• The Transactions of the Korean Institute of Electrical Engineers A
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• v.48 no.11
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• pp.1424-1428
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• 1999

This paper presents a mathematical decomposition coordination method to implementing the distributed optimal power flow (OPF), wherein a regional decomposition technique is adopted to parallelize the OPT. The proposed approach is based on the Alternating Direction Method (ADM), a variant of the conventional Augmented Lagrangian approach, and makes it possible the independent regional AC-OPF for each control area while the global optimum for the entire system is assured. This paper is an extension of our previous work based on the auxiliary problem principle (APP). The proposed approach in this paper is a completely new one, however, in that ADM is based on the Proximal Point Algorithm which has long been recognized as one of the attractive methods for convex programming and min-max-convex-concave programming. The proposed method was demonstrated with IEEE 50-Bus system.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1221-1230
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• 2018

We derive a sharp decomposition formula for the state polytope of the Hilbert point and the Hilbert-Mumford index of reducible varieties by using the decomposition of characters and basic convex geometry. This proof captures the essence of the decomposition of the state polytopes in general, and considerably simplifies an earlier proof by the authors which uses a careful analysis of initial ideals of reducible varieties.

• Proceedings of the Korean Operations and Management Science Society Conference
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• 1999.04a
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• pp.316-317
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• 1999

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.265-276
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• 2019

For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn-Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn-Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.823-833
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• 2009

It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.669-677
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• 1996

Let $M_n$ be the $C^*$-algebra of all $n \times n$ matrices over the complex field, and $P[M_m, M_n]$ the convex cone of all positive linear maps from $M_m$ into $M_n$ that is, the maps which send the set of positive semidefinite matrices in $M_m$ into the set of positive semi-definite matrices in $M_n$. The convex structures of $P[M_m, M_n]$ are highly complicated even in low dimensions, and several authors [CL, KK, LW, O, R, S, W]have considered the possibility of decomposition of $P[M_m, M_n] into subcones.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.759-772
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• 2012

In this paper, semi-infinite constrained programming, a class of constrained nonsmooth optimization problems, are transformed into unconstrained nonsmooth convex programs under the help of exact penalty function. The unconstrained objective function which owns the primal-dual gradient structure has connection with $\mathcal{VU}$-space decomposition. Then a $\mathcal{VU}$-space decomposition method can be applied for solving this unconstrained programs. Finally, the superlinear convergence algorithm is proved under certain assumption.

• Journal of the Korean Operations Research and Management Science Society
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• v.21 no.1
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• pp.1-17
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• 1996

There have been considerable researches for solving large-scale separable convex optimization ptoblems. In this paper we present a method for large-scale nonseparable smooth convex optimization problems with block-angular linear constraints. One of them is occurred in reconfiguration of the virtual path network which finds the routing path and assigns the bandwidth of the path for each traffic class in ATM (Asynchronous Transfer Mode) network [1]. The solution is approximated by solving a sequence of the block-angular structured separable quadratic programming problems. Bundle-based decomposition method [10, 11, 12]is applied to each large-scale separable quadratic programming problem. We implement the method and present some computational experiences.

• Korean Journal of Computational Design and Engineering
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• v.8 no.4
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• pp.270-277
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• 2003

In two dimensional mechanical design analysis, quadrilateral element mesh is preferred because it provides more accurate result than triangular element mesh. However, automation of quadrilateral element mesh generation is much more complex because of its geometrical complexities. In this study, an automatic quadrilateral element mesh generation algorithm based on the boundary normal offsetting method and the boundary decomposition method is developed. In so doing, nodes are automatically placed using the boundary normal offsetting method and the decomposition method is applied to decompose the designed domain into a set of convex subdomains. The generated elements are improved by relocation of the existing nodes based on the four criteria - uniformity, aspect ratio, skewness and taper degree. The developed algorithm requires minimal user inputs such as boundary data and the distance between nodes.

• The Journal of Korea Robotics Society
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• v.18 no.1
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• pp.1-9
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• 2023

This paper presents an optimal path planning strategy for aerial searching and surveying of a user-designated area using multiple Unmanned Aerial Vehicles (UAVs). The method is designed to deal with a single unseparated polygonal area, regardless of polygonal convexity. By defining the search area into a set of grids, the algorithm enables UAVs to completely search without leaving unsearched space. The presented strategy consists of two main algorithmic steps: cellular decomposition and path planning stages. The cellular decomposition method divides the area to designate a conflict-free subsearch-space to an individual UAV, while accounting the assigned flight velocity, take-off and landing positions. Then, the path planning strategy forms paths based on every point located in end of each grid row. The first waypoint is chosen as the closest point from the vehicle-starting position, and it recursively updates the nearest endpoint set to generate the shortest path. The path planning policy produces four path candidates by alternating the starting point (left or right edge), and the travel direction (vertical or horizontal). The optimal-selection policy is enforced to maximize the search efficiency, which is time dependent; the policy imposes the total path-length and turning number criteria per candidate. The results demonstrate that the proposed cellular decomposition method improves the search-time efficiency. In addition, the candidate selection enhances the algorithmic efficacy toward further mission time-duration reduction. The method shows robustness against both convex and non-convex shaped search area.