• East Asian mathematical journal
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• 제37권5호
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• pp.543-551
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• 2021

Recently, semidefinite optimization problems have been intensively studied since many optimization problem can be changed into the problems and the the problems are very computationable. In this paper, we consider a semidefinite linear vector optimization problem (VP) and we establish the optimality theorems for (VP), which holds without any constraint qualification.

• 한국정보과학회:학술대회논문집
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• 한국정보과학회 2005년도 한국컴퓨터종합학술대회 논문집 Vol.32 No.1 (A)
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• pp.892-894
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• 2005

Graph partitioning provides an important tool for data clustering, but is an NP-hard combinatorial optimization problem. Spectral clustering where the clustering is performed by the eigen-decomposition of an affinity matrix [1,2]. This is a popular way of solving the graph partitioning problem. On the other hand, semidefinite relaxation, is an alternative way of relaxing combinatorial optimization. issuing to a convex optimization[4]. In this paper we present a semidefinite programming (SDP) approach to graph equi-partitioning for clustering and then we use eigen-decomposition to obtain an optimal partition set. Therefore, the method is referred to as semidefinite spectral clustering (SSC). Numerical experiments with several artificial and real data sets, demonstrate the useful behavior of our SSC. compared to existing spectral clustering methods.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1245-1256
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• 2011

In this paper, we proposed a regularization interior point method for semidefinite programming with free variables which can be taken as an extension of the algorithm for standard semidefinite programming. Since an inexact search direction at each iteration is used, the computation of the designed algorithm is much less compared with the existing solution methods. The convergence analysis of the method is established under weak conditions.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.375-381
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• 2008

Boundedness for the set of all the $\epsilon$-approximate solutions for convex optimization problems are considered. We give necessary and sufficient conditions for the sets of all the $\epsilon$-approximate solutions of a convex optimization problem involving finitely many convex functions and a convex semidefinite problem involving a linear matrix inequality to be bounded. Furthermore, we give examples illustrating our results for the boundedness.

• 한국경영과학회지
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• 제26권2호
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• pp.13-46
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• 2001

SDP (Semidefinite Programming), as a sort of “cone-LP”, optimizes a linear function over the intersection of an affine space and a cone that has the origin as its apex. SDP, however, has been developed in the process of searching for better solution methods for NP-hard combinatorial optimization problems. We surveyed the basic theories necessary to understand SDP researches. First, We examined SDP duality, comparing it to LP duality, which is essential for the interior point method, Second, we showed that SDP can be optimized from an interior solution in polynomial time with a desired error limit. finally, we summarized several research papers that showed SDP can improve solution methods for some combinatorial optimization problems, and explained why SDP has become one of the most important research topics in optimization. We tried to integrate SDP theories. relatively diverse and complicated. to survey research papers with our own perspective, and thus to help researcher to pursue their SDP researches in depth.

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제1권1호
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• pp.35-43
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• 2001

In this paper, we propose a design method for GBSB (generalized brain-state-in-a-box) based associative memories. Based on the theoretical investigation about the properties of GBSB, we parameterize the solution space utilizing the limited number of parameters sufficient to represent the solution space and appropriate to be searched. Next we formulate the problem of finding a GBSB that can store the given pattern as stable states in the form of constrained optimization problems. Finally, we transform the constrained optimization problem into a SDP(semidefinite program), which can be solved by recently developed interior point methods. The applicability of the proposed method is illustrated via design examples.

• 한국정보과학회:학술대회논문집
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• 한국정보과학회 2004년도 가을 학술발표논문집 Vol.31 No.2 (1)
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• pp.697-699
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• 2004

Despite many successful spectral clustering algorithm (based on the spectral decomposition of Laplacian(1) or stochastic matrix(2) ) there are several unsolved problems. Most spectral clustering Problems are based on the normalized of algorithm(3) . are close to the classical graph paritioning problem which is NP-hard problem. To get good solution in polynomial time. it needs to establish its convex form by using relaxation. In this paper, we apply a novel optimization technique. semidefinite programming(SDP). to the unsupervised clustering Problem. and present a new multiple Partitioning method. Experimental results confirm that the Proposed method improves the clustering performance. especially in the Problem of being mixed with non-compact clusters compared to the previous multiple spectral clustering methods.

• East Asian mathematical journal
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• 제27권5호
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• pp.539-543
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• 2011

In this paper, we consider an optimization problem which consists a nonconvex quadratic objective function and two nonconvex quadratic constraint functions. We formulate its dual problem with semidefinite constraints, and we establish weak and strong duality theorems which hold between these two problems. And we give an example to illustrate our duality results. It is worth while noticing that our weak and strong duality theorems hold without convexity assumptions.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제5권2호
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• pp.137-147
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• 2001

Nonconvex Quadratic Optimization Problems (QOP) are solved approximately by SDP (semidefinite programming) relaxation and SOCP (second order cone programmming) relaxation. Nonconvex QOPs with special structures can be solved exactly by SDP and SOCP. We propose a method to formulate general nonconvex QOPs into the special form of the QOP, which can provide a way to find more accurate solutions. Numerical results are shown to illustrate advantages of the proposed method.

• Interaction and multiscale mechanics
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• 제1권1호
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• pp.103-121
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• 2008

This paper presents an optimization-based method for computing a minimal bounding ellipsoid that contains the set of static responses of an uncertain braced frame. Based on a non-stochastic modeling of uncertainty, we assume that the parameters both of brace stiffnesses and external forces are uncertain but bounded. A brace member represents the sum of the stiffness of the actual brace and the contributions of some non-structural elements, and hence we assume that the axial stiffness of each brace is uncertain. By using the $\mathcal{S}$-lemma, we formulate a semidefinite programming (SDP) problem which provides an outer approximation of the minimal bounding ellipsoid. The minimum bounding ellipsoids are computed for a braced frame under several uncertain circumstances.