We develop an algorithm for the linear ordering problem, which has a large number of applications such as triangulation of input-output matrices, minimizing total weighted completion time in one-machine scheduling, and aggregation of individual preferences. The algorithm is based on the Lagrangian relaxation of a binary integer linear programming formulation of the problem. Since the number of the constraints is proportional to the third power of the number of items and grows rapidly, we propose a modified subgradient method that temporarily ignores a large part of the constraints and gradually adds constraints whose Lagrangian multipliers are likely to be positive at an optimal multiplier vector. We also propose an improvement on the ordinary pegging test by using the problem structure.

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We propose a method to reduce the sizes of SDP relaxation problems for a given polynomial optimization problem (POP). This method is an extension of the elimination method for a sparse SOS polynomial in [8] and exploits sparsity of polynomials involved in a given POP. In addition, we show that this method is a partial application of a facial reduction algorithm, which generates a smaller SDP problem with an interior feasible solution. In general, SDP relaxation problems for POPs often become highly degenerate because of a lack of interior feasible solutions. As a result, the resulting SDP relaxation problems obtained by this method may have an interior feasible solution, and one may be able to solve the SDP relaxation problems effectively. Numerical results in this paper show that the resulting SDP relaxation problems obtained by this method can be solved fast and accurately.

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Kitahara and Mizuno [3] get upper bounds for the maximum number of different basic feasible solutions generated by the simplex method with the most negative pivoting rule. In this paper, we obtain lower bounds of the maximum number to show how tight the upper bounds are. Part of the results in this paper are shown in Kitahara and Mizuno [4] as a quick report without proof. They present a simple variant of Klee-Minty's LP and get a lower bound. In this paper, we explain and prove the properties of the variant more precisely. We also show a new lower bound by using a simple example of LP.

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We build a computational framework for determining an optimal dynamic asset allocation over multiple periods. To do this, we use a nonlinear control policy, which is a function of past returns of investable assets. By employing a kernel method, the problem of selecting the best control policy from among nonlinear functions can be formulated as a convex quadratic optimization problem. Furthermore, we reduce the problem to a linear optimization problem by employing L1-norm regularization. A numerical experiment was conducted wherein scenarios of the rate of return of investable assets were generated by using a one-period autoregressive model, and the results showed that our investment strategy improves an investment performance more than other strategies from previous studies do.

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In this paper, we deal with the node capacitated in-tree packing problem. The input consists of a directed graph, a root node, a node capacity function and edge consumption functions for heads and tails. The problem is to find a subset of rooted spanning in-trees and their packing numbers, where the packing number of an in-tree is the number of times it is packed, so as to maximize the sum of packing numbers under the constraint that the total consumption of the packed in-trees at each node does not exceed the capacity of the node. This problem is known to be NP-hard. Previously, we proposed a two-phase heuristic algorithm for this problem. The algorithm generates promising candidate in-trees to be packed in the first phase and computes the packing number of each in-tree in the second phase. In this paper, we improve the first phase algorithm by using Lagrangian relaxation instead of LP (linear programming) relaxation. We conducted computational experiments on graphs used in related papers and on randomly generated instances. The results indicate that our new algorithm generates in-trees faster than our previous algorithm and obtains better solutions than existing algorithms without generating many in-trees.

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This paper proposes the Maximum Flow-Covering Location and service Start Time Problem (MFCLSTP). The problem seeks to determine locations and service start time of p facilities which provide a service for a given duration, so as to maximally cover flows. Each flow is specified by a pair of origin-destination (OD) nodes and the departure time of the origin node. We assume that a given commuter flow is covered when commuters can stop at a facility, fully consume that facility's service, and arrive at the destination node by a given time. Two models are considered for MFCLSTP: MFCLSTP1, in which the service start time of each facility can be independently determined; and MFCLSTP2, in which all facilities have the same service start time. We provide integer programming formulations and propose heuristic solution algorithms. The proposed models are applied to a case study of the Tokyo metropolitan railway network using census data for commuter traffic. The solutions obtained by the heuristic algorithms for both models are compared. Solutions of MFCLSTP2 show that selected locations are spatially dispersed to cover different types of flows, whereas solutions of MFCLSTP1 closely locates some facilities having different start times in the central area of Tokyo.

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The use of network analysis has increased in various fields. In particular, a lot of attention has been paid to centrality metrics using shortest paths, which require a comparatively smaller amount of computation, and the global characteristic features within the network. While theoretical and experimental progress has enabled greater control over networks, large amounts of computation required for dealing with large-scale networks is a major hurdle. This research is on high-performance network analysis considering the memory hierarchy in a computer; it targets extremely important kernel types called shortest paths and centrality. Our implementation, called NETAL (NETwork Analysis Library), can achieve high efficiency in parallel processing using many-core processors such as the AMD Opteron 6174, which has the NUMA architecture. We demonstrated through tests on real-world networks that NETAL is faster than previous implementations. In the all-pairs shortest paths for the weighted graph USA-road-d.NY.gr (n=264K, m=734K), our implementation solved the shortest path distance labels in 44.4 seconds and the shortest paths with multiple predecessors in 411.2 seconds. Compared with the 9th DIMACS benchmark solver, our implementation is, respectively, 302.7 times and 32.7 times faster. NETAL succeeded in solving the shortest path distance labels for the USA-road-d.USA.gr (n=24M, m=58M) without preprocessing in 7.75 days. Numerical results showed that our implementation performance was 432.4 times that of the Δ-stepping algorithm and 228.9 times that of the 9th DIMACS benchmark solver. Furthermore, while GraphCT took 18 hours to compute the betweenness of web-BerkStan, our implementation computed multiple centrality metrics (closeness, graph, stress, and betweenness) simultaneously within 1 hour. A performance increase of 2.4-3.7 times compared with R-MAT graph was confirmed for SSCA#2.

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