دوگانگی نوع ولف برای برنامه‌‌های ریاضی با قیود تعادلی ناهموار
محورهای موضوعی : آمار
علی انصاری اردلی
1
(گروه ریاضی کاربردی، دانشکده علوم ریاضی، دانشگاه شهرکرد، شهرکرد، ایران)
کلید واژه: Wolfe dual problem, optimality conditions, optimization problem with equilibrium constraints, convexificators,
چکیده مقاله :
یک برنامه‌ی ریاضی باقیود تعادلی یکی از مسائل بهینه‌سازی است که قیود آن برای مدل‌سازی تعادل‌های معینی در کاربردهای علوم مهندسی و اقتصاد مورد استفاده قرار می‌گیرد. هدف ما در این مقاله بررسی شرایط لازم بهینگی و بدست آوردن دوگان ولف برای این گونه مسائل است. برای این منظور یک مساله‌ی بهینه‌سازی با قیود تعادلی را در حالت ناهموار و غیرمحدب در نظر گرفته و فرض می‌کنیم توابعی که در مساله وجود دارند الزاما مشتق‌پذیر و یا محدب نیستند. به کمک مفهوم محدب‌کننده‌ها که تعمیمی از زیردیفرانسیل‌ها هستند، مفاهیم ایستایی تعمیم یافته، تحدب تعمیم یافته و برخی از توصیف‌های قیدی را برای این‌گونه از مسائل تعریف می‌کنیم. مساله‌ی دوگان وُلف را برای یک مساله‌ی بهینه‌سازی با قیود تعادلی معرفی می‌کنیم و برای این مساله با استفاده از مفهوم محدب‌کننده‌ها، قضایای دوگانگی ضعیف و دوگانگی قوی 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.
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