• Journal of applied mathematics & informatics
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• 제20권1_2호
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• pp.209-224
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• 2006

This paper presents an optimization model with performance constraints for two kinds of graph elements layout problem. The layout problem is partitioned into finite subproblems by using graph theory and group theory, such that each subproblem overcomes its on-off nature about optimal variable. Furthermore each subproblem is relaxed and the continuity about optimal variable doesn't change. We construct a min-max problem which is locally equivalent to the relaxed subproblem and develop the first order necessary and sufficient conditions for the relaxed subproblem by virtue of the min-max problem and the theories of convex analysis and nonsmooth optimization. The global optimal solution can be obtained through the first order optimality conditions.

• Journal of applied mathematics & informatics
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• 제27권1_2호
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• pp.365-374
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• 2009

Under certain conditions, we use augmented Lagrangians to transform a class of generalized semi-infinite min-max problems into common semi-infinite min-max problems, with the same set of local and global solutions. We give two conditions for the transformation. One is a necessary and sufficient condition, the other is a sufficient condition which can be verified easily in practice. From the transformation, we obtain a new first-order optimality condition for this class of generalized semi-infinite min-max problems.

• 대한수학회보
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• 제23권2호
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• pp.141-148
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• 1986

In [1], M. Guignard considered a constraint set in a Banach space, which is similar to that in [2] and gave a first order necessary optimality condition which generalized the Kuhn-Tucker conditions [3]. Sufficiency is proved for objective functions which is either pseudoconcave [5] or quasi-concave [6] where the constraint sets are taken pseudoconvex. In this note, we consider a psedoconvex programming problem in a Hilbert space. Constraint set in a Hillbert space being pseudoconvex and the objective function is restrained by an operator equation. Then we use the methods similar to that in [1] and [6] to obtain a necessary and sufficient optimality condition.

• Korean Journal of Mathematics
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• 제16권1호
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• pp.57-84
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• 2008

In this paper, a boundary control problem for a flow governed by the time-dependent two dimensional Navier-Stokes equations is considered. We derive a mathematical formulation and a relevant process for an appropriate control along the part of the boundary to minimize the drag due to the flow. After showing the existence of an optimal solution, the first order optimality conditions are derived. The strict differentiability of the state solution in regard to the control parameter shall be exposed rigorously, and the necessary conditions along with the system for the optimal solution shall be deduced in conjunction with the evaluation of the first order Gateaux derivative to the performance functional.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제19권3호
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• pp.327-364
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• 2015

In this paper, a generalized guidance law with an arbitrary pair of guidance coefficients for impact angle control is proposed. Under the assumptions of a stationary target and a lag-free missile with constant speed, necessary conditions for the guidance coefficients to satisfy the required terminal constraints are obtained by deriving an explicit closed-form solution. Moreover, optimality of the generalized impact-angle control guidance law is discussed. By solving an inverse optimal control problem for the guidance law, it is found that the generalized guidance law can minimize a certain quadratic performance index. Finally, analytic solutions of the generalized guidance law for a first-order lag system are investigated. By solving a third-order linear time-varying ordinary differential equation, the blowing-up phenomenon of the guidance loop as the missile approaches the target is mathematically proved. Moreover, it is found that terminal misses due to the system lag are expressed in terms of the guidance coefficients, homing geometry, and the ratio of time-to-go to system time constant.

• 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.