يافتن جواب‌هاي بهين دسته‌اي از مسائل بهينه‌سازي پارامتري بر حسب مقادير پارامتر با استفاده از شبكه‌هاي عصبي چندلايه

Finding Optimal Solutions to a Class of Parametric Optimization Problems in Terms of Parameter Values by using Multilayer Neural Networks

ﭼﮑﻴﺪﻩ: ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ، ﻣﺴﺎﺋﻞ ﺑﻬﻴﻨﻪ ﺳﺎﺯﻱ ﭘﺎﺭﺍﻣﺘﺮﻱ ﻣﻮﺭﺩ ﺑﺮﺭﺳﻲ ﻗﺮﺍﺭ ﮔﺮﻓﺘﻪ ﺍﻧﺪ. ﺩﺭ ﻳﮏ ﻣﺴﺎﻟﻪ ﺑﻬﻴﻨﻪ ﺳﺎﺯﻱ ﭘﺎﺭﺍﻣﺘﺮﻱ ﻓﺮﺽ ﻣﻲ ﺷﻮﺩλ ∈ Rn ﺑﺮﺩﺍﺭ ﭘﺎﺭﺍﻣﺘﺮﻫﺎ ﻭx∗ ∈ Rn ﺟﻮﺍﺏ ﺑﻬﻴﻦ ﻣﺘﻨﺎﻇﺮ ﺑﺎλ ﺑﺎﺷﺪ. ﻫﺪﻑ ﺍﻳﻦ ﺗﺤﻘﻴﻖ ﻣﺸﺨﺺ ﮐﺮﺩﻥ ﺗﺎﺑﻌﻲ ﻣﺎﻧﻨﺪ ﺍﺳﺖ ﮐﻪ ∗λ) = x( ﺟﻮﺍﺏ ﺑﻬﻴﻦ ﻣﺘﻨﺎﻇﺮ ﺑﺎλ ﺑﺎﺷﺪ. ﺑﺮﺍﻱ ﺍﻳﻦ ﮐﺎﺭ ﺍﺑﺘﺪﺍ ﺑﻪ ﺍﺯﺍﻱ ﻫﺮ ﻣﻘﺪﺍﺭλ ، ﺟﻮﺍﺏ ﺑﻬﻴﻦ ﻣﺘﻨﺎﻇﺮ ﻣﺤﺎﺳﺒﻪ ﻣﻲ ﺷﻮﺩ. ﺑﺪﻳﻦ ﺗﺮﺗﻴﺐ ﻳﮏ ﻣﺠﻤﻮﻋﻪ ﺍﺯ ﺩﺍﺩﻩ ﻫﺎﻱ ﺁﻣﻮﺯﺷﻲ ﻣﺘﺸﮑﻞ ﺍﺯ ﭘﺎﺭﺍﻣﺘﺮﻫﺎ ﻭ ﻣﻘﺎﺩﻳﺮ ﺑﻬﻴﻦ ﺁﻧﻬﺎ ﺑﻪ ﺩﺳﺖ ﻣﻲ ﺁﻳﺪ. ﻳﮏ ﺷﺒﮑﻪ ﻋﺼﺒﻲ ﭼﻨﺪﻻﻳﻪ ﺭﺍ ﺗﻮﺳﻂ ﺩﺍﺩﻩ ﻫﺎﻱ ﺁﻣﻮﺯﺷﻲ، ﺁﻣﻮﺯﺵ ﺩﺍﺩﻩ ﻭ ﺩﺭ ﻧﺘﻴﺠﻪ ﻋﻤﻠﮑﺮﺩ ﺩﺭ ﻳﮏ ﺩﺍﻣﻨﻪ ﻣﻌﻠﻮﻡ ﺗﻮﺳﻂ ﺍﻳﻦ ﺷﺒﮑﻪ ﻋﺼﺒﻲ ﻣﺸﺨﺺ ﻣﻲ ﺷﻮﺩ. ﺩﺭ ﻭﺍﻗﻊ ﺗﺎﺑﻊ ﺑﻪ ﺍﺯﺍﻱ ﻫﺮ ﻣﻘﺪﺍﺭ ﺍﺯ ﭘﺎﺭﺍﻣﺘﺮ )ﭘﺎﺭﺍﻣﺘﺮﻫﺎ(، ﺟﻮﺍﺏ ﺑﻬﻴﻦ ﻣﺘﻨﺎﻇﺮ ﺭﺍ ﺗﻮﺳﻂ ﺷﺒﮑﻪ ﭼﻨﺪﻻﻳﻪ ﺁﻣﻮﺯﺵ ﺩﻳﺪﻩ ﻣﺸﺨﺺ ﻣﻲ ﮐﻨﺪ. ﺩﺭ ﻧﻬﺎﻳﺖ ﭼﻨﺪ ﻣﺜﺎﻝ ﺑﺮﺍﻱ ﺑﺮﺭﺳﻲ ﮐﺎﺭﺍﻳﻲ ﺭﻭﺵ ﺍﺭﺍﺋﻪ ﻣﻲ ﺷﻮﺩ.

In this paper, parametric optimization problems are investigated. In a parametric optimization problem we assume $lambdainmathbb{R}^n$ is the vector of the parameters and $x^*$ is the optimal answer corresponding to it. The purpose of this paper is to determine a function such as $psi$ so that we have $psi(lambda)=x^*$. To do this, first for each $lambda$, the corresponding optimal answer is calculated. In this way, a set of data bases consisting of parameters and the corresponding optimal values are obtained. A multilayer network of data base is trained to obtain the function $psi$ in a domain. In fact, the function $psi$ for each value of the parameter specifies the corresponding answer by the trained multilayer network. Finally, we conduct several numerical examples to test our method.