• Nonlinear Functional Analysis and Applications
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• v.27 no.4
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• pp.869-879
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• 2022

In this paper, we consider matrix optimization problems. We investigate augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems. Following the proofs of Eckstein [3], and Eckstein and Yao [5], we prove convergence theorems for augmented Lagrangian methods of multipliers and alternating direction methods of multipliers for the problems.

• Journal of the Korean Operations Research and Management Science Society
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• v.29 no.4
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• pp.117-134
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• 2004

This study suggests a data driven optimization approach, which simulates the models of human learning processes from cognitive sciences. It shows how the human learning processes can be simulated and applied to solving combinatorial optimization problems. The main advantage of using this method is in applying it into problems, which are very difficult to simulate. 'Undecidable' problems are considered as best possible application areas for this suggested approach. The concept of an 'undecidable' problem is redefined. The learning models in human learning and decision-making related to combinatorial optimization in cognitive and neural sciences are designed, simulated, and implemented to solve an optimization problem. We call this approach 'SLO : simulated learning for optimization.' Two different versions of SLO have been designed: SLO with position & link matrix, and SLO with decomposition algorithm. The methods are tested for traveling salespersons problems to show how these approaches derive new solution empirically. The tests show that simulated learning for optimization produces new solutions with better performance empirically. Its performance, compared to other hill-climbing type methods, is relatively good.

• International Journal of Control, Automation, and Systems
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• v.1 no.3
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• pp.271-281
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• 2003

In this paper, a new nonlinear programming approach is suggested to solve biaffine matrix inequality (BMI) problems in multiobjective and structured controls. It is shown that these BMI problems are reduced to nonlinear minimization problems. An algorithm that is easily implemented with existing convex optimization codes is presented for the nonlinear minimization problem. The efficiency of the proposed algorithm is illustrated by numerical examples.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.375-381
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• 2008

Boundedness for the set of all the $\epsilon$-approximate solutions for convex optimization problems are considered. We give necessary and sufficient conditions for the sets of all the $\epsilon$-approximate solutions of a convex optimization problem involving finitely many convex functions and a convex semidefinite problem involving a linear matrix inequality to be bounded. Furthermore, we give examples illustrating our results for the boundedness.

• Current Optics and Photonics
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• v.5 no.3
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• pp.322-328
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• 2021

In this paper, a pretreatment method for a matrix in convex optimization is proposed to optimize the spectral reconstruction process of a disordered dispersion spectrometer. Unlike the reconstruction process of traditional spectrometers using Fourier transforms, the reconstruction process of disordered dispersion spectrometers involves solving a large-scale matrix equation. However, since the matrices in the matrix equation are obtained through measurement, they contain uncertainties due to out of band signals, background noise, rounding errors, temperature variations and so on. It is difficult to solve such a matrix equation by using ordinary nonstationary iterative methods, owing to instability problems. Although the smoothing Tikhonov regularization approach has the ability to approximatively solve the matrix equation and reconstruct most simple spectral shapes, it still suffers the limitations of reconstructing complex and irregular spectral shapes that are commonly used to distinguish different elements of detected targets with mixed substances by characteristic spectral peaks. Therefore, we propose a special pretreatment method for a matrix in convex optimization, which has been proved to be useful for reducing the condition number of matrices in the equation. In comparison with the reconstructed spectra gotten by the previous ordinary iterative method, the spectra obtained by the pretreatment method show obvious accuracy.

• Proceedings of the KSME Conference
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• 2001.06c
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• pp.408-413
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• 2001

The structural optimization is carried out in the continuous design space or discrete design space. Methods for discrete variables such as genetic algorithms are extremely expensive in computational cost. In this research, an iterative optimization algorithm using orthogonal arrays is developed for design in discrete space. An orthogonal array is selected on a discrete design space and levels are selected from candidate values. Matrix experiments with the orthogonal array are conducted. New results of matrix experiments are obtained with penalty functions for constraints. A new design is determined from analysis of means(ANOM). An orthogonal array is defined around the new values and matrix experiments are conducted. The final optimum design is found from iterative process. The suggested algorithm has been applied to various problems such as truss and frame type structures. The results are compared with those from a genetic algorithm and discussed.

• International Journal of Control, Automation, and Systems
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• v.1 no.2
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• pp.184-193
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• 2003

Biaffine Matrix Inequality (BMI) is known to provide the most general framework in control synthesis, but problems involving BMI's are very difficult to solve because nonconvex optimization should be solved. In the previous paper, we proposed a new solver for problems involving BMI's using Evolutionary Algorithms (EA). In this paper, we solve several control synthesis examples such as Reduced-order control, Simultaneous stabilization, Multi-objective control, $H_{\infty}$ optimal control, Maxed $H_2$ / $H_{\infty}$control design, and Robust $H_{\infty}$ control. Each of these problems is formulated as the standard BMI form, and solved by the proposed algorithm. The performance in each case is compared with those of conventional methods.

• Proceedings of the KSME Conference
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• 2000.11a
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• pp.818-823
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• 2000

In practical design studies, most of designers solve multidisciplinary problems with complex design structure. These multidisciplinary problems have hundreds of analysis and thousands of variables. The sequence of process to solve these problems affects the speed of total design cycle. Thus it is very important for designer to reorder original design processes to minimize total cost and time. This is accomplished by decomposing large multidisciplinary problem into several multidisciplinary analysis subsystem (MDASS) and processing it in parallel. This paper proposes new strategy for parallel decomposition of multidisciplinary problem to raise design efficiency by using genetic algorithm and shows the relationship between decomposition and multidisciplinary design optimization (MDO) methodology.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.3
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• pp.107-116
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• 2021

We introduce optimization algorithms using Bregman Divergence for solving non-negative matrix factorization (NMF) problems. Bregman divergence is known a generalization of some divergences such as Frobenius norm and KL divergence and etc. Some algorithms can be applicable to not only NMF with Frobenius norm but also NMF with more general Bregman divergence. Matrix Factorization is a popular non-convex optimization problem, for which alternating minimization schemes are mostly used. We develop the Bregman proximal gradient method applicable for all NMF formulated in any Bregman divergences. In the derivation of NMF algorithm for Bregman divergence, we need to use majorization/minimization(MM) for a proper auxiliary function. We present algorithmic aspects of NMF for Bregman divergence by using MM of auxiliary function.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.23 no.6
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• pp.683-691
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• 2010

In this paper, using an adjoint variable method, we develop a design sensitivity analysis(DSA) method applicable to heat conduction problems in steady state. Also, a topology design optimization method is developed using the developed DSA method. Design sensitivity expressions with respect to the thermal conductivity are derived. Since the already factorized system matrix is utilized to obtain the adjoint solution, the cost for the sensitivity computation is trivial. For the topology design optimization, the design variables are parameterized into normalized bulk material densities. The objective function and constraint are the thermal compliance of structures and allowable material volume respectively. Through several numerical examples, the developed DSA method is verified to yield very accurate sensitivity results compared with finite difference ones, requiring less than 0.25% of CPU time for the finite differencing. Also, the topology optimization yields physical meaningful results.