrary linear constraints into this framework, and evaluate our algorithm experimentally.

Greedy robust wind farm layout optimization with feasibility guarantee

ABSTRACTThis article proposes a method with light data requirements for generating robust wind farm layouts. Robustness in this work is quantified as the lowest energy conversion efficiency of the wind farm across all wind directions. A quadratic integer programming formulation for generating robustness-maximizing layouts is presented. Small instances of the proposed formulation can be solved to optimality using branch and bound. A modified greedy algorithm that guarantees solution feasibility with regards to inter-turbine safety distance is proposed to find solutions to larger problem instances. A series of experiments were conducted using real world wind data collected at two sites to demonstrate the trade-offs in power generation between robust layouts and power output maximizing layouts. The results show a loss of around 1.1% in hourly power generation in return for an increase in minimum power output of 1% to 45% across all directions for robust layouts generated in the experiments. The increase in robustness largely depends on the shape and orientation of the wind farm relative to the dominant wind direction, as well as the difference between the average wind speed at the site of the wind farm and rated wind speed of the turbines.

Decreasing height along continued fractions

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

Virtual Network Survivability Through Joint Spare Capacity Allocation and Embedding

A key challenge in network virtualization is to efficiently map a virtual network (VN) on a substrate network (SN), while accounting for possible substrate failures. This is known as the survivable VN embedding (SVNE) problem. The state-of-the-art literature has studied the SVNE problem from infrastructure providers’ (InPs’) perspective, i.e., provisioning backup resources in the SN. A rather unexplored solution spectrum is to augment the VN with sufficient spare backup capacity to survive substrate failures and embed the resulting VN accordingly. Such augmentation enables InPs to offload failure recovery decisions to the VN operator, thus providing more flexible VN management. In this paper, we study the problem of jointly optimizing spare capacity allocation in a VN and embedding the VN to guarantee full bandwidth in the presence of multiple substrate link failures. We formulate the optimal solution to this problem as a quadratic integer program that we transform into an integer linear program. We also propose a heuristic algorithm to solve larger instances of the problem. Based on analytical study and simulation, our key findings are: 1) provisioning shared backup resources in the VN can yield ~33% more resource efficient embedding compared to doing the same at the SN level and 2) our heuristic allocates ~21% extra resources compared to the optimal, while executing several orders of magnitude faster.

Optimizing allocation in a warehouse network

We study the allocation problem faced by an international retail company operating e-commerce: the firm wants to determine the initial allocation of goods to warehouses, so as to provide customers in local markets with high level of service. We represent the movement of goods through the warehouse network to the customers using linear and quadratic integer programming models, and provide a computational evaluation using real data.

A Fair Multi-Channel Assignment Algorithm With Practical Implementation in Distributed Cognitive Radio Networks

Designing an efficient spectrum assignment (SA) mechanism is a key issue for realizing dynamic spectrum access in cognitive radio network. In multi-channel selection based SA schemes, secondary users (SUs) are able to utilize multiple channels simultaneously to enhance the network throughput. However, a fairness problem may happen if few SUs utilize too many idle data channels that other SUs are left with no idle channels, thus increasing the blocking probability and reducing the fairness. Aiming at improving the network throughput with multi-channel selection capability while maintaining fairness among the SUs, in this paper, we propose a fair multi-channel assignment scheme (FMCA) for distributed cognitive radio networks. For the FMCA scheme, we design a new MAC framework for sensing and access contention resolution, which is integrated into the FMCA scheme. Channel-aggregation (CA) technique is used in each SU to enable the multi-channel selection ability. Considering both of the idle data channel utilization efficiency and the transmit power budget constrained CA ability of each SU, we analytically formulate a channel assignment problem according to the well-known Jain’s fairness criterion. Our objective is to find a channel assignment with maximal fairness index for all SUs. The optimization problem is turned out to be a quadratic integer programming (QIP). According to the definition of Jain’s fairness criterion, we design an algorithm to get the optimal solution of the QIP. With the optimal channel assignment solution, the FMCA scheme is realized in the channel assignment phase of the proposed MAC protocol. Extensive simulation results show that the proposed FMCA scheme gets a good tradeoff between throughput and fairness compared with the existing SA schemes.

The minimum distance superset problem: formulations and algorithms

The partial digest problem consists in retrieving the positions of a set of points on the real line from their unlabeled pairwise distances. This problem is critical for DNA sequencing, as well as for phase retrieval in X-ray crystallography. When some of the distances are missing, this problem generalizes into a “minimum distance superset problem”, which aims to find a set of points of minimum cardinality such that the multiset of their pairwise distances is a superset of the input. We introduce a quadratic integer programming formulation for the minimum distance superset problem with a pseudo-polynomial number of variables, as well as a polynomial-size integer programming formulation. We investigate three types of solution approaches based on an available integer programming solver: (1) solving a linearization of the pseudo-polynomial-sized formulation, (2) solving the complete polynomial-sized formulation, or (3) performing a binary search over the number of points and solving a simpler feasibility or optimization problem at each step. As illustrated by our computational experiments, the polynomial formulation with binary search leads to the most promising results, allowing to optimally solve most instances with up to 25 distance values and 8 solution points.

The minimum-cost transformation of graphs

A complete proof that algorithms proposed by the authors solve the problem of minimum-cost transformation of a graph into another graph is given. The problem is solved both by a direct algorithm of linear complexity and by a reduction to quadratic integer linear programming.

Extensions on ellipsoid bounds for quadratic integer programming

Ellipsoid bounds for strictly convex quadratic integer programs have been proposed in the literature. The idea is to underestimate the strictly convex quadratic objective function q of the problem by another convex quadratic function with the same continuous minimizer as q and for which an integer minimizer can be easily computed. We initially propose in this paper a different way of constructing the quadratic underestimator for the same problem and then extend the idea to other quadratic integer problems, where the objective function is convex (not strictly convex), and where the objective function is nonconvex and box constraints are introduced. The quality of the bounds proposed is evaluated experimentally and compared to the related existing methodologies.