Abstract
Given a setΩ ofR n and a functionf fromΩ intoR n we consider a problem of finding a pointx * inΩ such that(x−x *)t f(x *)≽0 holds for every pointx inΩ. This problem is called the stationary point problem and the pointx * is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytopeΩ defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point inΩ with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints ofΩ. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.
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Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.
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Dai, Y., Sekitani, K. & Yamamoto, Y. A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems. ZOR Zeitschrift für Operations Research Methods and Models of Operations Research 36, 23–53 (1992). https://doi.org/10.1007/BF01541030
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DOI: https://doi.org/10.1007/BF01541030