Shape Approximation and Size Difference of the Upper Part of the Talus: Implication for Implant Design of the Talar Component for Total Ankle Replacement

The implant design of the talar component for total ankle replacement (TAR) should match the surface morphology of the talus so that the replaced ankle can restore the natural motion of the tibiotalar joint and may reduce postoperative complications. The purpose of this study was to introduce a new 3D fitting method (the two-sphere fitting method of the talar trochlea with three fitting resection planes) to approximate the shape of the upper part of the talus for the Chinese population. 90 models of the tali from CT images of healthy volunteers were used in this study. Geometrical fitting and morphological measurements were performed for the surface morphology of the upper part of the talus. The accuracy of the two-sphere fitting method of the talar trochlea was assessed by a comparison of previously reported data. Parameters of the fitting geometries with different sizes were recorded and compared. Results showed that compared with previously reported one-sphere, cylinder, and bitruncated cone fitting methods, the two-sphere fitting method presented the smallest maximum distance difference, indicating that talar trochlea can be approximated well as two spheres. The radius of the medial fitting sphere R M was 20.69 ± 2.19 mm which was significantly smaller than the radius of the lateral fitting sphere R L of 21.32 ± 1.88 mm. After grouping all data by the average radius of fitting spheres, the result showed that different sizes of the upper part of the talus presented significantly different parameters except the orientation of the lateral cutting plane, indicating that the orientation of the lateral cutting plane may keep consistent for all upper part of the talus and have no relationship with the size. The linear regression analyses demonstrated a weak correlation ( R 2 < 0.5 ) between the majority of parameters and the average radius of the fitting spheres. Therefore, different sizes of the upper part of the talus presented unique morphological features, and the design of different sizes of talar components for TAR should consider the size-specific characteristics of the talus. The parameters measured in this study provided a further understanding of the talus and can guide the design of different sizes of the talar components of the TAR implant.

Efficient Solution Methods for a General r-Interdiction Median Problem with Fortification

This study generalizes the r-interdiction median (RIM) problem with fortification to simultaneously consider two types of risks: probabilistic exogenous disruptions and endogenous disruptions caused by intentional attacks. We develop a bilevel programming model that includes a lower-level interdiction problem and a higher-level fortification problem to hedge against such risks. We then prove that the interdiction problem is supermodular and subsequently adopt the cuts associated with supermodularity to develop an efficient cutting-plane algorithm to achieve exact solutions. For the fortification problem, we adopt the logic-based Benders decomposition (LBBD) framework to take advantage of the two-level structure and the property that a facility should not be fortified if it is not attacked at the lower level. Numerical experiments show that the cutting-plane algorithm is more efficient than benchmark methods in the literature, especially when the problem size grows. Specifically, with regard to the solution quality, LBBD outperforms the greedy algorithm in the literature with an up-to 13.2% improvement in the total cost, and it is as good as or better than the tree-search implicit enumeration method. Summary of Contribution: This paper studies an r-interdiction median problem with fortification (RIMF) in a supply chain network that simultaneously considers two types of disruption risks: random disruptions that occur probabilistically and disruptions caused by intentional attacks. The problem is to determine the allocation of limited facility fortification resources to an existing network. It is modeled as a bilevel programming model combining a defender’s problem and an attacker’s problem, which generalizes the r-interdiction median problem with probabilistic fortification. This paper is suitable for IJOC in mainly two aspects: (1) The lower-level attacker’s interdiction problem is a challenging high-degree nonlinear model. In the literature, only a total enumeration method has been applied to solve a special case of this problem. By exploring the special structural property of the problem, namely, the supermodularity of the transportation cost function, we developed an exact cutting-plane method to solve the problem to its optimality. Extensive numerical studies were conducted. Hence, this paper fits in the intersection of operations research and computing. (2) We developed an efficient logic-based Benders decomposition algorithm to solve the higher-level defender’s fortification problem. Overall, this study generalizes several important problems in the literature, such as RIM, RIMF, and RIMF with probabilistic fortification (RIMF-p).

An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

We propose an analytic center cutting plane method to determine whether a matrix is completely positive and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman et al. [Berman A, Dur M, Shaked-Monderer N (2015) Open problems in the theory of completely positive and copositive matrices. Electronic J. Linear Algebra 29(1):46–58]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia et al. [Xia W, Vera JC, Zuluaga LF (2020) Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1):40–56]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like [Formula: see text] for d × d matrices. The method is implemented in Julia and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl . Summary of Contribution: Completely positive matrices play an important role in operations research. They allow many NP-hard problems to be formulated as optimization problems over a proper cone, which enables them to benefit from the duality theory of convex programming. We propose an analytic center cutting plane method to determine whether a matrix is completely positive by solving an optimization problem over the copositive cone. In fact, we can use our method to solve any copositive optimization problem, provided we know the radius of a ball containing an optimal solution. We emphasize numerical performance and stability in developing this method. A software implementation in Julia is provided.

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

A New Method of Central Axis Extracting for Pore Network Modeling in Rock Engineering

Characterizing internal microscopic structures of porous media is of vital importance to simulate fluid and electric current flow. Compared to traditional rock mechanics and geophysical experiments, digital core and pore network modeling is attracting more interests as it can provide more details on rock microstructure with much less time needed. The axis extraction algorithm, which has been widely applied for pore network modeling, mainly consists of a reduction and burning algorithm. However, the commonly used methods in an axis extraction algorithm have the disadvantages of complex judgment conditions and relatively low operating efficiency, thus losing the practicality in application to large-scale pore structure simulation. In this paper, the updated algorithm proposed by Palágyi and Kuba was used to perform digital core and pore network modeling. Firstly, digital core was reconstructed by using the Markov Chain Monte Carlo (MCMC) method based on the binary images of a rock cutting plane taken from heavy oil reservoir sandstone. The digital core accuracy was verified by comparing porosity and autocorrelation function. Then, we extracted the central axis of the digital pore space and characterize structural parameters through geometric transformation technology and maximal sphere method. The obtained geometric parameters were further assigned to the corresponding nodes of pore and throat on the central axis of the constructed model. Moreover, the accuracy of the new developed pore network model was measured by comparing pore/throat parameters, curves of mercury injection, and oil-water relative permeability. The modeling results showed that the new developed method is generally effective for digital core and pore network simulation. Meanwhile, the more homogeneity of the rock, which means the stronger “representative” of binary map the rock cutting plane, the more accurate simulated results can be obtained.