• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.35 no.3
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• pp.212-217
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• 2007

The Alienor method is a powerful tool for solving global optimization problems. It allows the transformation of a multi-variable problem into a new one that depends on a single variable. Any one-dimensional global optimization method can then be used to solve the transformed problem. Several one-dimensional global optimization methods coupled with the Alienor method have been suggested by mathematicians and it is shown that the suggested methods are successful for test functions. However, there are problems with these methods in engineering practice. In this paper, Lipschitzian optimization without using the Lipschitz constant is coupled with the Alienor method and applied to the test functions. Using test functions, it is shown that the suggested method can be successfully applied to global optimization problems.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.431-438
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• 2010

In this paper, using nonsmooth analysis, we established equivalence results between a locally Lipschitz vector optimization problem and its associated approximated problem under the proper efficiency.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.667-677
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• 2001

We consider a nonsmooth multiobjective opimization problem(PE) involving locally Lipschitz functions and define gen-eralized invexity for locally Lipschitz functions. Using Fritz John type optimality conditions, we establish Fritz John type sufficient optimality theorems for (PE) under generalized invexity.

• Journal of applied mathematics & informatics
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• v.15 no.1_2
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• pp.313-321
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• 2004

In this paper we study the problem of three-dimensional layout optimization on the simplified rotating vessel of satellite. The layout optimization model with behavioral constraints is established and some effective and convenient conditions of performance optimization are presented. Moreover, we prove that the performance objective function is locally Lipschitz continuous and the results on the relations between the local optimal solution and the global optimal solution are derived.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.1
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• pp.31-40
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• 2017

In this paper, we consider a nonsmooth multiobjective robust optimization problem with more than two locally Lipschitz objective functions and locally Lipschitz constraint functions in the face of data uncertainty. We prove a nonsmooth sufficient optimality theorem for a weakly robust efficient solution of the problem. We formulate a Wolfe type dual problem for the problem, and establish duality theorems which hold between the problem and its Wolfe type dual problem.

• Journal of the Chungcheong Mathematical Society
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• v.37 no.1
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• pp.41-55
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• 2024

The nonconvex and nonsmooth optimization problem has been widely applicable in image processing and machine learning. In this paper, we propose an extension of the proximal iteratively reweighted ℓ₁ algorithm for nonconvex and nonsmooth minmization problem. We assume the co-coerciveness of a term of objective function instead of Lipschitz gradient condition, which is generalized property of Lipschitz continuity. We prove the global convergence of the proposed algorithm. Numerical results show that the proposed algorithm converges faster than original proximal iteratively reweighed algorithm and existing algorithms.

• Journal of Institute of Control, Robotics and Systems
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• v.22 no.6
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• pp.411-416
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• 2016

This paper presents the design methodology of an unknown input observer for Lipschitz nonlinear systems with unknown inputs in the framework of convex optimization. We use an unknown input observer (UIO) to consider both nonlinearity and disturbance. By deriving a sufficient condition for exponential stability in the linear matrix inequality (LMI) form, existence of a stabilizing observer gain matrix of UIO will be assured by checking whether the quadratic stability margin of the error dynamics is greater than the Lipschitz constant or not. If quadratic stability margin is less than a Lipschitz constant, the coordinate transformation may be used to reduce the Lipschitz constant in the new coordinates. Furthermore, to reduce the maximum singular value of the observer gain matrix elements, an object function to minimize it will be optimally designed by modifying its magnitude so that amplification of sensor measurement noise is minimized via multi-objective optimization algorithm. The performance of UIO is compared to a nonlinear observer (Luenberger-like) with an application to a flexible joint robot system considering a change of load and disturbance. Finally, it is validated via simulations that the estimated angular position and velocity provide true values even in the presence of unknown inputs.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1527-1534
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• 2010

In this paper, we consider nonsmooth optimization problem of which objective and constraint functions are locally Lipschitz. We establish sufficient optimality conditions and duality results for nonsmooth vector optimization problem given under nearly strict invexity and near invexity-infineness assumptions.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.287-301
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• 2014

In this paper, we prove a necessary optimality theorem for a nonsmooth optimization problem in the face of data uncertainty, which is called a robust optimization problem. Recently, the robust optimization problems have been intensively studied by many authors. Moreover, we give examples showing that the convexity of the uncertain sets and the concavity of the constraint functions are essential in the optimality theorem. We present an example illustrating that our main assumptions in the optimality theorem can be weakened.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.353-359
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• 2007

A necessary and sufficient condition for Demyanov difference and Minkowski difference of compact convex subsets in $R^2$ being equal is given in this paper. Several examples are computed by Matlab to test our result. The necessary and sufficient condition makes us to compute Clarke subdifferential by quasidifferential for a special of Lipschitz functions.