共查询到20条相似文献,搜索用时 62 毫秒
1.
Pierre Maréchal 《Optimization Letters》2012,6(2):357-362
We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function
g : \mathbbRm ? [0, ¥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β
1 + ··· + β
n
. We also provide further generalization to functions of the form (x,y1, . . . , yn)? g(x)1+af1(y1)-b1 ···fn(yn)-bn{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f
k
concave, positively homogeneous and nonnegative on their domains. 相似文献
2.
In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where
f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and
g(y), y ? \mathbb Rn{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions
for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes
are significantly tighter than popular factorable programming relaxations. 相似文献
3.
Jean B. Lasserre 《Mathematical Programming》2009,120(2):457-477
We provide a sufficient condition on a class of compact basic semialgebraic sets for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g
j
that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed , there is a convex set such that (where B is the unit ball of ), and has an explicit SDr in terms of the g
j
’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L
f
associated with K and any linear is a sum of squares. We also provide an approximate SDr specific to the convex case.
相似文献
4.
Let
X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional
\mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2 dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx, 相似文献
5.
Bart De Bruyn 《Annals of Combinatorics》2010,14(3):307-318
Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}
6.
Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if
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