Cutting-plane Research Articles

Shape Analysis of the Patellar Bone Surface and Cutting Plane for Knee Replacement Surgery.

Geometry of the patella (kneecap) remains poorly understood yet is highly relevant to performing the correct patellar cut to reduce pain and to improve function and satisfaction after knee replacement surgery. Although studies routinely refer to “parallel to the anterior surface” and “the patellar horizon,” a quantitative definition of these is lacking and significant variability exists between observers for this irregularly-shaped bone. A 2D-3D shape analysis technique was developed to determine the optimal device configuration for contacting the patellar surface. Axial and sagittal pseudo-X-rays were created from 18 computed tomography (CT) scans of cadaveric knees. Four expert surgeons reviewed three repetitions of the X-rays in randomized order, marking their desired cut plane and their estimate of the anterior surface. These 2D results were related back to the 3D model to create the desired plane. There was considerable variability in perceptions, with intra- and intersurgeon repeatability (standard deviations) ranging from 1.3° to 2.4°. The best configuration of contact points to achieve the desired cutting plane was three pegs centred on the patellar surface, two superior and one inferior, forming a 16 mm equilateral triangle. This configuration achieved predicted cut planes within 1° of the surgeon ranges on all 18 patellae. Implementing this, as was done in a subsequent prototype surgical device, should help improve the success and satisfaction of knee replacement surgery.

MPTV: Matching Pursuit-Based Total Variation Minimization for Image Deconvolution.

Total variation (TV) regularization has proven effective for a range of computer vision tasks through its preferential weighting of sharp image edges. Existing TV-based methods, however, often suffer from the over-smoothing issue and solution bias caused by the homogeneous penalization. In this paper, we consider addressing these issues by applying inhomogeneous regularization on different image components. We formulate the inhomogeneous TV minimization problem as a convex quadratic constrained linear programming problem. Relying on this new model, we propose a matching pursuit-based total variation minimization method (MPTV), specifically for image deconvolution. The proposed MPTV method is essentially a cutting-plane method that iteratively activates a subset of nonzero image gradients and then solves a subproblem focusing on those activated gradients only. Compared with existing methods, the MPTV is less sensitive to the choice of the trade-off parameter between data fitting and regularization. Moreover, the inhomogeneity of MPTV alleviates the over-smoothing and ringing artifacts and improves the robustness to errors in blur kernel. Extensive experiments on different tasks demonstrate the superiority of the proposed method over the current state of the art.

Robust Optimization for Electricity Generation

We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current (AC) optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxations, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear, etc.) in electric power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem, and we establish convergence and other desirable properties. Experimental results indicate that our robust convex relaxation of the ACOPF problem can provide a tight lower bound.

The p-arborescence star problem: Formulations and exact solution approaches

We consider a problem arising in the design of ad-hoc wireless sensor networks. Given a connected bidirectional digraph with n sensors and a fixed sink, the p-arborescence star problem is the problem of clustering the sensors into p disjoint subsets and defining a directed backbone that spans the sink and all p cluster-heads, one for each cluster. The overall objective of the problem is to minimize the sum of intra and inter-cluster costs. We introduce and compare formulations based on directed cutset constraints and multicommodity flow variables. Valid inequalities to strengthen the models are also given. To solve these models, we develop efficient cutting-plane methods. We carry out extensive computational experiments and comparisons to illustrate the performance of our methods.

A cutting plane method for risk-constrained traveling salesman problem with random arc costs

In this manuscript, we consider a stochastic traveling salesman problem with random arc costs and assume that the travel cost of each arc follows a normal distribution. All the other parameters in the problem are considered deterministic. In the presence of uncertainty, the optimal route achieved from solving the deterministic model might be exposed to a high risk that the actual cost exceeds the available resource. In this respect, we present the stochastic model incorporating risk management, and the Value at Risk and Conditional Value at Risk techniques are applied as the risk measures to assess and control the risk associated with the uncertainty. A novel cutting plane algorithm is developed to deal with the difficulty of solving such model, and exhibits superior computational performance in our numerical experiments over other solution approaches.

Programmatic review of safety and effectiveness of isoniazid preventive therapy in HIV-infected pregnant women on ART in routine care

The aim of Network Design Problem with Vulnerability Constraints (NDPVC), is to design survivable telecommunications networks that impose length bounds on the communication paths of each commodity pair, before and after the failure of any k links. This problem was proposed as an alternative to the Hop-Constrained Survivable Network Design Problem (kHSNDP), which addresses similar issues, but imposes very conservative constraints, possibly leading to unnecessarily expensive solution or even rendering instances infeasible. In fact, it is known that the cost of the optimal solutions of the NDPVC never exceeds that of the related kHSNDP. However, previous results using the standard methods of a general-purpose integer linear (ILP) solver, combined with several ILP formulations, show that such methods fail to solve most instances in the benchmarking test set, within a time limit of two hours.In this paper, we propose three branch-and-cut algorithms, which are significantly more efficient in solving the NDPVC. The first algorithm is a cutting-plane method devised in the context of a new layered graph ILP formulation, whereas the other two are based on Benders decomposition methods of previously known formulations. With the proposed new methods, we are able to solve substantially more instances of the NDPVC and therefore able to provide a more complete comparison of its solutions to those of the kHSNDP.

PENGGUNAAN METODE CUTTING PLANE UNTUK MENYELESAIKAN MINIMUM SPANNING TREE DENGAN KENDALA BOBOT PADA GRAF K_n

This study aims to determine the minimum spanning tree of a complete graph K_n with weight constraints and completion using the cutting plane method. The cutting plane method is one of the algorithms included in the exact method. This algorithm works by reducing the solution area so that it becomes narrower. As a result, the feasible solutions that will be investigated become less and less. This is because the cutting plane method works based on the optimal linear programming solution of relaxation solved by the simplex method. In this paper we give illustration of the algorithm applied for two cases, one for K_4 and one for K_5.

“Facet” separation with one linear program

Given polyhedron P and and a point $$x^*$$ , the separation problem for polyhedra asks to certify that $$x^*\in P$$ and if not, to determine an inequality that is satisfied by P and violated by $$x^*$$ . This problem is repeatedly solved in cutting plane methods for Integer Programming and the quality of the violated inequality is an essential feature in the performance of such methods. In this paper we address the problem of finding efficiently an inequality that is violated by $$x^*$$ and either defines an improper face or a facet of P. We show that, by solving a single linear program, one almost surely obtains such an improper face or facet.

A numerical study of the return mapping application for the subloading surface model

PurposeMany practical problems in engineering require fast, accurate numerical results. In particular, in cyclic plasticity or fatigue simulations, the high number of loading cycles increases the computation effort and time. The purpose of this study is to show that the return mapping technique in the framework of unconventional plasticity theories is a good compromise between efficiency and accuracy in finite element analyses.Design/methodology/approachThe accuracy of the closest point projection method and the cutting plane method implementations for the subloading surface model are discussed under different loading conditions by analyzing the error as a function of the input step size and the efficiency of the algorithms.FindingsMonotonic tests show that the two different implicit integration schemes have the same accuracy and are in good agreement with the solution obtained using an explicit forward Euler scheme, even for large input steps. However, the closest point projection method seems to describe better the evolution of the similarity centre in the cyclic loading analyses.Practical implicationsThe purpose of this work is to show two alternative implicit integration schemes of the extended subloading surface method for metallic materials. The backward Euler integrations can guarantee a good description of the material behaviour and, at the same time, reduce the computational cost. This aspect is particularly important in the field of low or high cycle fatigue, because of the large number of cycles involved.Originality/valueA detailed description of both the cutting plane and closest point projection methods is offered in this work. In particular, the two integrations schemes are compared in terms of accuracy and computation time for monotonic and cyclic loading tests.

Оценка параметров червячных передач на основе методов 3D компьютерной графики

The current state of the machine-building production market is such that various modifications of its products are produced by single specimens. This requires that production should, on the one hand, reduce the terms for preparation and production of new products along with increasing the same products’ nomenclature, and on the other hand, it is economically necessary to reduce these products’ factory labor hours and cost along with high quality of manufacturing thereof. Such production’s profitability depends largely on effectiveness of the use of computer equipment and software that will allow virtualization of design and technological processes, which in the final development will allow transfer all stages related to preparing for manufacturing of a new product to mathematical and geometric computer models of manufactured products. This permits to leave only one stage of preparing for manufacturing of a new product — geometric synthesis, taking into account the workspace configuration. Due to the complexity of processes for tooth-profiling and engagement, their exact calculation can be only performed by a computer. A mathematical model of engagement, for example, a worm pair, which takes into account not only geometric parameters, but also the enterprise’s machine capabilities, makes it possible, without manufacturing the worm pair in a metal, to obtain reliable information about all features of the worm pair operation. The modern computers’ processing speed allows in a short time to choose the option that gives the best approximation to consumer’s desired requirements. This paper shows an example for application of the method of geometric computer solid-state modeling for the technological process of worm gear shaping as an envelope of the tool model’s producing surface. The algorithm work consists in the mutual motion and rotation of a part blank and the tool. As a result of mutual intersection of the part blank and the tool surfaces, final shapes of tooth surfaces are formed as precise computer solid-state models. In this paper has been shown the final result for computer shaping of the worm’s working surface. One of the stages for determining the gaps between the working surfaces of the worm and the worm wheel has been described. One of variants for interactive analysis of the worm pair’s computer model is application of the cutting-plane method to obtain a set of sections. The mutual intersection of the worm and worm wheel’s planes and working surfaces makes it possible to determine points’ coordinates and thereon to define the gaps between the working surfaces of the product.

Analytic center cutting plane methods for variational inequalities over convex bodies

An analytic center cutting plane method is an iterative algorithm based on the computation of analytic centers. In this paper, we propose some analytic center cutting plane methods for solving quasimonotone or pseudomonotone variational inequalities whose domains are bounded or unbounded convex bodies.

Reformulations for utilizing separability when solving convex MINLP problems

Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.

Service system design with immobile servers, stochastic demand and concave-cost capacity selection

The service system design problem is a location-allocation problem with service quality considerations that is often modeled as a network of M/M/1 queues to minimize facility setup, customer access, and waiting costs. Traditionally, capacity decisions are either ignored or modeled as a selection among discrete capacity levels. In this work, we study the general continuous capacity case and account for economies-of-scale in its cost through an increasing concave function. We focus on the special square-root case that has been shown to model capacity in terms of the number of servers needed under Poisson arrivals and exponential service times. The problem is formulated as a mixed-integer nonlinear program with concave and convex terms in the objective function. Two novel resolution approaches are proposed: In the first, the problem is reformulated as a mixed-integer quadratic program with fourth-degree polynomial equality constraints. These constraints and the quadratic objective function are approximated using piecewise-linear segments. In the second, we use Lagrangian relaxation to decompose the problem and reformulate the subproblems as second-order cone programs that are solved at multiple utilization levels. The Lagrangian multipliers are updated using a cutting-plane method and a feasible solution is obtained by solving the corresponding set-covering formulation. The solution approaches are tested and compared. The linearization approach provides high quality solutions within short computational times for small instances and lower accuracy; whereas the Lagrangian approach scales well as size increases.

Benders decomposition for the Hazmat Transport Network Design Problem

We propose a new method for solving the Hazmat Transport Network Design Problem. In this problem, the government wants to reduce the risk of hazardous accidents for the population by restricting the shipment of hazardous goods on roads. When taking that decision, the government has to anticipate the reaction of the carriers who want to minimize the transportation costs by solving a shortest path problem. We use a bilevel formulation that guarantees stable solutions and transform this model into a mixed-integer linear program by applying the Karush–Kuhn–Tucker conditions. This model is solved to optimality with a multi-cut Benders decomposition. Valid inequalities and an acceleration approach for the master problem further reduce both the iteration number and the run time. Moreover, a partial decomposition technique for bilevel problems is introduced. The numerical study shows the computational benefits of the method and run time savings of more than 50% for large instances compared to a cutting plane method from the literature. The method is especially efficient for large number of shipments. Finally, we show that the bilevel model reduces the risk by 35% on average compared to a two-step decision process that does not anticipate the carriers’ reaction.

Optimizing Cutting Planes for Advanced Joining and Additive Manufacturing

While additive manufacturing allows more complex shapes than conventional manufacturing processes, there is a clear benefit in leveraging both new and old processes in the definition of metal parts. For example, one could create complex part shapes where the main “body” is defined by extrusion and machining, while small protruding features are defined by additive manufacturing. This paper looks at how optimization and geometric reasoning can be combined to identify cutting planes within complex three-dimensional (3D) shapes. These cutting planes are used to divide realistic mechanical parts into subparts that can be joined together through additive manufacturing or linear friction welding (LFW). The optimization method presents possible manufacturing alternatives to an engineering designer where optimality is defined as a minimization of cost. The paper presents and compares several cutting planes identification methods and describes how the optimization finds the optimal results for several example parts.

MTZ-primal-dual model, cutting-plane, and combinatorial branch-and-bound for shortest paths avoiding negative cycles

Let $$D=(V,A)$$ be a digraph with a set of vertices V, and a set of arcs A, with $$c_{ij} \in {\mathbb {R}}$$ representing the cost of each arc $$(i,j) \in A$$. The problem of finding the shortest-path avoiding negative cycles (SPNC) is NP-hard and consists in determining, if it exists, a path of minimum cost between two distinguished vertices $$s \in V$$, and $$t \in V$$. We propose three exact solution approaches for SPNC, including a compact primal-dual model, a combinatorial branch-and-bound algorithm, and a cutting-plane method. Extensive computational experiments performed on both benchmark and randomly generated instances indicate that our approaches either outperform or are competitive with existing mixed-integer programming models for the SPNC while providing optimal solutions for challenging instances in small execution times.

Algorithmic innovations and software for the dual decomposition method applied to stochastic mixed-integer programs

We present algorithmic innovations for the dual decomposition method to address two-stage stochastic programs with mixed-integer recourse and provide an open-source software implementation that we call DSP. Our innovations include the incorporation of Benders-like cuts in a dual decomposition framework to tighten Lagrangian subproblems and aid the exclusion of infeasible first-stage solutions for problems without (relative) complete recourse. We also use an interior-point cutting-plane method with new termination criteria for solving the Lagrangian master problem. We prove that the algorithm converges to an optimal solution of the Lagrangian dual problem in a finite number of iterations, and we also prove that convergence can be achieved even if the master problem is solved suboptimally. DSP can solve instances specified in C code, SMPS files, and Julia script. DSP also implements a standard Benders decomposition method and a dual decomposition method based on subgradient dual updates that we use to perform benchmarks. We present extensive numerical results using SIPLIB instances and a large unit commitment problem to demonstrate that the proposed innovations provide significant improvements in the number of iterations and solution times. The software reviewed as part of this submission has been given the Digital Object Identifier (DOI) https://doi.org/10.5281/zenodo.998971 .

Scalable Planning for Energy Storage in Energy and Reserve Markets

Energy storage can facilitate the integration of renewable energy resources by providing arbitrage and ancillary services. Jointly optimizing energy and ancillary services in a centralized electricity market reduces the system's operating cost and enhances the profitability of energy storage systems. However, achieving these objectives requires that storage be located and sized properly. We use a bilevel formulation to optimize the location and size of energy storage systems, which perform energy arbitrage and provide regulation services. Our model also ensures the profitability of investments in energy storage by enforcing a rate of return constraint. Computational tractability is achieved through the implementation of a primal decomposition and a subgradient-based cutting-plane method. We test the proposed approach on a 240-bus model of the Western Electricity Coordinating Council system and analyze the effects of different storage technologies, rate of return requirements, and regulation market policies on energy storage participation on the optimal storage investment decisions. We also demonstrate that the proposed approach outperforms exact methods in terms of solution quality and computational performance.

Notions of Maximality for Integral Lattice-Free Polyhedra: The Case of Dimension Three

Lattice-free sets and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. The family of all integral lattice-free polyhedra that are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra ℤd-maximal. For fixed d, the family of ℤd-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. This is a challenging task already for small dimensions. In contrast, the subfamily of all integral lattice-free polyhedra that are not properly contained in any other lattice-free set, which we call ℝd-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of ℤd-maximality and ℝd-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, for d ≥ 4 there exist integral lattice-free polyhedra that are ℤd-maximal but not ℝd-maximal. We consider the remaining case d = 3 and prove that for integral lattice-free polyhedra the notions of ℝ3-maximality and ℤ3-maximality are equivalent. This allows to complete the classification of all ℤ3-maximal integral lattice-free polyhedra.

Branch-and-cut methods for the Network Design Problem with Vulnerability Constraints

The aim of Network Design Problem with Vulnerability Constraints (NDPVC), is to design survivable telecommunications networks that impose length bounds on the communication paths of each commodity pair, before and after the failure of any k links. This problem was proposed as an alternative to the Hop-Constrained Survivable Network Design Problem (kHSNDP), which addresses similar issues, but imposes very conservative constraints, possibly leading to unnecessarily expensive solution or even rendering instances infeasible. In fact, it is known that the cost of the optimal solutions of the NDPVC never exceeds that of the related kHSNDP. However, previous results using the standard methods of a general-purpose integer linear (ILP) solver, combined with several ILP formulations, show that such methods fail to solve most instances in the benchmarking test set, within a time limit of two hours.In this paper, we propose three branch-and-cut algorithms, which are significantly more efficient in solving the NDPVC. The first algorithm is a cutting-plane method devised in the context of a new layered graph ILP formulation, whereas the other two are based on Benders decomposition methods of previously known formulations. With the proposed new methods, we are able to solve substantially more instances of the NDPVC and therefore able to provide a more complete comparison of its solutions to those of the kHSNDP.