• Journal of Advanced Marine Engineering and Technology
• /
• 제28권5호
• /
• pp.801-810
• /
• 2004

The Primal-Dual Interior-Point (PDIP) method is currently one of the fastest emerging topics in optimization. This method has become an effective solution algorithm for large scale nonlinear optimization problems. such as the electric Optimal Power Flow (OPF) and natural gas and electricity OPF. This study describes major theoretical developments of the PDIP method as well as practical issues related to implementation of the method. A simple quadratic problem with linear equality and inequality constraints

• International Journal of Aeronautical and Space Sciences
• /
• 제11권1호
• /
• pp.10-18
• /
• 2010

The purpose of this paper deals with direct multiple-shooting method (DMS) to resolve helicopter maneuver problems of helicopters. The maneuver problem is transformed into nonlinear problems and solved DMS technique. The DMS method is easy in handling constraints and it has large convergence radius compared to other strategies. When parameterized with piecewise constant controls, the problems become most effectively tractable because the search direction is easily estimated by solving the structured Karush-Kuhn-Tucker (KKT) system. However, generally the computation of function, gradients and Hessian matrices has considerably time-consuming for complex system such as helicopter. This study focused on the approximation of the KKT system using the matrix exponential and its integrals. The propose method is validated by solving optimal control problems for the linear system where the KKT system is exactly expressed with the matrix exponential and its integrals. The trajectory tracking problem of various maneuvers like bob up, sidestep near hovering flight speed and hurdle hop, slalom, transient turn, acceleration and deceleration are analyzed to investigate the effects of algorithmic details. The results show the matrix exponential approach to compute gradients and the Hessian matrix is most efficient among the implemented methods when combined with the mixed time integration method for the system dynamics. The analyses with the proposed method show good convergence and capability of tracking the prescribed trajectory. Therefore, it can be used to solve critical areas of helicopter flight dynamic problems.

• 한국통신학회논문지
• /
• 제34권7A호
• /
• pp.562-567
• /
• 2009

본 논문에서는 여러 송수신 안테나를 갖는 송신기(source node), 중계기(relay node), 수신기(destination node)로 구성된 두 홉(two-hop) 빔포밍(beamforming) 중계 시스템을 고려한다. 송수신 심벌간 평균제곱오차(mean square error: MSE)를 최소화하는 송수신기 빔포밍 벡터와 중계기 가중치 행렬을 설계한다. 이때, 송신기 또는 중계기에 송신 전력을 줄이기 위하여, 국소(local)부동식전력제약(inequality power constraint)을 사용한다. 제약식이 있는 평균제곱오차 최소화 문제는 라그랑즈(Lagrange) 방법을 써 제약식이 없는 최적화 문제로 바꿀 수 있고, Karush-Kuhn-Tucker (KKT) 조건으로부터 그 해를 얻을 수 있다. 제안한 중계 시스템에 송신기와 중계기 송신전력을 각각 국소부동식으로 제약하여, 그 결과 두 홉에 채널 상태가 다를 경우, 최대 신호대잡음비(signal-to-noise ratio: SNR)를 얻는 기존 방식과 대등한 성능을 내며, 동시에 송신기 또는 중계기 송신 전력을 줄일 수 있다. 이를 모의실험을 통해 확인하였다.

• Journal of applied mathematics & informatics
• /
• 제33권5_6호
• /
• pp.627-633
• /
• 2015

In this paper, we generalize the concept of the energy of Seidel matrix S(G) which denoted by S^α(G) and obtain some results related to this matrix. Also, we obtain an upper and lower bound for S^α(G) related to all of graphs with |detS(G)| ≥ (n - 1); n ≥ 3.

• 한국수학교육학회지시리즈E:수학교육논문집
• /
• 제37권1호
• /
• pp.65-84
• /
• 2023

라그랑주 승수법(Method of Lagrange Multipliers)은 등식 제약조건하에서 미분가능한 함수의 최대, 최소를 구하는 대표적인 방법이다. 선형대수학, 최적화 이론, 제어 이론을 포함하여 최근에는 인공지능 기초수학에서도 널리 활용되고 있다. 특히 라그랑주 승수법은 미분적분학과 선형대수학을 연결하는 중요한 도구이며, 주성분 분석(Principal Component Analysis, PCA)을 포함한 인공지능 알고리즘에 많이 활용되고 있다. 따라서 교수자는 대학 미분적분학에서 처음 라그랑주 승수법을 접하는 학생들에게 구체적인 학습 동기를 제공할 필요가 생겼다. 이에 본 논문에서는 교수자가 학생들에게 라그랑주 승수법을 효과적으로 교육하는데 필요한 통합적인 시야를 제공한다. 먼저 다양한 전공의 학생들이 계산에 대한 부담을 덜고 원리를 쉽게 이해할 수 있도록 개발한 시각화 자료 및 파이썬(Python) 기반의 SageMath 코드를 제공한다. 또한 라그랑주 승수법으로 행렬의 고윳값과 고유벡터를 유도하는 과정을 상세히 소개한다. 그리고 라그랑주 승수법을 간단한 경우에 대한 증명에서 시작하여 일반화된 최적화 문제로 확장하고, 수업에서 학생들이 라그랑주 승수와 PCA를 활용하여 실제 데이터를 분석한 결과를 추가하였다. 본 연구는 대학수학을 지도하는 다양한 전공의 교수자들에게 도움이 될 기초자료가 될 것이다.

• Journal of applied mathematics & informatics
• /
• 제29권5_6호
• /
• pp.1395-1407
• /
• 2011

When a Sequential Quadratic Programming (SQP) method is used to solve the nonlinear programming problems, one of the main difficulties is that the Quadratic Programming (QP) subproblem may be incompatible. In this paper, an SQP algorithm is given by modifying the traditional QP subproblem and applying a class of $l_{\infty}$ penalty function whose penalty parameters can be adjusted automatically. The new QP subproblem is compatible. Under the extended Mangasarian-Fromovitz constraint qualification condition and the boundedness of the iterates, the algorithm is showed to be globally convergent to a KKT point of the non-linear programming problem.

• 대한기계학회논문집A
• /
• 제29권12권
• /
• pp.1629-1637
• /
• 2005

Optimum sensitivity analysis (OSA) is the process to find the sensitivity of optimum solution with respect to the parameter in the optimization problem. The prevalent OSA methods calculate the optimum sensitivity as a post-processing. In this research, a simple technique is proposed to obtain optimum sensitivity as a result of the original optimization problem, provided that the optimum sensitivity of objective function is required. The parameters are considered as additional design variables in the original optimization problem. And then, it is endowed with equality constraints to penalize the additional variables. When the optimization problem is solved, the optimum sensitivity of objective function is simultaneously obtained as Lagrange multiplier. Several mathematical and engineering examples are solved to show the applicability and efficiency of the method compared to other OSA ones.

• Journal of applied mathematics & informatics
• /
• 제23권1_2호
• /
• pp.269-279
• /
• 2007

This paper describes a direct search method for a class of linearly constrained optimization problem. Through research we find it can be treated as an unconstrained optimization problem. And with the decrease of dimension of the variables need to be computed in the algorithms, the implementation of convergence to KKT points will be simplified to some extent. Convergence is shown under mild conditions which allow successive frames to be rotated, translated, and scaled relative to one another.

• Journal of Communications and Networks
• /
• 제18권4호
• /
• pp.629-638
• /
• 2016

In this paper, we investigate an energy-efficient resource allocation problem in a two-way relay (TWR) network consisting of multiple user pairs and an amplify-and-forward (AF) relay. As the users and relay have individual energy efficiencies (EE), we formulate a multi-objective optimization problem (MOOP). A single-objective optimization problem (SOOP) of the MOOP is introduced using a weighted-sum method, which achieves a single Pareto optimal point of the MOOP. To derive the algorithm for the SOOP, we propose a more tractable equivalent problem using the Karush-Kuhn-Tucker conditions of the SOOP, which guarantees convergence at the local optimal points. The proposed equivalent problem can be efficiently solved by the proposed iterative algorithm. Numerical results demonstrate the effectiveness of the proposed algorithm in achieving the optimal EE in multi-user AF TWR networks.

• 한국정밀공학회:학술대회논문집
• /
• 한국정밀공학회 2005년도 추계학술대회 논문집
• /
• pp.464-469
• /
• 2005

Optimum sensitivity analysis (OSA) is the process to find the sensitivity of optimum solution with respect to the parameter in the optimization problem. The prevalent OSA methods calculate the optimum sensitivity as a post-processing. In this research, a simple technique is proposed to obtain optimum sensitivity as a result of the original optimization problem, provided that the optimum sensitivity of objective function is required. The parameters are considered as additional design variables in the original optimization problem. And then, it is endowed with equality constraints to penalize the additional variables. When the optimization problem is solved, the optimum sensitivity of objective function is simultaneously obtained as Lagrange multiplier. Several mathematical and engineering examples are solved to show the applicability and efficiency of the method compared to other OSA ones.