是log-concave函数 对称正定矩阵的行列式detX的行列式是log-concave函数 对称正定矩阵的行列式detX与迹的比值:detX/trX 许多概率密度分布都是log-concave函数,例如多元高斯分布: 多元指数分布: p(x)=(∏i=1nλi)e−λTx Wishart分布: f(X)=a(detX)(p−n−1)/2e−12tr(Σ
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....
首先，log-convex函数定义为满足[公式]或[公式]，而log-concave函数则是满足[公式]或[公式]。log-convex性是凸函数的一个特性，如果一个函数是二阶可微的，可以通过二阶判定准则判断其为log-convex。
Moreover, the piecewise formulation that defines several popular activation functions, prevents the computation of convex/concave relaxations as they violate the factorable function requirement. To improve the performance of relaxations of ANNs for deterministic global optimization applications, this study ...
log-concave函数与log-concave函数在乘法、正数点乘以及积分运算下保持封闭性。这意味着如果两个函数都是log-concave函数，那么它们的乘积也是log-concave函数。log-convex函数的运算封闭性也适用于乘法、正数点乘和积分运算。但需要注意的是，两个log-concave函数直接相加不一定保持log-concave性质。结论 了解...
Convex and Concave Functions - Explore the concepts of convex and concave functions, their properties, and significance in optimization. Learn how these functions are used in various applications.
Convexandconcavefunctions 2.5.Convex and concave functions Let f be a real-valued function f:S→R.In what follows,we assume that S⊆R n is a convex set.The function f is convex on a set S if∀x1,x2∈S,∀λ∈[0,1]such thatλx1+(1−λ)x2∈S,f(λx1+(1−λ)x2)≤...
Convex optimization is the process of minimizing a convex objective function subject to convex constraints or, equivalently, maximizing a concave objective function subject to convex constraints. Points satisfying local optimality conditions can be found efficiently for many convex optimization problems. Becau...
Th. Composition of convex/concave functions Assume h:R→Rh:R→R and g:RN→Rg:RN→R are twice differentiable. Define f(x)=h(g(x)),∀x∈RNf(x)=h(g(x)),∀x∈RN, thenhh is convex & non-decreasing, gg is convex ⟹⟹ ff is convex hh is convex & non-increasing, gg is...
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...