Cutting Plane Generation Research Articles

Asynchronous distributed charging protocol for plug-in electric vehicles

The proliferation of plug-in electric vehicles (PEVs) advocates a distributed paradigm for coordinating PEV charging. Unlike existing primal-dual decomposition or consensus methods, this paper proposes a cutting-plane-based distributed algorithm considering a peer-to-peer (P2P) communication network, which enables asynchronous coordination among PEVs. First, an equivalent surrogate model is established by exploiting the duality of the original optimization problem, which masks the private information of individual users by equivalent transformation. Then, a cutting-plane based algorithm is derived to solve the surrogate problem in a distributed manner with intrinsic superiority to cope with various asynchrony. Critical implementation issues, including distributed initialization, distributed cutting-plane generation, and localized stopping criteria, are discussed in detail. Numerical tests on IEEE 37- and 123-node feeders with actual data illustrate that the proposed method is resilient to several asynchronies and admits the plug-and-play operation mode. The proposed methodology is expected to provide an alternative path toward the robust protocol for PEV charging.

Cutting Planes from the Branch-and-Bound Tree: Challenges and Opportunities

In this short paper, we argue that the standard approach adopted by modern mixed-integer linear programming solvers of using very little cutting plane generation in the branch-and-bound tree can be too conservative and lead to the loss of significant opportunities. Our observation is motivated by some relatively simple computational investigation on a couple of instances in the MIPlib 2010 collection for which the benefit of generating globally valid cuts in the tree is significant. History: This “Challenge” paper was invited by the Editor-in-Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida.

Cutting Plane Generation through Sparse Principal Component Analysis

Cutting Plane Generation through Sparse Principal Component Analysis

Efficient Topology Design Algorithms for Power Grid Stability

The dynamic response of power grids to small disturbances influences their overall stability. This letter examines the effect of network topology on the linearized time-invariant dynamics of electric power systems. The proposed framework utilizes <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm based stability metrics to study the optimal placement of lines on existing networks as well as the topology design of new networks. The design task is first posed as an NP-hard mixed-integer nonlinear program (MINLP) that is exactly reformulated as a mixed-integer linear program (MILP) using McCormick linearization. To improve computation time, graph-theoretic properties are exploited to derive valid inequalities (cuts) and tighten bounds on the continuous optimization variables. Moreover, a cutting plane generation procedure is put forth that is able to interject the MILP solver and augment additional constraints to the problem on-the-fly. The efficacy of our approach in designing optimal grid topologies is demonstrated through numerical tests on the IEEE 39-bus network.

Combining Polyhedral Approaches and Stochastic Dual Dynamic Integer Programming for Solving the Uncapacitated Lot-Sizing Problem Under Uncertainty

We study the uncapacitated lot-sizing problem with uncertain demand and costs. The problem is modeled as a multistage stochastic mixed-integer linear program in which the evolution of the uncertain parameters is represented by a scenario tree. To solve this problem, we propose a new extension of the stochastic dual dynamic integer programming algorithm (SDDiP). This extension aims at being more computationally efficient in the management of the expected cost-to-go functions involved in the model, in particular by reducing their number and by exploiting the current knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. The algorithm is based on a partial decomposition of the problem into a set of stochastic subproblems, each one involving a subset of nodes forming a subtree of the initial scenario tree. We then introduce a cutting plane–generation procedure that iteratively strengthens the linear relaxation of these subproblems and enables the generation of an additional strengthened Benders’ cut, which improves the convergence of the method. We carry out extensive computational experiments on randomly generated large-size instances. Our numerical results show that the proposed algorithm significantly outperforms the SDDiP algorithm at providing good-quality solutions within the computation time limit. Summary of Contribution: This paper investigates a combinatorial optimization problem called the uncapacitated lot-sizing problem. This problem has been widely studied in the operations research literature as it appears as a core subproblem in many industrial production planning problems. We consider a stochastic extension in which the input parameters are subject to uncertainty and model the resulting stochastic optimization problem as a multistage stochastic integer program. To solve this stochastic problem, we propose a novel extension of the recently published stochastic dual dynamic integer programming (SDDiP) algorithm. The proposed extension relies on two main ideas: the use of a partial decomposition of the scenario tree and the exploitation of existing knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. We provide the results of extensive computational experiments carried out on large-size randomly generated instances. These results show that the proposed extended algorithm significantly outperforms the SDDiP at providing good-quality solutions for the stochastic uncapacitated lot-sizing problem. Although the paper focuses on a basic lot-sizing problem, the proposed algorithmic framework may be useful to solve more complex practical production planning problems.

On the exact separation of cover inequalities of maximum-depth

We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly {mathcal {N}}{mathcal {P}}-hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

The minimum area spanning tree problem: Formulations, Benders decomposition and branch-and-cut algorithms

The Minimum Area Spanning Tree Problem (MASTP) is defined in terms of a complete undirected graph G, where every vertex represents a point in the two dimensional Euclidean plane. Associated to each edge, there is a disk placed right at its midpoint, with diameter matching the length of the edge. MASTP seeks a spanning tree of G that minimizes the area in the union of the disks associated to its edges. This paper presents Integer Programming (IP) approaches for MASTP, introduces pre-processing techniques to reduce the size of the formulations, and characterizes valid inequalities for reinforcing its Linear Programming Relaxation bounds. Several Branch-and-cut (BC) algorithms exploiting such ideas are introduced. Additionally, we also apply Benders Decomposition to one of these formulations. An accompanying BC method, that separates Benders optimality cuts, is also introduced. Aiming to save linear programming re-optimization times, that algorithm makes use of an early branching strategy. Given the set of valid inequalities used in the polyhedral representation of the problem and the best available upper bound for the optimal cost, it detects if the node cannot be pruned by bounds and then stops the cutting plane generation, in order to branch. Our algorithms manage to solve instances with up to only 15 vertices, suggesting thus that MASTP is hard to solve in practice, at least with the currently available methods. Thanks to the early branching strategy, the Benders based BC method obtained the best computational results, by far. It solved more instances to proven optimality than the other algorithms (31 out of 45 cases) and it is about three times faster than the second best performing algorithm.

A multi-stage stochastic integer programming approach for a multi-echelon lot-sizing problem with returns and lost sales

We consider an uncapacitated multi-item multi-echelon lot-sizing problem within a remanufacturing system involving three production echelons: disassembly, refurbishing and reassembly. We seek to plan the production activities on this system over a multi-period horizon. We consider a stochastic environment, in which the input data of the optimization problem are subject to uncertainty. We propose a multi-stage stochastic integer programming approach relying on scenario trees to represent the uncertain information structure and develop a branch-and-cut algorithm in order to solve the resulting mixed-integer linear program to optimality. This algorithm relies on a new set of tree inequalities obtained by combining valid inequalities previously known for each individual scenario of the scenario tree. These inequalities are used within a cutting-plane generation procedure based on a heuristic resolution of the corresponding separation problem. Computational experiments carried out on randomly generated instances show that the proposed branch-and-cut algorithm performs well as compared to the use of a stand-alone mathematical solver. Finally, rolling horizon simulations are carried out to assess the practical performance of the multi-stage stochastic planning model with respect to a deterministic model and a two-stage stochastic planning model.

On the impact of running intersection inequalities for globally solving polynomial optimization problems

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations of the form $$z_e = \prod _{v \in e} {z_v}$$, $$e \in E$$, where E denotes a set of subsets of cardinality at least two of a ground set. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.

Global optimization of nonconvex problems with convex-transformable intermediates

This paper addresses the global optimization of problems which contain convex-transformable functions. We present algorithms for identification of convex-transformable functions in general nonconvex problems, and introduce a new class of cutting planes based on recently developed relaxations for convex-transformable functions. These cutting planes correspond to the supporting hyperplanes of these convex relaxations. We integrate our recognition and cutting plane generation algorithms into the global solver BARON, and test our implementation by conducting numerical experiments on a large collection of nonconvex problems. Results demonstrate that the proposed implementation accelerates the convergence speed of the branch-and-bound algorithm, by significantly reducing computational time, number of nodes in the search tree, and required memory.

Multi-product valid inequalities for the discrete lot-sizing and scheduling problem

We consider a problem arising in the context of industrial production planning, namely the multi-product discrete lot-sizing and scheduling problem with sequence-dependent changeover costs. We aim at developing an exact solution approach based on a Cut & Branch procedure for this combinatorial optimization problem. To achieve this, we propose a new family of multi-product valid inequalities which corresponds to taking into account the conflicts between different products simultaneously requiring production on the resource. We then present both an exact and a heuristic separation algorithm which form the basis of a cutting-plane generation algorithm. We finally discuss computational results which confirm the practical usefulness of the proposed inequalities at strengthening the MILP formulation and at reducing the overall computation time.

Valid inequalities and lifting procedures for the shortest path problem in digraphs with negative cycles

In this paper, we discuss exact algorithm for the shortest elementary path problem in digraphs containing negative directed cycles. We investigate various classes of valid inequalities for the polytope of \(s-t\) elementary paths in digraphs. The problem of separation of these valid inequalities can be shown to be NP-hard, thus only solvable for small sized problems. To deal with larger problems, lifting techniques are proposed. We provide results of computational experiments to show the efficiency of lifted inequalities in reducing the integrality gap. Indeed, considering a series of difficult test examples, the integrality gap is shown to be \(100\,\%\) closed in about half of the cases within no more than ten rounds of cutting plane generation.

Global optimization of nonconvex problems with multilinear intermediates

We consider global optimization of nonconvex problems containing multilinear functions. It is well known that the convex hull of a multilinear function over a box is polyhedral, and the facets of this polyhedron can be obtained by solving a linear optimization problem (LP). When used as cutting planes, these facets can significantly enhance the quality of conventional relaxations in general-purpose global solvers. However, in general, the size of this LP grows exponentially with the number of variables in the multilinear function. To cope with this growth, we propose a graph decomposition scheme that exploits the structure of a multilinear function to decompose it to lower-dimensional components, for which the aforementioned LP can be solved very efficiently by employing a customized simplex algorithm. We embed this cutting plane generation strategy at every node of the branch-and-reduce global solver BARON, and carry out an extensive computational study on quadratically constrained quadratic problems, multilinear problems, and polynomial optimization problems. Results show that the proposed multilinear cuts enable BARON to solve many more problems to global optimality and lead to an average 60 % CPU time reduction.

Local cuts for mixed-integer programming

A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane, if one exists, and via substitution this gives a cutting plane in the original space. We extend this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes. Some of these possibilities are explored computationally, both in floating-point arithmetic and in rational arithmetic.

Coordinated cutting plane generation via multi-objective separation

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a diversity measure, we introduce a form of coordination between successive cuts. Our focus is on valid inequalities with 0–1 coefficients in the left-hand side and a constant right-hand side, which encompasses several families of valid inequalities. As cut diversity measure, we consider an aggregate of the 1-norm distances w.r.t. the normal vectors of the previous cuts. In this case, our lexicographic multi-objective separation problem reduces to the standard separation problem with different values for the objective function coefficients. The impact of our coordinated cutting plane generation scheme is assessed in a pure cutting plane setting when separating stable set and cut set inequalities for, respectively, the max clique and min Steiner tree problems. Compared to the standard separation of undominated maximally violated cuts, we close the same fraction of the duality gap in a considerably smaller number of rounds and cuts. The potential of our scheme is also indicated by the results obtained in a cut-and-branch setting for max clique, where cut coordination allows for a substantial reduction, on average, of the number of branch-and-bound nodes.

On coordinated cutting plane generation and mixed integer programs with nonconvex 2-norm constraints

On coordinated cutting plane generation and mixed integer programs with nonconvex 2-norm constraints

A revised model for compressor design and scheduling in gas-lifted oil fields

The design and real-time scheduling of lift-gas compressors in oil fields entails solving a mixed-integer non-linear problem that generalizes the facility location problem. This article presents a revised formulation that represents the constraints on compressor discharge pressure as a family of linear inequalities, which is shown to be tighter than a previous formulation. Mixed-integer linear approximations for these formulations are obtained by piecewise linearizing the non-linear functions, using binary variables and specially ordered set variables. Valid inequalities for compressor capacity along with separation and lifting procedures are proposed. The article also presents computational experiments comparing the formulations and evaluating the impact of cutting plane generation on solution speed.

Mixed-integer linear optimization for optimal lift-gas allocation with well-separator routing

The lift-gas allocation problem with well-separator routing constraints is a mixed-integer nonlinear program of considerable complexity. To this end, a mixed-integer linear formulation (compact) is obtained by piecewise-linearizing the nonlinear curves, using binary variables to express the linearization and routing decisions. A new formulation (integrated) combining the decisions on linearization and routing is developed by using a single binary variable. The structures of both formulations are explored to generate lifted cover cuts. Numerical tests show that the solution of the integrated formulation using cutting-plane generation is faster in spite of having more variables than the compact formulation.

Inequalities for the lattice width of lattice-free convex sets in the plane

A closed, convex set K in \({\mathbb{R}^2}\) with non-empty interior is called lattice-free if the interior of K is disjoint with \({\mathbb{Z}^2}\). In this paper we study the relation between the area and the lattice width of a planar lattice-free convex set in the general and centrally symmetric case. A correspondence between lattice width on the one hand and covering minima on the other, allows us to reformulate our results in terms of covering minima introduced by Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988). We obtain a sharp upper bound for the area for any given value of the lattice width. The lattice-free convex sets satisfying the upper bound are characterized. Lower bounds are studied as well. Parts of our results are applied in Averkov et al. (Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three, http://arxiv.org/abs/1010.1077, 2010) for cutting plane generation in mixed integer linear optimization, which was the original inducement for this paper. We further rectify a result of Kannan and Lovász (Ann Math (2) 128(3):577–602, 1988) with a new proof.

A Combined Parallel Lagrangian Decomposition and Cutting-Plane Generation for Maximum Stable Set Problems

We propose an integer programming formulation for the problem of finding the maximum k -partite induced sub-graph of a graph G based on representatives of stable sets. We investigate upper bounds provided by the solution, via a parallel sub-gradient algorithm, of a Lagrangian decomposition that breaks up this formulation into maximum weighted stable set problems for sub-graphs of G . Some computational experiments were carried out with an effective multi-threaded parallel implementation in a multi-core system, and their results are presented.