ﻋﻠﯽ اﻧﺼﺎري اردﻟﯽ * اﺳﺘﺎدﯾﺎر، ﮔﺮوه رﯾﺎﺿﯽ ﮐﺎرﺑﺮدي، داﻧﺸﮑﺪه ﻋﻠﻮم رﯾﺎﺿﯽ، داﻧﺸﮕﺎه ﺷﻬﺮﮐﺮد،

Wolf-type duality for nonsmooth mathematical programs with equilibrium constraints

ﯾﮏ ﺑﺮﻧﺎﻣﻪي رﯾﺎﺿﯽ ﺑﺎﻗﯿﻮد ﺗﻌﺎدﻟﯽ ﯾﮑﯽ از ﻣﺴﺎﺋﻞ ﺑﻬﯿﻨﻪﺳﺎزي اﺳﺖ ﮐﻪ ﻗﯿﻮد آن ﺑﺮاي ﻣﺪلﺳﺎزي ﺗﻌﺎدلﻫﺎي ﻣﻌﯿﻨﯽ در ﮐﺎرﺑﺮدﻫﺎي ﻋﻠﻮم ﻣﻬﻨﺪﺳﯽ و اﻗﺘﺼﺎد ﻣﻮرد اﺳﺘﻔﺎده ﻗﺮار ﻣﯽﮔﯿﺮد. ﻫﺪف ﻣﺎ در اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﺮرﺳﯽ ﺷﺮاﯾﻂ ﻻزم ﺑﻬﯿﻨﮕﯽ و ﺑﺪﺳﺖ آوردن دوﮔﺎن وﻟﻒ ﺑﺮاي اﯾﻦ ﮔﻮﻧﻪ ﻣﺴﺎﺋﻞ اﺳﺖ. ﺑﺮاي اﯾﻦ ﻣﻨﻈﻮر ﯾﮏ ﻣﺴﺎﻟﻪ ي ﺑﻬﯿﻨﻪﺳﺎزي ﺑﺎ ﻗﯿﻮد ﺗﻌﺎدﻟﯽ را در ﺣﺎﻟﺖ ﻧﺎﻫﻤﻮار و ﻏﯿﺮﻣﺤﺪب در ﻧﻈﺮ ﮔﺮﻓﺘﻪ و ﻓﺮض ﻣﯽﮐﻨﯿﻢ ﺗﻮاﺑﻌﯽ ﮐﻪ در ﻣﺴﺎﻟﻪ وﺟﻮد دارﻧﺪ اﻟﺰاﻣﺎً ﻣﺸﺘﻖﭘﺬﯾﺮ و ﯾﺎ ﻣﺤﺪب ﻧﯿﺴﺘﻨﺪ. ﺑﻪ ﮐﻤﮏ ﻣﻔﻬﻮم ﻣﺤﺪبﮐﻨﻨﺪهﻫﺎ ﮐﻪ ﺗﻌﻤﯿﻤﯽ از زﯾﺮدﯾﻔﺮاﻧﺴﯿﻞﻫﺎ ﻫﺴﺘﻨﺪ، ﻣﻔﺎﻫﯿﻢ اﯾﺴﺘﺎﯾﯽ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ، ﺗﺤﺪب ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ و ﺑﺮﺧﯽ از ﺗﻮﺻﯿﻒﻫﺎي ﻗﯿﺪي را ﺑﺮاي اﯾﻦﮔﻮﻧﻪ از ﻣﺴﺎﺋﻞ ﺗﻌﺮﯾﻒ ﻣﯽﮐﻨﯿﻢ. ﻣﺴﺎﻟﻪي دوﮔﺎن وُﻟﻒ را ﺑﺮاي ﯾﮏ ﻣﺴﺎﻟﻪي ﺑﻬﯿﻨﻪﺳﺎزي ﺑﺎ ﻗﯿﻮد ﺗﻌﺎدﻟﯽ ﻣﻌﺮﻓﯽ ﻣﯽﮐﻨﯿﻢ و ﺑﺮاي اﯾﻦ ﻣﺴﺎﻟﻪ ﺑﺎ اﺳﺘﻔﺎده از ﻣﻔﻬﻮم ﻣﺤﺪبﮐﻨﻨﺪهﻫﺎ، ﻗﻀﺎﯾﺎي دوﮔﺎﻧﮕﯽ ﺿﻌﯿﻒ و دوﮔﺎﻧﮕﯽ ﻗﻮي را ﺑﯿﺎن و اﺛﺒﺎت ﻣﯽﮐﻨﯿﻢ .

Mathematical program with equilibrium constraints is one of the optimization problems whose constraints are used to model certain equilibria in the applications of engineering sciences and economics. Our main aim in the present paper is to investigate the necessary optimality conditions and create a Wolfe type dual problem for such problems. To investigate these conditions, we consider non smooth and non convex optimization problem with equilibrium constraints and suppose that all functions are not necessarily differentiable or convex. For this optimization problem, using the notion of convexificator, which is viewed as a generalization of the idea of subdifferential, we remind some constraint qualifications, stationary conditions, and generalized convexity. Finally, weak duality theorem and strong duality theorem are established under appropriate generalized convexity assumptions and a constraint qualification for an optimization problem with equilibrium constraints based on the notion of convexificators. We also illustrate some of our results by an example.