• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.647-654
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• 2017

Duality methods based on modified Lagrangian functional for solving a model crack problem is considered. Without additional assumptions of regularity of the solution of an initial problem duality ratio is established for initial and dual problem.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.9 no.13
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• pp.45-50
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• 1986

본 연구에서 고찰한 surrogate relaxation은 Lagrangian relaxation 방법과는 달리 제약식들을 선형조합으로 묶어 문제를 푼다. 수리계획 분계가 convexity를 만족하지 못하는 경우에는 Lagrangian의 경우와 마찬가지로 surrogate gap이 발생한다. Lagrangian 쌍대이론을 토대로 surrogate optimality condition을 알아보고 수리계획법의 특별 형태인 정수선형계획법에 적용해 보았다. 일반적으로 surrogate gap은 Lagrangian gap 보다 작기 때문에 좀더 근사하게 원 문제의 최적 목적 함수값에 접근할 수 있다. 따라서 branch and bound 알고리즘을 개발할 때 중요한 정보를 제공하는 것이다.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.843-854
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• 2012

In this paper, the iterative Uzawa method with a modified Lagrangian functional is investigated to seek a saddle point for the semicoercive variational Signorini inequality.

• Journal of Communications and Networks
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• v.11 no.2
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• pp.166-174
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• 2009

In this paper, a distributed power and end-to-end rate control algorithm is proposed in the presence of licensed users. By Lagrangian duality theory, the optimal power and rate control solution is given for the unlicensed users while satisfying the interference temperature limits to licensed users. It is obtained that transmitting with either 0 or the maximum node power is the optimal scheme. The synchronous and asynchronous distributed algorithms are proposed to be implemented at the nodes and links. The convergence of the proposed algorithms are proved. Finally, further discussion on the utility-based fairness is provided for the proposed algorithms. Numerical results show that the proposed algorithm can limit the interference to licensed user under a predefined threshold.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.11A
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• pp.878-891
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• 2011

We consider the throughput-maximization problem for both the up- and downlink in a wireless network with interference channels. For this purpose, we design an iterative and distributive uplink algorithm based on Lagrangian relaxation. Using the uplink power prices and network duality, we achieve throughput-maximization in the dual downlink that has a symmetric channel and an equal power budget compared to the uplink. The network duality we prove here is a generalized version of previous research [10], [11]. Computational tests show that the performance of the up- and downlink throughput for our algorithms is close to the optimal value for the channel orthogonality factor, ${\theta}{\in}$(0.5, 1]. On the other hand, when the channels are slightly orthogonal (${\theta}{\in}$(0, 0.5]), we observe some throughput degradation in the downlink. We have extended our analysis to the real downlink that has a nonsymmetric channel and an unequal power budget compared to the uplink. It is shown that the modified duality-based approach is thoroughly applied to the real downlink. Considering the complexity of the algorithms in [6] and [18], we conclude that these results are quite encouraging in terms of both performance and practical applicability of the generalized duality theorem.

• Korean Journal of Mathematics
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• v.6 no.1
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• pp.27-46
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• 1998

We are concerned with an optimal control problem governed by a Poisson equation in which body force acts like a control parameter. The cost functional to be optimized is taken to represent the error from the desired observation and the cost due to the control. We recast the problem into the mixed formulation to take advantage of the minimax principle for the duality method. The existence of a saddle point for the Lagrangian shall be shown and the optimality system will be derived therein. Finally, to attain an optimal control, we combine the optimality system with an operational technique. By achieving the gradient of the cost functional, a convergent algorithm based on the projected gradient method is established.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.185-200
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• 2004

A stochastic optimal LQR control problem under some integral quadratic (IQ) constraints is studied, with cross terms in both the cost and the constraint functionals, allowing all the control weighting matrices being indefinite. Sufficient conditions for the well-posedness of this problem are given. When these conditions are satisfied, the optimal control is explicitly derived via dual theory.

• Nonlinear Functional Analysis and Applications
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• v.26 no.3
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• pp.611-628
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• 2021

In this paper, we propose a new branch-reduction-and-bound algorithm to solve the nonlinear knapsack problems by using general discrete monotonic optimization techniques. The specific properties of the problem are exploited to increase the efficiency of the algorithm. Computational experiments of the algorithm on problems with up to 30 variables and 5 different constraints are reported.

• Journal of the Korean Operations Research and Management Science Society
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• v.23 no.2
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• pp.187-200
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• 1998

This paper presents an original approach for solving short-term power scheduling in extended power system with two fuels in a unit and a limited fuel using Lagrangian relaxations. The underlying model incorporates the full set of costs and constraints including setup, production, ramping, and operational status, and takes the form of a mixed integer nonlinear control problem. Moreover, the mathematical model developed includes two fuels in a unit and a limited fuel, regulation reserve requirements of prespecified group of units. Lagrangian relaxation is used to disaggregate the model by generator into separate subproblems which are then solved with a nested dynamic program including empirical knowledges. The strength of the methodology lies partially in its ability to construct good feasible solutions from information provided by the dual. Thus, the need for branch-and-bound is eliminated. In addition, the inclusion of two fuels in a unit and a limited fuel provides new insight into the limitations of current techniques. Computational experience with the proposed algorithm indicates that Problems containing up to 23 units including 8 unit used two fuels and 24 time periods can be readily solved in reasonable times. Duality gaps of less than 4％ were achieved.

• Structural Engineering and Mechanics
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• v.7 no.4
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• pp.345-360
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• 1999

A general framework for the nonlinear geometric analysis of elastic space trusses is presented. Both total Lagrangian and finite incremental formulations are derived from the three key ingredients of statics, kinematics and constitutive law. Particular features of the general methodology include the preservation of static-kinematic duality through the concept of fictitious forces and deformations, and an exact description for arbitrarily large displacements, albeit small strain, that can be specialized to any order of geometrical nonlinearity. As for the numerical algorithm, we consider specifically the finite incremental case and suggest the use of a conventional, simple and flexible arc-length based method. Numerical examples are presented to illustrate and validate the accuracy of the approach.