Convex sets and convex functions are studied in this chapter in the setting of n -dimensional Euclidean space R n . Convexity is an attractive subject to study, for many reasons; it draws upon geometry, analysis
The link between convex sets and convex functions is via the epigraph. A function is convex iff. its epigraph is a convex set. Similarly, a function f is concave iff. its hypograph defined as: \bold{hypo}\; f = \{(x, t) | t \le f(x)\} is a convex set....
ConvexFunctions Definition f(θx+(1-θ)y)<=θf(x)+(1-θ)f(y) Alternatedefinitionintermsofepigraph Relationtoconvexsets + Provingafunctionisconvex It’softeneasierthanprovingsetsareconvexbecausethere aremoretools Firstorder Taylorexpansion(alwaysunderestimates) ...
Over the last 20 years, parallel to the theory of monotone operators, a calculus for the investigation of convex functionals designated by convex analysis has emerged, which allows one to solve a number of problems in a simple way. To this calculus belon
2.1.1 定义(凸集)空间 \mathbb{R}^n 的一个子集C ,若对于\forall x, y \in C ,\forall \lambda \in (0,1) ,有(1-\lambda)x+\lambda y \in C ,则称子集C 为一个凸的(convex)。根据前述,显然,所有仿射集(…
Convex Functions 2.1 Convex Sets and Convex Functions A C in a Hilbert space X is a set with the following property: for every x,y∈C,C contains the segment [x,y]={tx+(1-t)y:t∈[0,1]} (see for instance, Fig. 2.1). The next proposition summarizes some elementary properties of ...
(2001). Separation Theorems for Convex Sets and Convex Functions with Invariance Properties. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, ...
4. 凸集(Convex sets) 凸集定义: 如果集合\(C\)中的任意两点之间的线段(上的所有点)在\(C\)上,也就是说如果\(\forall{x_1,x_2∈C},0≤\theta≤1\),都有\(\theta x_1+(1-\theta)x_2∈C\),那么集合\(C\)为凸集。 注意要区分凸集和仿射集定义,前者是线段,后者是直线。
ChapterVConvex SetsandConvex FunctionsThischapteris asystematicandself-contained expositionof thebackgroundonconvexsetsandconvexfunctionsin finite-dimensionsthatarerequiredinotherpartsof thebook.It is of aratherelementary natureandrequiresnotmorethansomebasic factsonanalysisin finite-dimensionalvector spaces.The ...
Convex sets and convex functions: the fundamentals.- Continuity and ?(X).- The derivatives and the subdifferential.- Minima and quasi minima.- The Fenchel conjugate.- Duality.- Linear programming and game theory.- Hypertopologies, hyperc... ...