A CLASS OF NONLINEAR LAGRANGIANS: THEORY AND ALGORITHM

Li-Wei Zhang (), Yong-Hong Ren, Yue Wu and Xian-Tao Xiao
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Li-Wei Zhang: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China
Yong-Hong Ren: School of Mathematics, Liaoning Normal University, Dalian 116029, P. R. China
Yue Wu: School of Management, University of Southampton, Southampton, Highfield Southampton, SO17 1BJ, United Kingdom
Xian-Tao Xiao: Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2008, vol. 25, issue 03, 327-371

Abstract: This paper establishes a theory framework of a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. A set of conditions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms, to analyze condition numbers of nonlinear Lagrangian Hessians as well as to develop the dual approaches. These conditions are satisfied by well-known nonlinear Lagrangians appearing in literature. The convergence theorem shows that the dual algorithm based on any nonlinear Lagrangian in the class is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions and the error bound solution, depending on the penalty parameter, is also established. The paper also develops the dual problems based on the proposed nonlinear Lagrangians, and the related duality theorem and saddle point theorem are demonstrated. Furthermore, it is shown that the condition numbers of Lagrangian Hessians at optimal solutions are proportional to the controlling penalty parameters. We report some numerical results obtained by using nonlinear Lagrangians.

Keywords: Nonconvex optimization; nonlinear Lagrangian; dual algorithm; condition number; dual function (search for similar items in EconPapers)
Date: 2008
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Citations: View citations in EconPapers (1)

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DOI: 10.1142/S021759590800178X

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