Cutting Plane Algorithm Research Articles

On the relation between the extended supporting hyperplane algorithm and Kelley\u2019s cutting plane algorithm

Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm (J Soc Ind Appl Math 8(4):703–712, 1960) applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by a class of general, not necessarily convex nor differentiable, functions.

A physician planning framework for polyclinics under uncertainty

In this paper, we present a comprehensive two-level physician planning framework for polyclinics under uncertainty. The first level focuses on clinic scheduling and capacity planning decisions, whereas the second level deals with physician scheduling and operational adjustments decisions. In order to protect the generated schedules against demand uncertainty, the first level is modeled as an adjustable robust scheduling problem, which is solved using an ad hoc cutting plane algorithm. To cope with variability in patients’ treatment times, we formulate the second level as a two-stage stochastic problem and use a sample average approximation scheme to obtain solutions with small optimality gaps. We use a Monte-Carlo simulation algorithm and data obtained from a university health center in Montreal, Canada, to demonstrate the benefits of our planning framework. In particular, we show that the schedule generated by our approach is superior in terms of total cost as compared with the one obtained from a single-level deterministic model.

A multitask multiple kernel learning formulation for discriminating early- and late-stage cancers.

Genomic information is increasingly being used in diagnosis, prognosis and treatment of cancer. The severity of the disease is usually measured by the tumor stage. Therefore, identifying pathways playing an important role in progression of the disease stage is of great interest. Given that there are similarities in the underlying mechanisms of different cancers, in addition to the considerable correlation in the genomic data, there is a need for machine learning methods that can take these aspects of genomic data into account. Furthermore, using machine learning for studying multiple cancer cohorts together with a collection of molecular pathways creates an opportunity for knowledge extraction. We studied the problem of discriminating early- and late-stage tumors of several cancers using genomic information while enforcing interpretability on the solutions. To this end, we developed a multitask multiple kernel learning (MTMKL) method with a co-clustering step based on a cutting-plane algorithm to identify the relationships between the input tasks and kernels. We tested our algorithm on 15 cancer cohorts and observed that, in most cases, MTMKL outperforms other algorithms (including random forests, support vector machine and single-task multiple kernel learning) in terms of predictive power. Using the aggregate results from multiple replications, we also derived similarity matrices between cancer cohorts, which are, in many cases, in agreement with available relationships reported in the relevant literature. Our implementations of support vector machine and multiple kernel learning algorithms in R are available at https://github.com/arezourahimi/mtgsbc together with the scripts that replicate the reported experiments. Supplementary data are available at Bioinformatics online.

Robust portfolio optimization with second order stochastic dominance constraints

A portfolio optimization problem equipped with stochastic dominance constraints creates optimal portfolio ideal for rational and risk-averse investors. This paper proposes a robust portfolio optimization model involving second-order stochastic dominance in constraints. The input returns of the assets are considered as uncertain parameters and are varied in symmetric and bounded intervals to construct an optimal robust portfolio. Although the resulting optimization model is a linear program, it involves a large number of constraints, thereby urging us to apply the cutting plane algorithm. We experimentally examine the performance of our model on datasets drawn from S&P 500, S&P BSE 500, Nikkei 225, S&P Global 100, FTSE 100, and BOVESPA index, and compare it with the corresponding non-robust counterpart model. The optimal portfolios from the proposed model are shown to yield better performance on several performance measures, including Sharpe ratio, STARR ratio, standard deviation, worst return, violation area in SSD, value at risk, and conditional value at risk. The results demonstrate the effectiveness and efficiency of the presented robust model.

Outer-product-free sets for polynomial optimization and oracle-based cuts

This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $$S\cap P$$ , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form $$xx^T$$ . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach.

A Multi-Stage Stochastic Programming Approach to the Optimal Surveillance and Control of Emerald Ash Borer in Cities

Emerald ash borer (EAB), a wood-boring insect native to Asia and invading North America, has killed untold millions of high-value ash trees that shade streets, homes, and parks, and caused significant economic damage in cities of the United States. Local actions to reduce damage include surveillance to find EAB and control to slow its spread. We present a multi-stage stochastic mixed-integer programming (M-SMIP) model for the optimization of surveillance, treatment, and removal of ash trees in cities. Decision-dependent uncertainty is modeled by representing surveillance decisions and the realizations of the uncertain infestation parameter contingent upon surveillance as branches in the M-SMIP scenario tree. The objective is to allocate resources to surveillance and control over space and time to maximize public benefits. We develop a new cutting plane algorithm to strengthen the M-SMIP formulation and facilitate optimal solution. We calibrate and validate our model of ash dynamics using seven-years of observational data and apply the optimization model to a possible infestation in Burnsville, Minnesota. Proposed cutting planes improve the solution time by an average of seven times over solving the original M-SMIP model without cutting planes. Our comparative analysis shows that the M-SMIP model outperforms six different heuristic approaches proposed for the management of EAB. Results from optimally solving our M-SMIP model imply that, under a belief of infestation, it is critical to apply surveillance immediately to locate EAB and then prioritize treatment of minimally infested trees followed by removal of highly infested trees.

Potential theory and quadratic programming

We extend the notion of some energy-type expressions based on two sets, developed in the abstract potential theory. We also give the discretized version of the quantities defined, similar to Chebyshev constant. This extension allows to apply the potential-theoretic results to infinite quadratic programming problems. Together with a cutting plane algorithm, the Chebyshev-constant method ensures that under certain conditions, the infinite problem can be reduced to semi-infinite or to finite problems.

Exact robust solutions for the combined facility location and network design problem in hazardous materials transportation

We consider a leader-follower game in the form of a bi-level optimization problem that simultaneously optimizes facility locations and network design in hazardous materials transportation. In the upper level, the leader intends to reduce the facility setup cost and the hazmat exposure risk, by choosing facility locations and road segments to close for hazmat transportation. When making such decisions, the leader anticipates the response of the followers who want to minimize the transportation costs. Considering uncertainty in the hazmat exposure and the hazmat transport demand, we consider a robust optimization approach with multiplicative uncertain parameters and polyhedral uncertainty sets. The resulting problem has a min-max problem in the upper level and a shortest-path problem in the lower level. We devise an exact algorithm that combines a cutting plane algorithm with Benders decomposition

A two-stage robust optimisation for terminal traffic flow problem

Airport congestion witnesses potential conflicts: insufficient terminal airspace and delay propagation within scrambled the competition in the terminal manoeuvring area. Re-scheduling of flights is needed in numerous situations, heavy traffic in air segments, holding patterns, runway schedules and airport surface operations. Robust optimisation for terminal traffic flow problem, providing a practical point of view in hedging uncertainty, can leverage the adverse effect of uncertainty and schedule intervention. To avoid delay propagation throughout the air traffic flow network and reduce the vulnerability to disruption, this research adopts a two-stage robust optimisation approach in terminal traffic flow. It further enhances the quality of Pareto-optimality Benders-dual cutting plane based on core point approximation in the second stage recourse decision. The efficiency of the cutting plane algorithm is evaluated by a set of medium sized real-life scenarios. The numerical results show that the proposed scheme outperforms the well-known Pareto-optimal cuts in Benders-dual method from the literature.

An exact method for disassembly line balancing problem with limited distributional information

As an important part in product recycling, disassembly line balancing problem (DLBP) has attracted a large amount of attention. Stochastic DLBP is now a hot and challenging research topic, due to its wide applications. This work investigates a DLBP with uncertain task times, where the distributional information is limited, i.e. only the mean values and standard deviations are given, due to the lack of data. From a two-stage perspective, a disassembly process is determined and the disassembly tasks are assigned to workstations in the first stage, and the penalty cost (i.e. the recourse cost) for exceeding the cycle time is minimised in the second stage. The objective is to minimise the expected system cost. For the problem, a two-stage distributionally robust formulation is devised, to minimise the worst-case expected system cost out of all possible probability distributions. Different from literature that focuses on approximation methods to tackle the limited distributional information, an exact method, i.e. the cutting-plane algorithm, is developed. Numerical results show that compared with the state-of-the-art solution methods, our developed cutting-plane algorithm can provide more reliable solutions in term of the robustness, especially in some extreme cases.

From Classification to Optimization: A Scenario-based Robust Optimization Approach

This paper addresses data-driven decision-making problems under categorical uncertainty. Consider a two-stage optimization problem with first-stage planning and second-stage routing decisions, under uncertainty regarding which customers to visit. Classification models can estimate the probability of each customer visit from covariates but how to embed these predictions into the optimization? We propose a scenario-based robust optimization approach that combines stochastic programming (by constructing probabilistic scenarios), robust optimization (by protecting against adversarial perturbations within discrete uncertainty sets), and data-driven optimization (by defining scenarios and uncertainty sets from classification outputs). We develop a cutting-plane algorithm that converges to the optimum in a finite number of iterations for a general class of problems. Using public datasets, results suggest that (i) our scenario-based robust optimization approach outperforms benchmarks based on deterministic, stochastic and robust optimization; (ii) data-driven opti- mization outperforms benchmarks that ignore covariate information; and (iii) our cutting-plane algorithm outperforms direct implementation by returning better solutions in shorter computational times.

Volumetric Barrier Cutting Plane Algorithms for Stochastic Linear Semi-Infinite Optimization

In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier cutting plane interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by showing that the volumetric barrier associated with the recourse function of stochastic linear semi-infinite programs is a strongly self-concordant barrier and forms a self-concordant family on the first-stage solutions. The dominant terms in the complexity expressions obtained in this paper are given in terms of the problem dimension and the number of realizations. The novelty of our algorithms lies in their ability to kill the effect of the radii of the largest Euclidean balls contained in the feasibility sets on the dominant complexity terms.

A mixed-integer SDP solution approach to distributionally robust unit commitment with second order moment constraints

A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of stochastic parameters can be inferred from historical data, and then employed to model the set of probability distributions. The resulting problem is a two-stage distributionally robust unit commitment with second order moment constraints, and we show that it can be recast as a mixed-integer semidefinite programming (MI-SDP) with finite constraints. The solution algorithm of the problem comprises solving a series of relaxed MI-SDPs and a subroutine of feasibility checking and vertex generation. Based on the verification of strong duality of the semidefinite programming (SDP) problems, we propose a cutting plane algorithm for solving the MI-SDPs; we also introduce a SDP relaxation for the feasibility checking problem, which is an intractable biconvex optimization. Experimental results on a IEEE 6-bus system are presented, showing that without any tunings of parameters, the real-time operation cost of distributionally robust UC method outperforms those of deterministic UC and two-stage robust UC methods in general, and our method also enjoys higher reliability of dispatch operation.

Stochastic Localization Methods for Discrete Convex Simulation Optimization

We propose a set of new algorithms based on stochastic localization methods for large-scale discrete simulation optimization problems with convexity structure. All proposed algorithms, with the general idea of localizing potential good solutions to an adaptively shrinking subset, are guaranteed with high probability to identify a solution that is close enough to the optimal given any precision level. Specifically, for one-dimensional large-scale problems, we propose an enhanced adaptive algorithm with an expected simulation cost asymptotically independent of the problem scale, which is proved to attain the best achievable performance. For multi-dimensional large-scale problems, we propose statistically guaranteed stochastic cutting-plane algorithms, the simulation costs of which have no dependence on model parameters such as the Lipschitz parameter, as well as low polynomial order of dependence on the problem scale and dimension. Numerical experiments are implemented to support our theoretical findings. The theory results, joint the numerical experiments, provide insights and recommendations on which algorithm to use in different real application settings.

A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems

We consider a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite (psd) constraint is relaxed, and the resultant mixed-integer linear optimization problem is solved repeatedly, imposing at each iteration a valid inequality for the psd constraint. We prove the convergence properties of the algorithm. Moreover, to speed up the computation, we devise a branch-and-cut algorithm, in which valid inequalities are dynamically added during a branch-and-bound procedure. We test the computational performance of our cutting-plane and branch-and-cut algorithms for three types of MISDO problem: random instances, computing restricted isometry constants, and robust truss topology design. Our experimental results demonstrate that, for many problem instances, our branch-and-cut algorithm delivered superior performance compared with general-purpose MISDO solvers in terms of computational efficiency and stability.

Bi-objective Branch-and-Cut Algorithms Based on LP Relaxation and Bound Sets

Most real-world optimization problems are multi-objective by nature, with conflicting and incomparable objectives. Solving a multi-objective optimization problem requires a method that can generate all rational compromises between the objectives. This paper proposes two distinct bound set-based branch-and-cut algorithms for general bi-objective combinatorial optimization problems based on implicit and explicit lower-bound sets. The algorithm based on explicit lower-bound sets computes, for each branching node, a lower-bound set and compares it with an upper-bound set. The other fathoms branching nodes by generating a single point on the lower-bound set for each local nadir point. We outline several approaches for fathoming branching nodes, and we propose an updating scheme for the lower-bound sets that prevents us from solving the bi-objective linear programming relaxation of each branching node. To strengthen the lower-bound sets, we propose a bi-objective cutting-plane algorithm that adjusts the weights of the objective functions such that different parts of the feasible set are strengthened by cutting planes. In addition, we suggest an extension of the branching strategy “Pareto branching.” We prove the effectiveness of the algorithms through extensive computational results.

Towards Resilient Operation of Multimicrogrids: An MISOCP-Based Frequency-Constrained Approach

High penetration of distributed energy resources (DERs) is transforming the paradigm in power system operation. The ability to provide electricity to customers while the main grid is disrupted has introduced the concept of microgrids with many challenges and opportunities. Emergency control of dangerous transients caused by the transition between the grid-connected and island modes in microgrids is one of the main challenges in this context. To address this challenge, this paper proposes a comprehensive optimization and real-time control framework for maintaining frequency stability of multi-microgrid networks under an islanding event and for achieving optimal load shedding and network topology control with AC power flow constraints. The paper also develops a strong mixed-integer second-order cone programming (MISOCP)-based reformulation and a cutting plane algorithm for scalable computation. We believe this is the first time in the literature that such a framework for multi-microgrid network control is proposed, and its effectiveness is demonstrated with extensive numerical experiments.

Strong bounds for resource constrained project scheduling: Preprocessing and cutting planes

Resource Constrained Project Scheduling Problems (RCPSPs) without preemption are well-known NP-hard combinatorial optimization problems. A feasible RCPSP solution consists of a time-ordered schedule of jobs with corresponding execution modes, respecting precedence and resources constraints. In this paper, we propose a cutting plane algorithm to separate five different cut families, as well as a new preprocessing routine to strengthen resource-related constraints. New lifted versions of the well-known precedence and cover inequalities are employed. At each iteration, a dense conflict graph is built considering feasibility and optimality conditions to separate cliques, odd-holes and strengthened Chvátal-Gomory cuts. The proposed strategies considerably improve the linear relaxation bounds, allowing a state-of-the-art mixed-integer linear programming solver to find provably optimal solutions for 754 previously open instances of different variants of the RCPSPs, which was not possible using the original linear programming formulations.

A biobjective chance constrained optimization model to evaluate the economic and environmental impacts of biopower supply chains

Generating electricity by co-combusting biomass and coal, known as biomass cofiring, is shown to be an economically attractive option for coal-fired power plants to comply with emission regulations. However, the total carbon footprint of the associated supply chain still needs to be carefully investigated. In this study we propose a stochastic biobjective optimization model to analyze the economic and environmental impacts of biopower supply chains. We use a life cycle assessment approach to derive the emission factors used in the environmental objective function. We use chance constraints to capture the uncertain nature of energy content of biomass feedstocks. We propose a cutting plane algorithm which uses the sample average approximation method to model the chance constraints and finds high confidence feasible solutions. In order to find Pareto optimal solutions we propose a heuristic approach which integrates the $$\epsilon $$ -constraint method with the cutting plane algorithm. We show that the developed approach provides a set of local Pareto optimal solutions with high confidence and reasonable computational time. We develop a case study using data about biomass and coal plants in North and South Carolina. The results indicate that, cofiring of biomass in these states can reduce emissions by up to 8%. Increasing the amount of biomass cofired will not result in lower emissions due to biomass delivery.

The Branch and Cut Method for the Clique Partitioning Problem

A numerical study is carried out of the branch and cut method adapted for solving the clique partitioning problem (CPP). The problem is to find a family of pairwise disjoint cliques with minimum total weight in a complete edge-weighted graph. The two particular cases of the CPP are considered: The first is known as the aggregating binary relations problem (ABRP), and the second is the graph approximation problem (GAP). For the previously known class of facet inequalities of the polytope of the problem, the cutting-plane algorithm is developed. This algorithm includes the two new basic elements: finding a solution with given guaranteed accuracy and a local search procedure to solve the problem of inequality identification. The proposed cutting-plane algorithm is used to construct lower bounds in the branch and cut method. Some special heuristics are used to search upper bounds for the exact solution. We perform a numerical experiment on randomly generated graphs. Our method makes it possible to find an optimal solution for the previously studied cases of the ABRP and for new problems of large dimension. The GAP turns out to be a more complicated case of the CPP in the computational aspect. Moreover, some simple and difficult classes of the GAPs are identified for our algorithm.