Dual and bidual problems for a Lipschitz optimization problem based on quasi-conjugation |
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Authors: | Syuuji Yamada Tamaki Tanaka |
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Institution: | a Graduate School of Science and Technology, Niigata University, Niigata-City 9502181, Japan b Graduate School of Engineering, Osaka University, Suita, Osaka 5650871, Japan |
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Abstract: | In this paper, we consider a Lipschitz optimization problem (LOP) constrained by linear functions in Rn. In general, since it is hard to solve (LOP) directly, (LOP) is transformed into a certain problem (MP) constrained by a ball in Rn+1. Despite there is no guarantee that the objective function of (MP) is quasi-convex, by using the idea of the quasi-conjugate function defined by Thach (1991) 1], we can construct its dual problem (DP) as a quasi-convex maximization problem. We show that the optimal value of (DP) coincides with the multiplication of the optimal value of (MP) by −1, and that each optimal solution of the primal and dual problems can be easily obtained by the other. Moreover, we formulate a bidual problem (BDP) for (MP) (that is, a dual problem for (DP)). We show that the objective function of (BDP) is a quasi-convex function majorized by the objective function of (MP) and that both optimal solution sets of (MP) and (BDP) coincide. Furthermore, we propose an outer approximation method for solving (DP). |
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Keywords: | Lipschitz optimization Duality Quasi-conjugate function Outer approximation method |
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