• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.121-144
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• 2005

In this paper, we propose unconstrained and constrained posynomial Geometric Programming (GP) problem with negative or positive integral degree of difficulty. Conventional GP approach has been modified to solve some special type of GP problems. In specific case, when the degree of difficulty is negative, the normality and the orthogonality conditions of the dual program give a system of linear equations. No general solution vector exists for this system of linear equations. But an approximate solution can be determined by the least square and also max-min method. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications.

• Journal of Communications and Networks
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• 제17권3호
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• pp.241-246
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• 2015

In this paper, we consider a relaying system which employs a single relay in a wireless network with distributed sources and destinations. Here, all source, destination, and relay nodes are equipped with multiple antennas. For amplify-and-forward relay systems, we confirm the achievable sum rate through a joint multiple source precoders and a single relay filter design. To this end, we propose a new linear processing scheme in terms of maximizing the sum rate performance by applying a blockwise relaying method combined with geometric programming techniques. By allowing the global channel knowledge at the source nodes, we show that this joint design problem is formulated as a standard geometric program, which can guarantees a global optimal value under the modified sum rate criterion. Simulation results show that the proposed blockwise relaying scheme with the joint power allocation method provides substantial sum rate gain compared to the conventional schemes.

• 대한수학회보
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• 제48권3호
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• pp.523-537
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• 2011

A receiver operating characteristic (ROC) curve plots the true positive rate of a classier against its false positive rate, both of which are accuracy measures of the classier. The ROC curve has several interesting geometrical properties, including concavity which is a necessary condition for a classier to be optimal. In this paper, we study the nonparametric maximum likelihood estimator (NPMLE) of a concave ROC curve and its modification to reduce bias. We characterize the NPMLE as a solution to a geometric programming, a special type of a mathematical optimization problem. We find that the NPMLE is close to the convex hull of the empirical ROC curve and, thus, has smaller variance but positive bias at a given false positive rate. To reduce the bias, we propose a modification of the NPMLE which minimizes the $L_1$ distance from the empirical ROC curve. We numerically compare the finite sample performance of three estimators, the empirical ROC curve, the NMPLE, and the modified NPMLE. Finally, we apply the estimators to estimating the optimal ROC curve of the variance-threshold classier to segment a low depth of field image and to finding a diagnostic tool with multiple tests for detection of hemophilia A carrier.

• 한국농공학회지
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• 제29권4호
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• pp.73-92
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• 1987

Formulation of the geometric optimization for truss structures based on the elasticity theory turn out to be the nonlinear programming problem which has to deal with the cross-sectional area of the member and the coordinates of its nodes simultaneously. A few techniques have been proposed and adopted for the analysis of this nonlinear programming problem for the time being. These techniques, however, bear some limitations on truss shapes, loading conditions and design criteria for the practical application to real structures. A generalized algorithm for the geometric optimization of the truss structures, which can eliminate the above mentioned limitations, is developed in this study. The algorithm proposed utilizes the two-levels technique. In the first level which consists of two phases, the cross-sectional area of the truss member is optimized by transforming the nonlinear problem into SUMT, and solving SUMT utilizing the modified Newton Raphson method. In the second level, which also consists of two phases the geometric shape is optimized utillzing the unindirectional search technique of the Powell method which make it possible to minimize only the objective functlon. The algorithm proposed in this study is numerically tested for several truss structures with various shapes, loading conditions and design criteria, and compared with the results of the other algorithms to examine its applicability and stability. The numerical comparisons show that the two- levels algorithm proposed in this study is safely applicable to any design criteria, and the convergency rate is relatively fast and stable compared with other iteration methods for the geometric optimization of truss structures. It was found for the result of the shape optimization in this study to be decreased greatly in the weight of truss structures in comparison with the shape optimization of the truss utilizing the algorithm proposed with the other area optimum method.

• 대한토목학회논문집
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• 제5권3호
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• pp.49-59
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• 1985

탄성(彈性) 이론(理論)에 의하여 트러스의 형상최적화(形狀最適化) 문제(問題)를 형성(形成)하게 되면 부재(部材)의 단면적(斷面積)과 절점(節點)의 좌표(座標)를 동시에 고려(考慮)해야 하는 복잡(複雜)한 비선형(非線型) 계획문제(計劃問題)가 된다. 이런 비선형(非線形) 계획문제(計劃問題)를 해석(解析)할 수 있도록 제시(提示)된 기법(技法)이 별로 없고 현재 사용(使用)하고 있는 기법(技法)들도 실제(實際)의 적용(適用)에 제한(制限)을 받는 경우가 많다. 그러므로 트러스의 형태(形態), 재하조건(載荷條件) 등에 구애됨이 없이 트러스의 형상(形狀)을 최적화(最適化)할 수 있는 일반(一般) 해석기법(解析技法)이 필요(必要)한 것이다. 이에 본연구(本硏究)에서는 전(全) 해석과정(解析過程) two-phases로 나누어 phase 1 에서는 단면(斷面)을 최적화(最適化)하고 phase 2 에서는 트러스의 절점좌표(節點座標)를 변수(變數)로 하여 형상(形狀)을 최적화(最適化)하는 알고리즘을 개발(開發)한 것이다. 이 알고리즘의 phase 1 에서 유도(誘導)된 비선형(非線型) 계획문제(計劃問題)를 SUMT 문제(問題)로 변환(變換)시켜 Modified Newton-Raphson Method에 의한 SUMT 법(法)을 채택(採擇)하고 phase 2 에서는 Rosenbrock Method의 일방향(一方向) 탐사기법(探査技法)에 의해 목적함수(目的凾數)만이 최소(最小)가 되도록 하는 기법(技法)을 도입(導入)하여 최적화(最適化) 알고리즘 개발(開發)하였다. 개발(開發)된 알고리즘을 트러스의 형태(形態), 설계제약조건(設計制約條件), 재하조건(載河條件) 등을 변화(變化)시켜 가면서 수종(數種)의 트러스에 적용(適用)하여 수치계산(數値計算)을 실시(實施)하고 그 결과(結果)를 다른 알고리즘의 결과(結果)와 정교(正較)하므로서 개발(開發)된 알고리즘의 타당성(妥當性) 안정성(安定性) 적용성(適用性)을 검토(檢討)하였다. 연구(硏究) 결과(結果) 개발(開發)된 이 two-phases 알고리즘은 트러스의 설계조건(設計條件)에 구애받지 않고 트러스의 형상최적화(形狀最適化)에 적용(適用)할 수 있으며 안정성(安定性)있게 빠른 속도(速度)로 최적해(最適解)에 수렴(收斂)한다는 사실(事實)이 확인(確認)되었다. 이에 본(本) 알고리즘을 트러스의 형상최적화(形狀最適化) 알고리즘으로 새로이 제안(提案)하고 본(本) 알고리즘이트러스의 경제적(經劑的)인 설계(設計)에 도움을 줄 수 있을 것으로 사료(思料)된다.