• Title, Summary, Keyword: variational

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SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

  • Lee, Byung-Soo
    • East Asian mathematical journal
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    • v.26 no.3
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    • pp.415-421
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    • 2010
  • Many kinds of Minty's lemmas show that Minty-type variational inequality problems are very closely related to Stampacchia-type variational inequality problems. Particularly, Minty-type vector variational inequality problems are deeply connected with vector optimization problems. Liu et al. [10] considered vector variational inequalities for setvalued mappings by using scalarization approaches considered by Konnov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vector variational inequalities by using four kinds of Stampacchia-type scalar variational inequalities and obtain the relations of the solution sets between the six variational inequalities, which are more generalized results than those considered in [10]. In this paper, the author considers the Minty-type case corresponding to the Stampacchia-type case considered in [9].

MULTIOBJECTIVE VARIATIONAL PROGRAMMING UNDER GENERALIZED VECTOR VARIATIONAL TYPE I INVEXITY

  • Kim, Moon-Hee
    • Communications of the Korean Mathematical Society
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    • v.19 no.1
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    • pp.179-196
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    • 2004
  • Mond-Weir type duals for multiobjective variational problems are formulated. Under generalized vector variational type I invexity assumptions on the functions involved, sufficient optimality conditions, weak and strong duality theorems are proved efficient and properly efficient solutions of the primal and dual problems.

GENERALIZED VECTOR-VALUED VARIATIONAL INEQUALITIES AND FUZZY EXTENSIONS

  • Lee, Byung-Soo;Lee, Gue-Myung;Kim, Do-Sang
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.609-624
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    • 1996
  • Recently, Giannessi [9] firstly introduced the vector-valued variational inequalities in a real Euclidean space. Later Chen et al. [5] intensively discussed vector-valued variational inequalities and vector-valued quasi variationl inequalities in Banach spaces. They [4-8] proved some existence theorems for the solutions of vector-valued variational inequalities and vector-valued quasi-variational inequalities. Lee et al. [14] established the existence theorem for the solutions of vector-valued variational inequalities for multifunctions in reflexive Banach spaces.

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GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS

  • Kim, Jong Kyu;Salahuddin, Salahuddin;Lim, Won Hee
    • Korean Journal of Mathematics
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    • v.25 no.4
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    • pp.469-481
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    • 2017
  • In this paper, we established a general nonconvex split variational inequality problem, this is, an extension of general convex split variational inequality problems in two different Hilbert spaces. By using the concepts of prox-regularity, we proved the convergence of the iterative schemes for the general nonconvex split variational inequality problems. Further, we also discussed the iterative method for the general convex split variational inequality problems.

PERTURBED PROXIMAL POINT ALGORITHMS FOR GENERALIZED MIXED VARIATIONAL INEQUALITIES

  • Jeong, Jae-Ug
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.95-109
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    • 2002
  • In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

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AN ITERATIVE METHOD FOR NONLINEAR MIXED IMPLICIT VARIATIONAL INEQUALITIES

  • JEONG, JAE UG
    • Honam Mathematical Journal
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    • v.26 no.4
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    • pp.391-399
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    • 2004
  • In this paper, we develop an iterative algorithm for solving a class of nonlinear mixed implicit variational inequalities in Hilbert spaces. The resolvent operator technique is used to establish the equivalence between variational inequalities and fixed point problems. This equivalence is used to study the existence of a solution of nonlinear mixed implicit variational inequalities and to suggest an iterative algorithm for solving variational inequalities. In our results, we do not assume that the mapping is strongly monotone.

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MIXED QUASI VARIATIONAL INEQUALITIES INVOLVING FOUR NONLINEAR OPERATORS

  • Pervez, Amjad;Khan, Awais Gul;Noor, Muhammad Aslam;Noor, Khalida Inayat
    • Honam Mathematical Journal
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    • v.42 no.1
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    • pp.17-35
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    • 2020
  • In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the fixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

PCA-based Variational Model Composition Method for Roust Speech Recognition with Time-Varying Background Noise (시변 잡음에 강인한 음성 인식을 위한 PCA 기반의 Variational 모델 생성 기법)

  • Kim, Wooil
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.17 no.12
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    • pp.2793-2799
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    • 2013
  • This paper proposes an effective feature compensation method to improve speech recognition performance in time-varying background noise condition. The proposed method employs principal component analysis to improve the variational model composition method. The proposed method is employed to generate multiple environmental models for the PCGMM-based feature compensation scheme. Experimental results prove that the proposed scheme is more effective at improving speech recognition accuracy in various SNR conditions of background music, compared to the conventional front-end methods. It shows 12.14% of average relative improvement in WER compared to the previous variational model composition method.

On the browder-hartman-stampacchia variational inequality

  • Chang, S.S.;Ha, K.S.;Cho, Y.J.;Zhang, C.J.
    • Journal of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.493-507
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    • 1995
  • The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

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AUXILIARY PRINCEPLE AND ERROR ESTIMATES FOR VARIATIONAL INEQUALITIES

  • NOOR, MUHAMMED ASLAM
    • Honam Mathematical Journal
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    • v.15 no.1
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    • pp.105-120
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    • 1993
  • The auxiliary principle technique is used to prove the uniqueness and the existence of solutions for a class of nonlinear variational inequalities and suggest an innovative iterative algorithm for computing the approximate solution of variational inequalities. Error estimates for the finite element approximation of the solution of variational inequalities are derived, which refine the previous known results. An example is given to illustrate the applications of the results obtained. Several special cases are considered and studied.

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