• East Asian mathematical journal
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• v.31 no.3
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• pp.371-377
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• 2015

We consider a nondifferentiable robust optimization problem, which has a maximum function of continuously differentiable functions and support functions as its objective function, continuously differentiable functions as its constraint functions. We prove optimality conditions for the nondifferentiable robust optimization problem. We formulate a Wolfe type dual problem for the nondifferentiable robust optimization problem and prove duality theorems.

• Management Science and Financial Engineering
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• v.21 no.2
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• pp.1-8
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• 2015

This paper gives a brief review of the theory and applications of the robust optimization. Major issues including tractability, relations with relevant optimization frameworks, and dynamic robust optimization are addressed and future research directions are presented.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.723-734
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• 2013

A robust optimization problem, which has a maximum function of continuously differentiable functions as its objective function, continuously differentiable functions as its constraint functions and a geometric constraint, is considered. We prove a necessary optimality theorem and a sufficient optimality theorem for the robust optimization problem. We formulate a Wolfe type dual problem for the robust optimization problem, which has a differentiable Lagrangean function, and establish the weak duality theorem and the strong duality theorem which hold between the robust optimization problem and its Wolfe type dual problem. Moreover, saddle point theorems for the robust optimization problem are given under convexity assumptions.

• Journal of the Korean Society of Manufacturing Process Engineers
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• v.7 no.3
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• pp.3-11
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• 2008

Robust design pioneered by Dr. G. Taguchi has been applied to versatile engineering problems for improving quality. Since 1980s, the Taguchi method has been introduced to numerical optimization, complementing the deficiencies of deterministic optimization, which is often called the robust optimization. In this study, the robust optimization strategy is proposed by considering the robustness of objective and constraint functions. The statistics of responses in the functions are surrogated by kriging models. In addition, objective and/or constraint function is represented by the probability of success, thus facilitating robust optimization. The mathematical problem and the two-bar design problem are investigated to show the validity of the proposed method.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.597-602
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• 2013

In this paper, Mond-Weir type duality results for a uncertain multiobjective robust optimization problem are given under generalized invexity assumptions. Also, weak vector saddle-point theorems are obtained under convexity assumptions.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.29 no.9 s.240
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• pp.1243-1252
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• 2005

A current trend of design methodologies is to make engineers objectify or automate the decision-making process. Numerical optimization is an example of such technologies. However, in numerical optimization, the uncertainties are uncontrollable to efficiently objectify or automate the process. To better manage these uncertainties, the Taguchi method, reliability-based optimization and robust optimization are being used. To obtain the target performance with the maximum robustness is the main functional requirement of a mechanical system. In this research, a design procedure for global robust optimization is developed based on the kriging and global optimization approaches. The DACE modeling, known as the one of Kriging interpolation, is introduced to obtain the surrogate approximation model of the function. Robustness is determined by the DACE model to reduce real function calculations. The simulated annealing algorithm of global optimization methods is adopted to determine the global robust design of a surrogated model. As the postprocess, the first order second-moment approximation method is applied to refine the robust optimum. The mathematical problems and the MEMS design problem are investigated to show the validity of the proposed method.

• Journal of Mechanical Science and Technology
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• v.20 no.8
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• pp.1169-1182
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• 2006

Robust design technology has been applied to versatile engineering problems to ensure consistency in product performance. Since 1980s, the concept of robust design has been introduced to numerical optimization field, which is called the robust optimization. The robustness in the robust optimization is determined by a measure of insensitiveness with respect to the variation of a response. However, there are significant difficulties associated with the calculation of variations represented as its mean and variance. To overcome the current limitation, this research presents an implementation of the approximate statistical moment method based on kriging metamodel. Two sampling methods are simultaneously utilized to obtain the sequential surrogate model of a response. The statistics such as mean and variance are obtained based on the reliable kriging model and the second-order statistical approximation method. Then, the simulated annealing algorithm of global optimization methods is adopted to find the global robust optimum. The mathematical problem and the two-bar design problem are investigated to show the validity of the proposed method.

• Journal of the Korean Operations Research and Management Science Society
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• v.32 no.2
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• pp.99-107
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• 2007

We consider (worst-case) robust optimization versions of the Economic Order Quantity (EOQ) model with decreasing cost functions. Two variants of the EOQ model are discussed, in which the purchasing costs are decreasing power functions in either the order quantity or demand rate. We develop the corresponding worst-case robust optimization models of the two variants, where the parameters in the purchasing cost function of each model are uncertain but known to lie in an ellipsoid. For the robust EOQ model with the purchasing cost being a decreasing function of the demand rate, we derive the analytical optimal solution. For the robust EOQ model with the purchasing cost being a decreasing function of the order quantity, we prove that it is a convex optimization problem, and thus lends itself to efficient numerical algorithms.

• Proceedings of the KSME Conference
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• 2004.11a
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• pp.870-875
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• 2004

Current trend of design technologies shows engineers to objectify or automate the given decision-making process. The numerical optimization is an example of such technologies. However, in numerical optimization, the uncertainties are uncontrollable to efficiently objectify or automate the process. To better manage these uncertainties, Taguchi method, reliability-based optimization and robust optimization are being used. To obtain the target performance with the maximum robustness is the main functional requirement of a mechanical system. In this research, the robust design strategy is developed based on the DACE and the global optimization approaches. The DACE modeling, known as the one of Kriging interpolation, is introduced to obtain the surrogate approximation model of the system. The robustness is determined by the DACE model to reduce the real function calculations. The simulated annealing algorithm of global optimization methods is adopted to determine the global robust design of a surrogated model. The mathematical problems and the MEMS design problem are investigated to show the validity of the proposed method.

• East Asian mathematical journal
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• v.31 no.5
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• pp.737-742
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• 2015

In this paper, we consider a generalized fractional robust optimization problem (FP). Establishing a nonfractional optimization problem (NFP) equivalent to (FP), we establish necessary optimality conditions and duality results.