S. Schaible

Chung Yuan Christian University, 臺中市, Taiwan, Taiwan

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Publications (103)95.37 Total impact

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    ABSTRACT: Let X be a uniformly smooth Banach space and A be an m-accretive operator on X with A (0) ≠ ∅. Assume that F: X → X is δ-strongly accretive and λ-strictly pseudocontractive with δ + λ > 1. This article proposes hybrid viscosity approximation methods which combine viscosity approximation methods with hybrid steepest-descent methods. For each t ∈ (0, 1) and each integer n ≥ 0, let {xt, n} be defined by xt, n = tf(xt, n) + (1 − t)[Jrnxt, n − θtF(Jrnxt, n)] where f: X → X is a contractive map, {rn} ⊂ [ϵ, ∞) for some ϵ > 0 and {θt: t ∈ (0, 1)} ⊂ [0, 1) with . We deduce that as t → 0, {xt, n} converges strongly to a zero p of A, which is a unique solution of some variational inequality. On the other hand, given a point x0 ∈ X and given sequences {λn}, {μn} in [0, 1], {αn}, {βn} in (0, 1], let the sequence {xn} be generated by It is proven that under appropriate conditions {xn} converges strongly to the same zero p of A. The results presented here extend, improve and develop some very recent theorems in the literature to a great extent.
    Numerical Functional Analysis and Optimization - NUMER FUNC ANAL OPTIMIZ. 01/2012; 33(2):142-165.
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    ABSTRACT: We consider a problem of solution of a multi-valued inclusion on a cone segment. In the case where the underlying mapping possesses Z type properties we suggest an extension of Gauss-Seidel algorithms from nonlinear equations. We prove convergence of a modified double iteration process under rather mild additional assumptions. Some results of numerical experiments are also presented.
    Journal of Global Optimization 01/2012; 53:97-105. · 1.31 Impact Factor
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    N. Hadjisavvas, S. Schaible, N.-C. Wong
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    ABSTRACT: The notion of pseudomonotone operator in the sense of Karamardian has been studied for 35 years and has found many applications in variational inequalities and economics. The purpose of this survey paper is to present the most fundamental results in this field, starting from the earliest developments and reaching the latest results and some open questions. The exposition includes: the relation of (generally multivalued) pseudomonotone operators to pseudoconvex functions; first-order characterizations of single-valued, differentiable pseudomonotone operators; application to variational inequalities; the notion of equivalence of pseudomonotone operators and its application to maximality; a generalization of paramonotonicity and its relation to the cutting-plane method; and the relation to the revealed preference problem of mathematical economics. KeywordsPseudomonotone operators–Variational inequalities–Pseudomonotone∗ operators
    Journal of Optimization Theory and Applications 01/2012; 152(1):1-20. · 1.42 Impact Factor
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    Lu-Chuan Ceng, Qamrul Hasan Ansari, Siegfried Schaible
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    ABSTRACT: In this paper, we introduce and analyze a new hybrid extragradient-like iterative algorithm for finding a common solution of a generalized mixed equilibrium problem, a system of generalized equilibrium problems and a fixed point problem of infinitely many non expansive mappings. Under some mild conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these three problems. Such solution also solves an optimization problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement and generalization of the previously known results in this area. KeywordsGeneralized mixed equilibrium problem–System of generalized equilibrium problems–Optimization problems–Hybrid extragradient-like iterative scheme–Fixed points–Nonexpansive mappings–Strong convergence
    Journal of Global Optimization 01/2012; 53(1):69-96. · 1.31 Impact Factor
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    ABSTRACT: “Market Coupling” is currently seen as the most advanced market design in the restructuring of the European electricity market. Market Coupling, by construction, introduces what is generally referred to as an incomplete market: it leaves several constraints out of the market and hence avoids pricing them. This may or may not have important consequences in practice depending on the case on hand. Quasi-Variational Inequality problems and the associated Generalized Nash Equilibrium can be used for representing incomplete markets. Recent papers propose methods for finding a set of solutions of Quasi-Variational Inequality problems. We apply one of these methods to a subproblem of market coupling namely the coordination of counter-trading. This problem is an illustration of a more general question encountered, for instance, in hierarchical planning in production management. We first discuss the economic interpretation of the Quasi-Variational Inequality problem. We then apply the algorithmic approach to a set of stylized case studies in order to illustrate the impact of different organizations of counter-trading. The paper emphasizes the structuring of the problem. A companion paper considers the full problem of Market Coupling and counter-trading and presents a more extensive numerical analysis. KeywordsGeneralized Nash Equilibrium–Quasi-Variational Inequalities–Market coupling–Counter-trading–European electricity market
    Networks and Spatial Economics 01/2011; · 1.23 Impact Factor
  • L.C. Ceng, S. Schaible, J.C. Yao
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    ABSTRACT: Let E be a uniformly convex and uniformly smooth Banach space with the dual E* and let T : E → 2 E* be a maximal monotone operator. By using the technique of resolvent operators and by using modified Ishikawa iteration and modified Halpern iteration for relatively non-expansive mappings, we suggest and analyse two iterative algorithms for finding an element x E such that 0 T(x). Strong convergence theorems for such iterative algorithms are proved. The ideas of these algorithms are applied to solve the problem of finding a minimizer of a convex function on E.
    Optimization 08/2010; 59(6):807-819. · 0.71 Impact Factor
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    Siegfried Schaible, Jen-Chih Yao
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    ABSTRACT: In this paper, we report recent developments in solution methods for finding a common element of the fixed point set of a mapping and the solution set of the variational inequality in a Hilbert space. Key–Words: Variational inequality; Asymptotically strict pseudocontractive mapping in the intermediate sense; Fixed point; α-inverse strongly monotone mapping.
  • J.-Y. Lin, S. Schaible, R.-L. Sheu
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    ABSTRACT: In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic “Dinkelbach-like” algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103–120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming. KeywordsSum-of-ratios problem-Min-max fractional programming-Isotonic function-Dinkelbach-type algorithm-Cutting plane method
    Journal of Optimization Theory and Applications 01/2010; 146(3):581-601. · 1.42 Impact Factor
  • E. Allevi, A. Gnudi, Siegfried Schaible, Maria Teresa Vespucci
    J. Global Optimization. 01/2010; 46:561-569.
  • E. Allevi, A. Gnudi, S. Schaible, M. Vespucci
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    ABSTRACT: The principal aim of this paper is to extend some recent results which concern problems involving bifunctions to similar generalized problems for multivalued bifunctions. To this end, by using the appropriate notions of strict pseudomonotonicity we establish the relationships between generalized vector equilibrium problems and generalized minimal element problems of feasible sets. Moreover relationships between generalized least element problems of feasible sets and generalized vector equilibrium problems are studied by employing the concept of Z-multibifunctions.
    Journal of Global Optimization 01/2010; 46(4):561-569. · 1.31 Impact Factor
  • L. C. Zeng, S. Schaible, J. C. Yao
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    ABSTRACT: In this paper, the hybrid steepest descent methods are extended to develop new iterative schemes for finding the zeros of bounded, demicontinuous and φ-strongly accretive mappings in uniformly smooth Banach spaces. Two iterative schemes are proposed. Strong convergence results are established and applications to variational inequalities are given.
    Journal of Optimization Theory and Applications 03/2009; 141(1):75-91. · 1.42 Impact Factor
  • L. C. Ceng, S. Schaible, J. C. Yao
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    ABSTRACT: This paper introduces an Ishikawa type iterative algorithm for finding approximating solutions of a class of multi-valued variational inclusion problems. Characterization of strong convergence of this iterative method is established.
    Mathematical Methods of Operational Research 01/2009; 70(1):1-12. · 0.31 Impact Factor
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    Nicolas Hadjisavvas, Siegfried Schaible
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    ABSTRACT: Pseudomonotone maps are a generalization of paramonotone maps which is very closely related to the cutting plane property in variational inequality problems (VIP). In this paper, we rst generalize the so-called mini- mum principle suciency and the maximum principle suciency for VIP with multivalued maps. Then we show that pseudomonotonicity of the map implies the \maximum principle suciency" and, in fact, is equivalent to it in a sense. We then present two applications of pseudomonotone maps. First we show that pseudomonotone maps can be used instead of the much more restricted class of pseudomonotone+ maps in a cutting plane method. Finally, an application to a proximal point method is given.
    Journal of Global Optimization 01/2009; 43(4):565-575. · 1.31 Impact Factor
  • Nicolas Hadjisavvas, Siegfried Schaible
    J. Global Optimization. 01/2009; 43:565-575.
  • Qamrul Hasan Ansari, Ali P. Farajzadeh, Siegfried Schaible
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    ABSTRACT: In this paper, we consider vector variational inequality and vector F-complementarity problems in the setting of topological vector spaces. We extend the concept of upper sign continuity for vector-valued functions and provide some existence results for solutions of vector variational inequalities and vector F-complementarity problems. Moreover, the nonemptyness and compactness of solution sets of these problems are investigated under suitable assumptions. We use a version of Fan-KKM theorem and Dobrowolski’s fixed point theorem to establish our results. The results of this paper generalize and improve several results recently appeared in the literature.
    Journal of Global Optimization 01/2009; 45:297-307. · 1.31 Impact Factor
  • L. C. Ceng, S. Schaible, J. C. Yao
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    ABSTRACT: Let T\mathcal{T} be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and T\mathcal{T} has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of T\mathcal{T}, which is a solution of a certain variational inequality.
    Journal of Optimization Theory and Applications 01/2009; 143(2):245-263. · 1.42 Impact Factor
  • L. C. Ceng, S. Schaible, J. C. Yao
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    ABSTRACT: Let C be a nonempty closed convex subset of a Banach space E with the dual E *, let T:C→E * be a Lipschitz continuous mapping and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI(T−f,C)): find x∈C such that áy-x,Tx-fñ ³ 0,for all y Î C,\langle y-x,Tx-f\rangle\geq0,\quad\mbox{for all }y\in C, where f∈E * is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L.Li’s (J.Math. Anal. Appl. 295:115–126, 2004) iterative algorithm for finding approximate solutions of VI(T−f,C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI(T−f,C), which is also a fixed point ofS.
    Journal of Optimization Theory and Applications 01/2009; 141(2):265-283. · 1.42 Impact Factor
  • H. J. Chen, S. Schaible, R. L. Sheu
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    ABSTRACT: We propose a unified framework to study various versions of Dinkelbach-type algorithms for solving the generalized fractional programming problem. Classical algorithms used carefully selected iterate points and incorporated subtle convergence proofs. Our generic algorithm, however, is natural and simple. Moreover, the convergence analysis can be carried out through geometric observations and fundamental properties of convex functions. Consequently, the classical results are either refined or strengthened with new insights.
    Journal of Optimization Theory and Applications 01/2009; 141(1):93-105. · 1.42 Impact Factor
  • H. C. Lai, J. C. Liu, S. Schaible
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    ABSTRACT: We prove that a minmax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. Using a parametric approach, we establish the Kuhn-Tucker type necessary optimality conditions and prove the existence theorem of optimality for complex minimax fractional programming in the framework of generalized convexity. Subsequently, we apply the optimality conditions to formulate a one-parameter dual problem and prove weak duality, strong duality, and strict converse duality theorems involving generalized convex complex functions.
    Journal of Optimization Theory and Applications 03/2008; 137(1):171-184. · 1.42 Impact Factor
  • L. C. Ceng, S. Schaible, J. C. Yao
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    ABSTRACT: We introduce an implicit iteration scheme with a perturbed mapping for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space. Then, we establish some convergence theorems for this implicit iteration scheme which are connected with results by Xu and Ori (Numer. Funct. Analysis Optim. 22:767–772, 2001), Zeng and Yao (Nonlinear Analysis, Theory, Methods Appl. 64:2507–2515, 2006) and Takahashi and Takahashi (J. Math. Analysis Appl. 331:506–515, 2007). In particular, necessary and sufficient conditions for strong convergence of this implicit iteration scheme are obtained.
    Journal of Optimization Theory and Applications 01/2008; 139(2):403-418. · 1.42 Impact Factor

Publication Stats

2k Citations
95.37 Total Impact Points


  • 2009–2012
    • Chung Yuan Christian University
      臺中市, Taiwan, Taiwan
  • 2008–2012
    • Shanghai Normal University
      • Department of Mathematics
      Shanghai, Shanghai Shi, China
  • 2000–2009
    • Aligarh Muslim University
      • Department of Mathematics
      Alīgarh, Uttar Pradesh, India
  • 1990–2009
    • University of California, Riverside
      • The A. Gary Anderson Graduate School of Management
      Riverside, California, United States
  • 1973–2006
    • University of Cologne
      Köln, North Rhine-Westphalia, Germany
  • 2005
    • King Fahd University of Petroleum and Minerals
      • Department of Mathematics and Statistics
      Az̧ Z̧ahrān, Eastern Province, Saudi Arabia
  • 2004
    • National Sun Yat-sen University
      • Department of Applied Mathematics
      Kaohsiung, Kaohsiung, Taiwan
    • Pukyong National University
      • Department of Applied Mathematics
      Pusan, Busan, South Korea
  • 1993–2004
    • University of the Aegean
      • Department of Mathematics
      Kastro, North Aegean, Greece
  • 1981–2004
    • University of Alberta
      • Alberta School of Forest Science and Management
      Edmonton, Alberta, Canada
  • 2003
    • University of Bergamo
      Bérgamo, Lombardy, Italy
  • 1995
    • Università di Pisa
      Pisa, Tuscany, Italy
  • 1983
    • University of Iowa
      Iowa City, Iowa, United States