Cutting-plane Research Articles

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most used method for proving size lower bounds, and it is known that for the mentioned proof systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the cutting planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2009 under the name of CP cutwidth. In this article, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs \(G\) and \(H\) , we show a direct connection between the Weisfeiler–Leman differentiation number \(\mathsf{WL}(G,H)\) of the graphs and the width of a CP refutation for the corresponding isomorphism formula \(\mathrm{Iso}(G,H)\) . In particular, we show that if \(\mathsf{WL}(G,H)\leq k\) , then there is a CP refutation of \(\mathrm{Iso}(G,H)\) with width \(k\) , and if \(\mathsf{WL}(G,H) \gt k\) , then there are no CP refutations of \(\mathrm{Iso}(G,H)\) with width \(k-2\) . Similar results are known for other proof systems, like Resolution, Sherali–Adams, or Polynomial Calculus. We also obtain polynomial-length CP refutations from our width bound for isomorphism formulas for graphs with constant Weisfeiler–Leman dimension. Furthermore, we notice that a length lower bound for refuting graph isomorphism formulas in the subsystem of tree-like cutting planes with polynomially bounded coefficients follows from known results.

Stability Analysis of Matrices and Single-Parameter Matrix Families on Special Regions

This paper proposes an efficient method for determining the D-stability of matrices using Kelley’s cutting-plane method. The study examines the D-stability of matrices in symmetric regions of the complex plane, defined by quadratic matrix inequalities (QMI) and polynomial functions. Using Kelley’s cutting-plane method, we present an iterative algorithm for determining the D-stability of a given matrix. Further, the robustness of single-parameter matrix families is analyzed, and a method is proposed for considering their D-stability. Through examples, we indicate the possible applicability of the proposed approach in addressing stability problems in linear systems.

Optimisasi Jumlah Produksi Roti Kacang Dengan Metode Cutting Plane

Abstract. This research aims to optimize the production of peanut bread at UKM Peanut Roti Hj. Eliya Lubis in Tebing Tinggi City. The main challenges faced are fluctuating raw material prices and unstable consumer demand, which affect the production process. The research uses the cutting plane method to complete an integer linear program to determine optimal production quantities. This method was chosen because it is effective in solving optimization problems with integer results, which are relevant for in-house production. The research results show that the cutting plane method is able to produce better optimal solutions than other methods, such as branch and bound, thereby increasing production efficiency and company profits. This research makes a significant contribution in helping SME entrepreneurs face intense business competition by managing resources more effectively and efficiently.

An exact method for trilevel hub location problem with interdiction

In this paper, we study the problem of designing a hub network that is robust against deliberate attacks (interdictions). The problem is modeled as a three-level, two-player Stackelberg game, in which the network designer (defender) acts first to locate hubs to route a set of flows through the network. The attacker (interdictor) acts next to interdict a subset of the located hubs in the designer’s network, followed again by the defender who routes the flows through the remaining hubs in the network. We model the defender’s problem as a trilevel optimization problem, wherein the attacker’s response is modeled as a bilevel hub interdiction problem. We study such a trilevel problem on three variants of hub location problems studied in the literature namely: p-hub median problem, p-hub center, and p-hub maximal covering problems. We present a cutting plane based exact method to solve the problem. The cutting plane method uses supervalid inequalities, which is obtained from the solution of the lower level interdiction problem. To solve the lower level hub interdiction problem efficiently, we propose a penalty-based reformulation of the problem. Using the reformulation, we present a branch-and-cut based exact approach to solve the problem efficiently. We conduct experiments to show the computational advantages of the above algorithm. To the best of our knowledge, the cutting plane approach proposed in this paper is among the first exact method to solve trilevel location–interdiction problems. Our computational results show interesting implications of incorporating interdiction risks in the hub location problem.

Data-driven contract design for supply chain coordination with algorithm sharing and algorithm competition

Supply chain members can intelligently learn their decisions based on historical data by using Machine-Learning (ML) algorithms. To coordinate the supply chain, the data-driven contract design problems for three contracts—buyback, quantity flexibility, and combined quantity flexibility and rebate—were investigated for a supply chain with one manufacturer and multiple retailers under algorithm sharing and algorithm competition. The problems were formulated as bi-level optimization models by introducing nonlinear mapping from historical demand data to ordering decisions and using ML algorithms to learn the mapping parameters. The bi-level optimization models were transformed into semi-infinite programming models and solved using the (nested) cutting plane methods. Empirical studies using data from two databases showed that algorithm sharing or algorithm competition, the type of contract used, and learning algorithms were the three factors influencing the performance of supply chain coordination when using a data-driven contract design. Algorithm sharing was found to be more beneficial to the supply chain members than algorithm competition in promoting supply chain coordination. An effective incentive mechanism, such as an individualized buyback ratio and a rebate from the manufacturer to the retailers with a good forecast performance, can encourage the retailers to participate in algorithm sharing and improvement.

Penyelesaian Metode Round Off dan Metode Cutting Plane dalam Optimalisasi Produksi Anyaman Rotan UD. Kirana

UD Kirana is a trading business that produces rattan weaving in Rumbai district which consists of 5 types of products produced, namely guest chairs, terrace chairs, tender chairs, serving hoods and baskets. In the rattan woven production process, UD Kirana cannot predict how many items can be produced each week so that maximum profits have not been achieved.This research aims to find out how goods can be produced every week in order to get maximum profits.The methods used are the Round Off method and the Cutting Plane method. Based on the research result, UD Kirana produced 10 tender chairs, 7 serving hoods and 21 baskets with a profit of Rp. 6.060.000. The Round Off method is more efficient than the Cutting Plane method, this can be seen from the addition of constraints, namely the Round Off method adds two constraints and the Cutting Plane method adds three constraints or gomory constraints.

FairWASP: Fast and Optimal Fair Wasserstein Pre-processing

Recent years have seen a surge of machine learning approaches aimed at reducing disparities in model outputs across different subgroups. In many settings, training data may be used in multiple downstream applications by different users, which means it may be most effective to intervene on the training data itself. In this work, we present FairWASP, a novel pre-processing approach designed to reduce disparities in classification datasets without modifying the original data. FairWASP returns sample-level weights such that the reweighted dataset minimizes the Wasserstein distance to the original dataset while satisfying (an empirical version of) demographic parity, a popular fairness criterion. We show theoretically that integer weights are optimal, which means our method can be equivalently understood as duplicating or eliminating samples. FairWASP can therefore be used to construct datasets which can be fed into any classification method, not just methods which accept sample weights. Our work is based on reformulating the pre-processing task as a large-scale mixed-integer program (MIP), for which we propose a highly efficient algorithm based on the cutting plane method. Experiments demonstrate that our proposed optimization algorithm significantly outperforms state-of-the-art commercial solvers in solving both the MIP and its linear program relaxation. Further experiments highlight the competitive performance of FairWASP in reducing disparities while preserving accuracy in downstream classification settings.

Robust Beamforming for Downlink Multi-Cell Systems: A Bilevel Optimization Perspective

Utilization of inter-base station cooperation for information processing has shown great potential in enhancing the overall quality of communication services (QoS) in wireless communication networks. Nevertheless, such cooperations require the knowledge of channel state information (CSI) at base stations (BSs), which is assumed to be perfectly known. However, CSI errors are inevitable in practice which necessitates beamforming technique that can achieve robust performance in the presence of channel estimation errors. Existing approaches relax the robust beamforming design problems into semidefinite programming (SDP), which can only achieve a solution that is far from being optimal. To this end, this paper views robust beamforming design problems from a bilevel optimization perspective. In particular, we focus on maximizing the worst-case weighted sum-rate (WSR) in the downlink multi-cell multi-user multiple-input single-output (MISO) system considering bounded CSI errors. We first reformulate this problem into a bilevel optimization problem and then develop an efficient algorithm based on the cutting plane method. A distributed optimization algorithm has also been developed to facilitate the parallel processing in practical settings. Numerical results are provided to confirm the effectiveness of the proposed algorithm in terms of performance and complexity, particularly in the presence of CSI uncertainties.

An Integrated Predictive Maintenance and Operations Scheduling Framework for Power Systems Under Failure Uncertainty

Maintenance planning plays a key role in power system operations under uncertainty as it helps providers and operators ensure a reliable and secure power grid. This paper studies a short-term condition-based integrated maintenance planning with operations scheduling problem while considering the possible unexpected failure of generators as well as transmission lines. We formulate this problem as a two-stage stochastic mixed-integer program with failure scenarios sampled from the sensor-driven remaining lifetime distributions of the individual system elements whereas a joint chance constraint consisting of Poisson Binomial random variables is introduced to account for failure risks. Because of its intractability, we develop a cutting-plane method to obtain an exact reformulation of the joint chance constraint by proposing a separation subroutine and deriving stronger cuts as part of this procedure. We also derive a second-order cone programming-based safe approximation of this constraint to solve large-scale instances. Furthermore, we propose a decomposition algorithm implemented in parallel fashion for solving the resulting stochastic program, which exploits the features of the integer L-shaped method and the special structure of the maintenance and operations scheduling problem to derive valid and stronger sets of optimality cuts. We further present preprocessing steps over transmission line flow constraints to identify redundancies. To illustrate the computational performance and efficiency of our algorithm compared with more conventional maintenance approaches, we design a computational study focusing on a weekly plan with daily maintenance and hourly operational decisions involving detailed unit commitment subproblems. Our computational results on various IEEE instances demonstrate the computational efficiency of the proposed approach with reliable and cost-effective maintenance and operational schedules. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0154 .

Robust Charging Network Planning for Metropolitan Taxi Fleets

We study the robust charging station location problem for a large-scale commercial taxi fleet. Vehicles within the fleet coordinate on charging operations but not on customer acquisition. We decide on a set of charging stations to open to ensure operational feasibility. To make this decision, we propose a novel solution method situated between the location routing problems with intraroute facilities and flow refueling location problems. Additionally, we introduce a problem variant that makes a station sizing decision. Using our exact approach, charging stations for a robust operation of citywide taxi fleets can be planned. We develop a deterministic core problem employing a cutting plane method for the strategic problem and a branch-and-price decomposition for the operational problem. We embed this problem into a robust solution framework based on adversarial sampling, which allows for planner-selectable risk tolerance. We solve instances derived from real-world data of the metropolitan area of Munich containing 1,000 vehicles and 60 potential charging station locations. Our investigation of the sensitivity of technological developments shows that increasing battery capacities shows a more favorable impact on vehicle feasibility of up to 10 percentage points compared with increasing charging speeds. Allowing for depot charging dominates both of these options. Finally, we show that allowing just 1% of operational infeasibility risk lowers infrastructure costs by 20%. Funding: This work was partially funded by the Deutsche Forschungsgemeinschaft [Project 277991500]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0207 .

A Neural Separation Algorithm for the Rounded Capacity Inequalities

The cutting plane method is a key technique for successful branch-and-cut and branch-price-and-cut algorithms that find the exact optimal solutions for various vehicle routing problems (VRPs). Among various cuts, the rounded capacity inequalities (RCIs) are the most fundamental. To generate RCIs, we need to solve the separation problem, whose exact solution takes a long time to obtain; therefore, heuristic methods are widely used. We design a learning-based separation heuristic algorithm with graph coarsening that learns the solutions of the exact separation problem with a graph neural network (GNN), which is trained with small instances of 50 to 100 customers. We embed our separation algorithm within the cutting plane method to find a lower bound for the capacitated VRP (CVRP) with up to 1,000 customers. We compare the performance of our approach with CVRPSEP, a popular separation software package for various cuts used in solving VRPs. Our computational results show that our approach finds better lower bounds than CVRPSEP for large-scale problems with 400 or more customers, whereas CVRPSEP shows strong competency for problems with less than 400 customers. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) [RS-2023-00259550] and the Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) [2022-0-01032, Development of Collective Collaboration Intelligence Framework for Internet of Autonomous Things]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0310 .

A joint solution framework for the optimal operation and evaluation of integrated energy systems based on exergy

Exergy analysis provides a unified scale for evaluating the quality and quantity of energy, and is suitable and essential for scientifically and rationally guiding the optimization and evaluation of integrated energy systems with intricate multi-component configurations. The present work proposes an exergy analysis model and joint solution framework for integrated energy systems based on exergy, and presents an efficient and accurate solution method. The exergy analysis model is analogous to the energy-based model and possesses a succinct form. This model is then applied to establish an input exergy model defining the exergy of inputs, a destroyed exergy model defining the exergy lost due to the conversions, transmissions, and storages, and a benefit exergy model defining the remaining useful exergy for the loads. The joint solution framework optimizes the system from the basis of exergy conservation, and evaluates the destroyed exergy of each link and the prices of the various exergy products from an economic perspective based on the established exergy analysis model. The complexity of the framework is then reduced by establishing a sequential optimization and evaluation model. The optimization model is solved accurately and efficiently using a stringent cutting plane method to address the problem of relaxation inaccuracy, and piecewise linearization to accommodate the non-convex constraints. Numerical computations on a 97-node system demonstrate the accuracy of the proposed models in exergy-based optimization, destruction, and price evaluation, preventing substantial deviations (e.g., a 23% discrepancy in total destroyed exergy). The adopted cost allocation method based on energy quality reflects fair pricing, rectifying conventional pricing disparities (e.g., a 47.56% underestimation of the electric load using energy-based pricing). Therefore, the accuracy and efficiency of the developed exergy analysis model, solution framework, and solution method are validated.

A cutting-plane method with internal iteration points for the general convex programming problem

A cutting method for solving the problem of convex programming was proposed. The method calculates iteration points based on approximation by polyhedral sets of the constraint region and the epigraph of the objective function. Its distinguishing feature is that the main sequence of approximations is constructed within the admissible region. At each step, it is also possible to assess how close the current value of the function is to the optimal value. The convergence of the method was proved. A few of its implementations were outlined.

Solving a Class of Cut-Generating Linear Programs via Machine Learning

Cut-generating linear programs (CGLPs) play a key role as a separation oracle to produce valid inequalities for the feasible region of mixed-integer programs. When incorporated inside branch-and-bound, the cutting planes obtained from CGLPs help to tighten relaxations and improve dual bounds. However, running the CGLPs at the nodes of the branch-and-bound tree is computationally cumbersome due to the large number of node candidates and the lack of a priori knowledge on which nodes admit useful cutting planes. As a result, CGLPs are often avoided at default settings of branch-and-cut algorithms despite their potential impact on improving dual bounds. In this paper, we propose a novel framework based on machine learning to approximate the optimal value of a CGLP class that determines whether a cutting plane can be generated at a node of the branch-and-bound tree. Translating the CGLP as an indicator function of the objective function vector, we show that it can be approximated through conventional data classification techniques. We provide a systematic procedure to efficiently generate training data sets for the corresponding classification problem based on the CGLP structure. We conduct computational experiments on benchmark instances using classification methods such as logistic regression. These results suggest that the approximate CGLP obtained from classification can improve the solution time compared with that of conventional cutting plane methods. Our proposed framework can be efficiently applied to a large number of nodes in the branch-and-bound tree to identify the best candidates for adding a cut. History: Accepted by Andrea Lodi, Area Editor/Design & Analysis of Algorithms — Discrete. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0241 .

Design of experiments for the stochastic unit commitment with economic dispatch models

We develop a Design and Analysis of the Computer Experiments (DACE) approach to the stochastic unit commitment problem for power systems with significant renewable integration. For this purpose, we use a two-stage stochastic programming formulation of the stochastic unit commitment-economic dispatch problem. Typically, a sample average approximation of the true problem is solved using a cutting plane method (such as the L-shaped method) or scenario decomposition (such as Progressive Hedging) algorithms. However, when the number of scenarios increases, these solution methods become computationally prohibitive. To address this challenge, we develop a novel DACE approach that exploits the structure of the first-stage unit commitment decision space in a design of experiments, uses features based upon solar generation, and trains a multivariate adaptive regression splines model to approximate the second stage of the stochastic unit commitment-economic dispatch problem. We conduct experiments on two modified IEEE-57 and IEEE-118 test systems and assess the quality of the solutions obtained from both the DACE and the L-shaped methods in a replicated procedure. The results obtained from this approach attest to the significant improvement in the computational performance of the DACE approach over the traditional L-shaped method.

A dual method for polar cuts in disjoint bilinear programming

As one branch of deterministic approaches to disjoint bilinear programming, cutting plane methods are renowned for its ability to systematically reduce the search space by adding cutting planes that are able to cut off regions deemed infeasible or suboptimal. Polar cuts have been widely utilized as a dominating type of cut in terms of deepness. During the establishment of a polar cut, the modified Newton's method is employed to derive the cutting points along the positive or negative extensions of edges emanating from a local solution. Nonetheless, its performance can be further improved along the positive extensions. Drawing inspiration from integer programming, we develop a new approach based on the LP duality theory for this purpose. It re-formulates the original program with a piece-wise linear concave objective function as a single LP. Moreover, we propose a new technique to derive the edges as accommodation to degeneracy. Numerical results show that, by utilizing our newly developed dual method, computing time can be gradually saved as the percentage of generated cutting points along the positive extensions of edges rises.

A Survey On Non – linear Optimization And Global Optimization Methods

bjective functions and constraints are not linear. Non-linear optimization involves finding the best solution in the presence of non-linear relationships among the variables. These problems arise in various fields such as engineering design, economics, machine learning, and operational research. The primary challenge in non-linear optimization is the potential existence of multiple local optima, which can make it difficult to identify the global optimum. Global optimization methods are designed to overcome this challenge by searching for the global optimum, which is the best possible solution among all local optima. These methods can be broadly categorized into deterministic and stochastic approaches. Deterministic methods, such as branch and bound, interval analysis, and cutting-plane methods, systematically explore the search space and provide guarantees of finding the global optimum or bounds on the global optimum. These methods are often rigorous but can be computationally expensive, especially for high-dimensional problems. On the other hand, stochastic methods, such as genetic algorithms, simulated annealing, and particle swarm optimization, use probabilistic rules to explore the search space. These methods are inspired by natural processes and heuristics, offering flexibility and often being more efficient for large-scale problems. While they do not guarantee finding the global optimum, they are widely used due to their ability to escape local optima and explore the search space effectively. Hybrid methods that combine deterministic and stochastic approaches have also gained popularity. These methods leverage the strengths of both approaches, enhancing the robustness and efficiency of the optimization process. For example, a hybrid algorithm might use a stochastic method to explore the search space broadly and then apply a deterministic method to refine the search around promising areas. Recent advancements in computational power and algorithmic development have significantly improved the efficiency and applicability of non-linear and global optimization methods. Techniques such as machine learning-based optimization and parallel computing have enabled the handling of more complex and larger-scale optimization problems. Moreover, these advancements have facilitated the integration of optimization methods into real-time and adaptive systems, further broadening their applications.

A Survey On Non – linear Optimization And Global Optimization Methods

Non-linear optimization and global optimization methods are crucial in solving complex real-world problems where the objective functions and constraints are not linear. Non-linear optimization involves finding the best solution in the presence of non-linear relationships among the variables. These problems arise in various fields such as engineering design, economics, machine learning, and operational research. The primary challenge in non-linear optimization is the potential existence of multiple local optima, which can make it difficult to identify the global optimum. Global optimization methods are designed to overcome this challenge by searching for the global optimum, which is the best possible solution among all local optima. These methods can be broadly categorized into deterministic and stochastic approaches. Deterministic methods, such as branch and bound, interval analysis, and cutting-plane methods, systematically explore the search space and provide guarantees of finding the global optimum or bounds on the global optimum. These methods are often rigorous but can be computationally expensive, especially for high-dimensional problems. On the other hand, stochastic methods, such as genetic algorithms, simulated annealing, and particle swarm optimization, use probabilistic rules to explore the search space. These methods are inspired by natural processes and heuristics, offering flexibility and often being more efficient for large-scale problems. While they do not guarantee finding the global optimum, they are widely used due to their ability to escape local optima and explore the search space effectively. Hybrid methods that combine deterministic and stochastic approaches have also gained popularity. These methods leverage the strengths of both approaches, enhancing the robustness and efficiency of the optimization process. For example, a hybrid algorithm might use a stochastic method to explore the search space broadly and then apply a deterministic method to refine the search around promising areas. Recent advancements in computational power and algorithmic development have significantly improved the efficiency and applicability of non-linear and global optimization methods. Techniques such as machine learning-based optimization and parallel computing have enabled the handling of more complex and larger-scale optimization problems. Moreover, these advancements have facilitated the integration of optimization methods into real-time and adaptive systems, further broadening their applications.

An exact second order cone programming approach for traffic assignment problems

Demographic changes, urbanization and increasing vehicle ownership at unprecedented rates put a lot of strain on cities particularly on urban mobility and transportation and overwhelm transportation network infrastructures and current transportation systems, which are not built to cope with such a fast increasing demand. Traffic congestion is considered as the most difficult challenge to tackle for sustainable urban mobility and is aggravated by the increased freight activity due to e-commerce and on-demand delivery and the explosive growth in transportation network companies and ride-hailing services. There is a need to implement a combination of policies to ensure that increased urban traffic congestion does not lower the quality of life and threaten global climate and human health and to prevent further economic losses. This study aims to contribute to the United Nations (UN) climate action and sustainable development goals in tackling recurring traffic congestion problem in urban areas to achieve a sustainable urban mobility in that it offers a solution methodology for traffic assignment problem. We introduce an exact generalized solution methodology based on reformulation of existing traffic assignment problems as a second order cone programming (SOCP) problem and propose column generation (CG) and cutting plane (CP) algorithms to solve the problem effectively for large scale road network instances. We conduct numerical experiments to test the performance of the proposed algorithms on realistic road networks.

Optimalisasi Perencanaan Dan Pengendalian Produksi Rak Palet Menggunakan Metode Cutting Plane

PT Atlantic Anugrah Metalindo is a manufacturing company that produces costumize products, steel door systems and fireproof doors. Currently PT Atlantic Anugrah Metalindo is producing 100 units of pallet rack manufacturing with customers from PT X so that proper material requirements planning is needed. The purpose of this research is to determine the material requirements of pallet racks optimally and efficiently. This research requires calculating the Bill of Materials (BOM) using the Cutting plane method and controlling the production process to run effectively and efficiently. Based on the calculations that have been carried out, the results show that for 2 mm thick mild steel material requirements, 6 sheets of plate are needed, 3 mm thick mild steel requires 173 sheets of plate, 3.6 mm thick mild steel requires 12 sheets of plate and 5 mm thick mild steel requires 12 sheets of plate. So that from these calculations PT Atlantic Anugrah Metalindo is expected to minimize the material requirements of pallet rack products.