یک مدل عصبی پویای کارا برای حل مسایل بهینه سازی ناهموار مقید با محدودیت‌های آفین و کران‌دار

در اﯾﻦ ﻣﻘﺎﻟﻪ ﯾﮏ ﻣﺪل ﻏﯿﺮﺟﺮﯾﻤﻪای ﻣﺒﺘﻨﯽ ﺑﺮ ﺷﻤﻮل دﯾﻔﺮاﻧﺴﯿﻠﯽ ﺑﺮای ﺣﻞ ﻣﺴﺎﯾﻞ ﺑﻬﯿﻨﻪﺳﺎزی ﻣﻘﯿﺪ ﺑﺎ ﻗﯿﻮد ﮐﺮاندار و ﺗﺴﺎوی ﺧﻄﯽ ﭘﯿﺸﻨﻬﺎد ﮔﺮدﯾﺪه اﺳﺖ. ﻫﻤﮕﺮاﯾﯽ ﺧﻂ ﺳﯿﺮﻫﺎ ﺑﻪ ﻧﺎﺣﯿﻪی ﺷﺪﻧﯽ ﻣﺴﺎوی در زﻣﺎن ﻣﺘﻨﺎﻫﯽ را اﺛﺒﺎت ﻧﻤﻮدهاﯾﻢ. ﻫﻢﭼﻨﯿﻦ ﻣﻌﺎدل ﺑﻮدن ﻧﻘﻄﻪ ﺗﻌﺎدﻟﯽ ﺷﺒﮑﻪ ﻋﺼﺒﯽ ﭘﯿﺸﻨﻬﺎدی ﺑﺎ ﺟﻮاب ﺑﻬﯿﻨﻪی ﻣﺴﺄﻟﻪ ﺑﻬﯿﻨﻪﺳﺎزی اﺻﻠﯽ را ﻧﺸﺎن دادهاﯾﻢ. ﺑﻪﻋ ﻼوه ﭘﺎﯾﺪاری ﺷﺒﮑﻪی ﭘﯿﺸﻨﻬﺎدی ﺑﻪ ﻣﻔﻬﻮم ﻟﯿﺎﭘﺎﻧﻮف و ﻫﻤﮕﺮاﯾﯽ ﺳﺮاﺳﺮی آن ﺑﻪ ﺟﻮاب ﺑﻬﯿﻨﻪی دﻗﯿﻖ ﻣﺴﺄﻟﻪ ﺑﻬﯿﻨﻪﺳﺎزی اﺻﻠﯽ اﺛﺒﺎت ﮔﺮدﯾﺪه اﺳﺖ. ﻣﺪل ﭘﯿﺸﻨﻬﺎدی در ﻣﻘﺎﯾﺴﻪ ﺑﺎ ﻣﺪلﻫﺎی ﻣﻮﺟﻮد ﺑﺮای ﺣﻞ ﻣﺴﺎﯾﻞ ﺑﻬﯿﻨﻪﺳﺎزی ﻣﺤﺪب ﻧﺎﻫﻤﻮار ﻓﺎﻗﺪ ﭘﺎراﻣﺘﺮﺟﺮﯾﻤﻪ ﯾﺎ ﺗﺎﺑﻊ ﺟﺮﯾﻤﻪ ﺑﻮده و ﭘﯿﺎدهﺳﺎزی آن آﺳﺎنﺗﺮ ﻣﯽﺑﺎﺷﺪ. ﺑﻪﻋﻨﻮان ﮐﺎرﺑﺮد، ﺷﺒﮑﻪ ﻋﺼﺒﯽ ﭘﯿﺸﻨﻬﺎدی ﺑﻪ ﯾﮏ ﻣﺪل ﺑﺮای ﺣﻞ ﻣﺴﺎﯾﻞ ﺑﻬﯿﻨﻪﺳﺎزی ﻣﺤﺪب ﻧﺎﻫﻤﻮار ﻣﻘﯿﺪ ﺑﺎ ﻗﯿﻮد ﻣﺴﺎوی ﺧﻄﯽ و ٠ ≤ x ﺗﺒﺪﯾﻞ ﺷﺪه اﺳﺖ. در اﻧﺘﻬﺎ ﺑﺮای ﻧﺸﺎن دادن ﮐﺎراﯾﯽ ﻣﺪل ﭘﯿﺸﻨﻬﺎدی ﺗﻌﺪادی ﻣﺜﺎل اراﯾﻪ ﮔﺮدﯾﺪه اﺳﺖ.

A neurodynamic model for solving nonsmooth constrained optimization problems with affine and bound constraints

In this paper, a nonpenalty differential inclusionbased model is proposed for solving nonsmooth convex optimization problems subject to linear equality and bound constraints. We prove the convergence of the trajectories to the equality feasible region in finite time. Also, the equivalence of the equilibrium point of the proposed neural network and the optimal solution of the original optimization problem is shown. In addition, we prove the stability of the proposed neural network in the sense of Lyapunov and the global convergence to an exact optimal solution to the original problem. In comparison with some existing models for solving nonsmooth convex optimization problems, there does not exist any penalty parameter or penalty function in the model’s structure and the implementation of the proposed model is easier. In the end, as an application, the proposed neural network is reduced to a model for solving nonsmooth convex optimization problems subject to linear equality and $xgeq0$ constraints. Also, some illustrative examples are given to show the effectiveness of the proposed neural networks.