Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions
نویسندگان
چکیده
We provide a rule to calculate the subdifferential of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions, and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namely chain rule formulas of standard type, are obtained from our main result via new and direct proofs.
منابع مشابه
Subdifferential Calculus Rules for Supremum Functions in Convex Analysis
Extending and improving some recent results of Hantoute, López, and Zălinescu and others, we provide characterization conditions for subdifferential formulas to hold for the supremum function of a family of convex functions on a real locally convex space.
متن کاملVariational representation , HCR and CR lower bounds
It should be noted that the requirement of f to be convex in the definition of f -divergence is essential. In Euclidean spaces any convex function can be represented as the pointwise supremum of a family of affine functions and vice versa, every supremum of a family of affine functions produces a convex function. Take f(x) = 12 |x− 1| as an example. We see that it can be written as a pointwise ...
متن کاملElements of Quasiconvex Subdifferential Calculus
A number of rules for the calculus of subdifferentials of generalized convex functions are displayed. The subdifferentials we use are among the most significant for this class of functions, in particular for quasiconvex functions: we treat the Greenberg-Pierskalla’s subdifferential and its relatives and the Plastria’s lower subdifferential. We also deal with a recently introduced subdifferentia...
متن کاملPartial second-order subdifferentials of -prox-regular functions
Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2 functions. The class of prox-regular functions covers all convex functions, lower C2 functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regula...
متن کاملNew results in subdifferential calculus with applications to convex optimization
Chain and addition rules of subdifferential calculus are revisited in the paper and new proofs, providing local necessary and sufficient conditions for their validity, are presented. A new product rule pertaining to the composition of a convex functional and a Young function is also established and applied to obtain a proof of Kuhn-Tucker conditions in convex optimization under minimal assumpti...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM Journal on Optimization
دوره 19 شماره
صفحات -
تاریخ انتشار 2008