يك الگوريتم جديد براي مسائل نامساوي تغييراتي با كاربرد در مسأله تعادل ترافيك نامتقارن

A New Algorithm of the Variational Inequality Problems with Application on the Asymmetric Traffic Equilibrium Problem

در اﯾﻦ ﻣﻘﺎﻟﻪ ﯾﮏ اﻟﮕﻮرﯾﺘﻢ ﺗﺼﻮﯾﺮ دوﮔﺎﻣﯽ ﻣﺒﺘﻨﯽ ﺑﺮ روش ﮔﺮادﯾﺎن اﻓﺰوده ﺑﺮاي ﺣﻞ ﻣﺴﺎﺋﻞ ﻧﺎﻣﺴﺎوي ﺗﻐﯿﯿﺮاﺗﯽ ﻣﻌﺮﻓﯽ ﻧﻤﻮده و ﺑﻪ اﺛﺒﺎت ﻗﻀﯿﻪ ﻫﻤﮕﺮاﯾﯽ اﻟﮕﻮرﯾﺘﻢ ﭘﯿﺸﻨﻬﺎدي، ﻣﯽﭘﺮدازﯾﻢ. ﯾﮑﯽ از ﭘﺎراﻣﺘﺮﻫﺎﯾﯽ ﮐﻪ ﮐﺎراﯾﯽ و دﻗﺖ اﻟﮕﻮرﯾﺘﻢﻫﺎي ﺗﺼﻮﯾﺮ را ﺗﻌﯿﯿﻦ ﻣﯽﮐﻨﺪ اﻧﺘﺨﺎب اﻧﺪازه ﮔﺎم اﻟﮕﻮرﯾﺘﻢ اﺳﺖ. اﯾﻦ اﻧﺘﺨﺎب واﺑﺴﺘﻪ ﺑﻪ وﯾﮋﮔﯽﻫﺎي اﻧﻘﺒﺎﺿﯽ ﻧﮕﺎﺷﺘﯽ اﺳﺖ ﮐﻪ در اﻟﮕﻮرﯾﺘﻢ ﺑﻪ ﻧﺎﺣﯿﻪ ﺷﺪﻧﯽ ﺗﺼﻮﯾﺮ ﻣﯽ ﺷﻮد. اﮔﺮ ﺑﻪ ﻋﻨﻮان ﻧﻤﻮﻧﻪ ﺛﺎﺑﺖ ﻟﯿﭗﺷﯿﺘﺰ ﻧﮕﺎﺷﺖ ﻣﺸﺨﺺ ﻧﺒﺎﺷﺪ در اﻧﺘﺨﺎب اﻧﺪازه ﮔﺎم اﻟﮕﻮرﯾﺘﻢ دﭼﺎر ﻣﺸﮑﻞ ﻣﯽﺷﻮﯾﻢ. در اﻟﮕﻮرﯾﺘﻢ ﭘﯿﺸﻨﻬﺎدي ﻧﯿﺎز ﺑﻪ داﻧﺴﺘﻦ ﺛﺎﺑﺖ ﻟﯿﭗﺷﯿﺘﺰ ﺑﺮداﺷﺘﻪ ﺷﺪه اﺳﺖ و روﺷﯽ اراﺋﻪ ﺷﺪه اﺳﺖ ﮐﻪ اﻧﺘﺨﺎب اﻧﺪازه ﮔﺎم را آﺳﺎن ﻣﯽﮐﻨﺪ. ﻣﺴﺄﻟﻪ ﺗﻌﺎدل ﺷﺒﮑﻪ ﺗﺮاﻓﯿﮏ ﻧﺎﻣﺘﻘﺎرن را ﺑﻪ ﺻﻮرت ﯾﮏ ﻣﺴﺄﻟﻪ ﻧﺎﻣﺴﺎوي ﺗﻐﯿﯿﺮاﺗﯽ روي ﻓﻀﺎي ﺟﺮﯾﺎن ﻣﺴﯿﺮﻫﺎي ﺷﺒﮑﻪ ﻣﺪلﺳﺎزي ﻧﻤﻮده و ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺳﺎﺧﺘﺎر ﺗﺠﺰﯾﻪﭘﺬﯾﺮ ﻣﺠﻤﻮﻋﻪ ﺷﺪﻧﯽ اﯾﻦ ﻣﺪل، ﺣﺎﻟﺖ ﺗﻌﺎدل ﺷﺒﮑﻪ ﺗﺮاﻓﯿﮏ را ﺑﺎ اﺳﺘﻔﺎده ازاﻟﮕﻮرﯾﺘﻢ ﭘﯿﺸﻨﻬﺎدي، ﺑﻪ دﺳﺖ ﻣﯽآورﯾﻢ. در ﻧﻬﺎﯾﺖ، ﻧﺘﺎﯾﺞ ﻋﺪدي ﺣﺎﺻﻞ از اﺟﺮاي اﯾﻦ اﻟﮕﻮرﯾﺘﻢ را ﺑﺮ روي ﺷﺒﮑﻪ ﺗﺮاﻓﯿﮏ آزﻣﺎﯾﺸﯽ ﺳﺎﯾﻮﮐﺲﻓﺎل اراﺋﻪ ﻣﯽﮐﻨﯿﻢ .

In this paper, we introduce a double projection algorithm based on extragradient method for solving the variational inequality problems and we prove the convergence theorem of proposed algorithm. One of the parameters that determine the efficiency and accuracy of the projection method is a properly selected step size. This selection is based on the contractive properties of operator which is projected on the feasible region. For example, if the Lipschitz constant is not known, we have trouble choosing step size of the algorithm. The proposed algorithm eliminates the need to know the Lipschitz constant and provides a method that facilitates step size selection.We formulate the asymmetric traffic equilibrium problem as a variational inequality on the path flows space. According to the decomposable structure of the feasible set of this model, we obtain the traffic network equilibrium state, by using the proposed algorithm. Finally, we present the numerical results of using this algorithm on the Sioux-Falls test traffic network.