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....
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−1x) ...
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。
Quasiconvex & Quasiconcave functions Differentiable Quasiconvex Function Strictly Quasiconvex Function Strongly Quasiconvex Function Pseudoconvex Function Convex Programming Problem Fritz-John Conditions Karush-Kuhn-Tucker Optimality Necessary Conditions Algorithms for Convex Problems Convex Optimization - Quick Gui...
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 ...
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)≤...
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...
,m are convex functions.Linear programs (LP) and convex quadratic programs (QP) are convex optimization problems. Conic optimization problems, where the inequality constraints are convex cones, are also convex optimization problems. Problems with linear or convex quadratic objectives and linear and ...
Convexoptimization problems 4–2Optimal and locally optimal points x is feasible if x∈domf0 and it satis,es the constraints a feasible x is optimal if f0(x)=p⋆;Xopt is the set of optimal points x is locally optimal if there is an R>0 such that x is optimal for ...