• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1157-1165
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• 2011

The Karush-Kuhn-Tucker (KKT) necessary optimality conditions for nonlinear differentiable programming problems are also sufficient under suitable convexity assumptions. The KKT conditions in multiobjective programming problems with interval-valued objective and constraint functions are derived in this paper. The main contribution of this paper is to obtain the Pareto optimal solutions by resorting to the sufficient optimality condition.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.685-692
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• 2009

We consider the linearized (approximated) problem for differentiable vector optimization problem, and then we establish equivalence results between a differentiable vector optimization problem and its associated linearized problem under the proper efficiency.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.149-159
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• 2009

In this paper, we introduce some approximate optimal solutions and an augmented Lagrangian function in nonlinear programming, establish dual function and dual problem based on the augmented Lagrangian function, discuss the relationship between the approximate optimal solutions of augmented Lagrangian problem and that of primal problem, obtain approximate KKT necessary optimality condition of the augmented Lagrangian problem, prove that the approximate stationary points of augmented Lagrangian problem converge to that of the original problem. Our results improve and generalize some known results.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1395-1407
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• 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
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• 제29권12권
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• pp.1629-1637
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• 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.

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

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