Convex Functions and SetsELSEVIERControl Theory and Design
(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, ...
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 ...
像perspective function一样, linear-fractional functions也维护了凸性。如果C是凸的并且位于f的定义域中( i.e., c^Tx + d \gt 0 对于 x \in C) ,那么它的图像f(C)就是凸的。同样的,如果 C\subseteq R^m 那么f^{-1}(x) 的反图像时也是凸的。
Quasiconvex functionsDefinitionA function \mathcal R^n\to \mathcal R is quasiconvex if its domain and all sublevel sets S_\alpha =\{x\in dom f|f(x)\leq \alpha \},\forall \alpha \in \mathcal Rare con…
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... ...
Thereby, a function $f:Xtimes Yto{mathbb{R}}$ is called biconvex, if f(x,y) is convex in y for fixed x∈X, and f(x,y) is convex in x for fixed y∈Y. This paper presents a survey of existing results concerning the theory of biconvex sets and biconvex functions and gives ...
strong quasiconvex functionsnon empty interior/ C1160 Combinatorial mathematicsIt is shown that the level sets are bounded for any xC if the function f:C R is strong quasiconvex and C R n is a convex set with nonempty interior. Thus we have generalized one of the results from Vial [3]....
Chapter 2:Convex Sets(凸集) Chapter 3:Convex functions(凸函数) Chapter 4:Convex optimization problems Chapter 5: Lagrangian duality (拉格朗日对偶) Part II: Applications(主要介绍凸优化是如何应用在实际中的) Part III: Algorithms unconstrained optimization ...
第二个图可能是凹的,也可能是 quasiconcave,但不能是凸的或 quasiconvex 因为其 sublevel sets 不是凸的。3.22证明以下函数是凸函数:(a)f(x)=−log(−log(∑i=1meaiTx+bi))f(x)=−log(−log(∑i=1meaiTx+bi)) ,定义域为 {x∣∑i=1meaiT+bi<1}{x∣∑i=1mea...