• Title, Summary, Keyword: robust optimization problem

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OPTIMALITY CONDITIONS AND DUALITY IN NONDIFFERENTIABLE ROBUST OPTIMIZATION PROBLEMS

  • Kim, Moon Hee
    • 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.

ON DUALITY THEOREMS FOR ROBUST OPTIMIZATION PROBLEMS

  • Lee, Gue Myung;Kim, Moon Hee
    • 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.

OPTIMALITY CONDITIONS AND DUALITY IN FRACTIONAL ROBUST OPTIMIZATION PROBLEMS

  • Kim, Moon Hee;Kim, Gwi Soo
    • East Asian mathematical journal
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    • v.31 no.3
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    • pp.345-349
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    • 2015
  • In this paper, we consider a fractional robust optimization problem (FP) and give necessary optimality theorems for (FP). Establishing a nonfractional optimization problem (NFP) equivalent to (FP), we formulate a Mond-Weir type dual problem for (FP) and prove duality theorems for (FP).

ROBUST DUALITY FOR NONSMOOTH MULTIOBJECTIVE OPTIMIZATION PROBLEMS

  • Lee, Gue Myung;Kim, Moon Hee
    • 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.

ON NONSMOOTH OPTIMALITY THEOREMS FOR ROBUST OPTIMIZATION PROBLEMS

  • Lee, Gue Myung;Son, Pham Tien
    • 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.

ON SUFFICIENCY AND DUALITY FOR ROBUST OPTIMIZATION PROBLEMS INVOLVING (V, ρ)-INVEX FUNCTIONS

  • Kim, Moon Hee;Kim, Gwi Soo
    • East Asian mathematical journal
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    • v.33 no.3
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    • pp.265-269
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    • 2017
  • In this paper, we formulate a sufficient optimality theorem for the robust optimization problem (UP) under (V, ${\rho}$)-invexity assumption. Moreover, we formulate a Mond-Weir type dual problem for the robust optimization problem (UP) and show that the weak and strong duality hold between the primal problems and the dual problems.

A Note on Robust Combinatorial Optimization Problem

  • Park, Kyung-Chul;Lee, Kyung-Sik
    • Management Science and Financial Engineering
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    • v.13 no.1
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    • pp.115-119
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    • 2007
  • In [1], robust combinatorial optimization problem is introduced, where a positive integer $\Gamma$ is used to control the degree of robustness. The proposed algorithm needs solutions of n+1 nominal problems. In this paper, we show that the number of problems needed reduces to $n+1-\Gamma$.

A Robust Design Using Approximation Model and Probability of Success (근사모델 및 성공확률을 이용한 강건설계)

  • Song, Byoung-Cheol;Lee, Kwon-Hee
    • 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.

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A Robust Optimization Using the Statistics Based on Kriging Metamodel

  • Lee Kwon-Hee;Kang Dong-Heon
    • 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.