Abstract
This paper describes a non-monotone direct search method (NMDSM) that finds a stationary point of linearly constrained minimization problems. At each iteration the algorithm uses NMDSM techniques on the Euclidean space \({\mathbb {R}}^n\) spanned by n variables carefully selected from the \(n+m\) variables formulated by the model under analysis. These variables are obtained by simple rules and are handled with pivot transformations frequently used in the solution of linear systems. A new weaker 0-order non smooth necessary condition is suggested, which transmute to other stationarity conditions, depending upon the kind of differentiability present in the system. Convergence with probability 1 is proved for non smooth functions. The algorithm is tested numerically on a set of small to medium size problems that have exhibited serious difficulties for their solution by other optimization techniques. The paper also considers possible extensions to non-linearly constrained problems via exact penalty function and a slightly modified algorithm satisfactorily solved a multi-batch multi-product plant that was modeled as a MINLP.
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Acknowledgements
This research has been partially funded by the Universidad Simón Bolívar and by the Universidade de Vigo through grant R2014/037 from the Galician Regional Government. Spain. Thanks to an anonymous referee who read the previous version and suggested modifications that enhanced the presentation of the paper.
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García-Palomares, U.M. Non-monotone derivative-free algorithm for solving optimization models with linear constraints: extensions for solving nonlinearly constrained models via exact penalty methods. TOP 28, 599–625 (2020). https://doi.org/10.1007/s11750-020-00549-y
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DOI: https://doi.org/10.1007/s11750-020-00549-y
Keywords
- Non monotone direct search method
- 0-Order stationarity
- Linear and non linear constraints
- Exact penalty functions