• Journal of Advanced Marine Engineering and Technology
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• v.28 no.5
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• pp.801-810
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• 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
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• v.11 no.1
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• pp.10-18
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• 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.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.34 no.7A
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• pp.562-567
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• 2009

In this paper, two-hop beamforming relay system, with source, relay, and destination nodes, is considered and the transmit- and receive-beamforming vectors and the relay processing matrix are designed for minimizing a mean square error (MMSE) between the transmit and receive signals. Here, to reduce the transmit power of the source or the relay, two local inequality constraints are involved with MMSE problem. By adopting the Lagrange method, closed formed Karush-Kuhn-Tucker (KKT) conditions (equalities) are derived and an iterative algorithm is developed to solve the entangled KKT equalities. Due to the inequality power constraints, the source or the relay can reduce its transmit power when the received signal-to-noise ratios (SNRs) of the first- and the second-hop are different. Meanwhile, the destination can achieve almost identical bit-error-rate performance compared to an optimal beamforming system maximizing the received SNR. This claim is supported by a computer simulation.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.627-633
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• 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.

• Communications of Mathematical Education
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• v.37 no.1
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• pp.65-84
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• 2023

The method of Lagrange multipliers, one of the most fundamental algorithms for solving equality constrained optimization problems, has been widely used in basic mathematics for artificial intelligence (AI), linear algebra, optimization theory, and control theory. This method is an important tool that connects calculus and linear algebra. It is actively used in artificial intelligence algorithms including principal component analysis (PCA). Therefore, it is desired that instructors motivate students who first encounter this method in college calculus. In this paper, we provide an integrated perspective for instructors to teach the method of Lagrange multipliers effectively. First, we provide visualization materials and Python-based code, helping to understand the principle of this method. Second, we give a full explanation on the relation between Lagrange multiplier and eigenvalues of a matrix. Third, we give the proof of the first-order optimality condition, which is a fundamental of the method of Lagrange multipliers, and briefly introduce the generalized version of it in optimization. Finally, we give an example of PCA analysis on a real data. These materials can be utilized in class for teaching of the method of Lagrange multipliers.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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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.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.12 s.243
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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.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.269-279
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• 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
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• v.18 no.4
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• pp.629-638
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• 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.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2005.10a
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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.