• Title, Summary, Keyword: duality

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DUALITY IN THE OPTIMAL CONTROL PROBLEMS OF NONLINEAR PARABOLIC SYSTEMS

  • Lee, Mi-Jin
    • East Asian mathematical journal
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    • v.16 no.2
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    • pp.267-275
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    • 2000
  • In this paper, we study the duality theory of nonlinear parabolic systems. The main objective is to prove the duality theorem under general conditions within an infinite-dimensional framework. As an application, an example is given.

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ON DUALITY FOR NONCONVEX QUADRATIC OPTIMIZATION PROBLEMS

  • Kim, Moon-Hee
    • East Asian mathematical journal
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    • v.27 no.5
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    • pp.539-543
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    • 2011
  • In this paper, we consider an optimization problem which consists a nonconvex quadratic objective function and two nonconvex quadratic constraint functions. We formulate its dual problem with semidefinite constraints, and we establish weak and strong duality theorems which hold between these two problems. And we give an example to illustrate our duality results. It is worth while noticing that our weak and strong duality theorems hold without convexity assumptions.

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR MINMAX FRACTIONAL OPTIMAL CONTROL PROBLEMS CONTAINING ARBITRARY NORMS

  • G. J., Zalmai
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.821-864
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    • 2004
  • Both parametric and parameter-free necessary and sufficient optimality conditions are established for a class of nondiffer-entiable nonconvex optimal control problems with generalized fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Based on these optimality results, ten Wolfe-type parametric and parameter-free duality models are formulated and weak, strong, and strict converse duality theorems are proved. These duality results contain, as special cases, similar results for minmax fractional optimal control problems involving square roots of positive semi definite quadratic forms, and for optimal control problems with fractional, discrete max, and conventional objective functions, which are particular cases of the main problem considered in this paper. The duality models presented here contain various extensions of a number of existing duality formulations for convex control problems, and subsume continuous-time generalizations of a great variety of similar dual problems investigated previously in the area of finite-dimensional nonlinear programming.

A Dual Problem and Duality Theorems for Average Shadow Prices in Mathematical Programming

  • Cho, Seong-Cheol
    • Journal of the Korean Operations Research and Management Science Society
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    • v.18 no.2
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    • pp.147-156
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    • 1993
  • Recently a new concept of shadow prices, called average shadow price, has been developed. This paper provides a dual problem and the corresponding duality theorems justifying this new shadow price. The general duality framework is used. As an important secondary result, a new reduced class of price function, the pp. h.-class, has been developed for the general duality theory. This should be distinguished from other known reductions achieved in some specific areas of mathematical programming, in that it sustains the strong duality property in all the mathematical programs. The new general dual problem suggested with this pp. h.-class provides, as an optimal solution, the average shadow prices.

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Duality in an Optimal Harvesting Problem by a Nonlinear Age-Spatial Structured Population Dynamic System

  • Kim, Yong-Kuk;Lee, Mi-Jin;Jung, Il-Hyo
    • Kyungpook Mathematical Journal
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    • v.51 no.4
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    • pp.353-364
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    • 2011
  • Duality in the optimal harvesting for a nonlinear age-spatial structured population dynamic model is studied in the framework of optimal control problem. In this paper the duality theory that displays the conjugacy of the primal problem is established and an application is given. Duality theory plays an important role in both optimization theory and methodology and the results may be applied to a realistic biological system on the point of optimal harvesting.

MIXED TYPE SECOND-ORDER DUALITY WITH SUPPORT FUNCTION

  • Husain, I.;Ahmed, A.;Masoodi, Mashoob
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1381-1395
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    • 2009
  • Mixed type second order dual to the non-differentiable problem containing support functions is formulated and duality theorems are proved under generalized second order convexity conditions. It is pointed out that the mixed type duality results already reported in the literature are the special cases of our results.

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A CHARACTERIZATION OF THE GENERALIZED PROJECTION WITH THE GENERALIZED DUALITY MAPPING AND ITS APPLICATIONS

  • Han, Sang-Hyeon;Park, Sung-Ho
    • Communications of the Korean Mathematical Society
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    • v.27 no.2
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    • pp.279-296
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    • 2012
  • In this paper, we define a generalized duality mapping, which is a generalization of the normalized duality mapping and using this, we extend the notion of a generalized projection and study their properties. Also we construct an approximating fixed point sequence using the generalized projection with the generalized duality mapping and prove its strong convergence.

SYMMETRIC DUALITY FOR NONLINEAR MIXED INTEGER PROGRAMS WITH A SQUARE ROOT TERM

  • Kim, Do-Sang;Song, Young-Ran
    • Journal of the Korean Mathematical Society
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    • v.37 no.6
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    • pp.1021-1030
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    • 2000
  • We formulate a pair of symmetric dual mixed integer programs with a square root term and establish the weak, strong and converse duality theorems under suitable invexity conditions. Moreover, the self duality theorem for our pair is obtained by assuming the kernel function to be skew symmetric.

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SYMMETRIC DUALITY FOR A CLASS OF NONDIFFERENTIABLE VARIATIONAL PROBLEMS WITH INVEXITY

  • LEE, WON JUNG
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.6 no.1
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    • pp.67-80
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    • 2002
  • We formulate a pair of nondifferentiable symmetric dual variational problems with a square root term. Under invexity assumptions, we establish weak, strong, converse and self duality theorems for our variational problems by using the generalized Schwarz inequality. Also, we give the static case of our nondifferentiable symmetric duality results.

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