Differentiable Convex Function Sufficient & Necessary Conditions for Global Optima Quasiconvex & Quasiconcave functions Differentiable Quasiconvex Function Strictly Quasiconvex Function Strongly Quasiconvex Function Pseudoconvex Function Convex Programming Problem Fritz-John Conditions Karush-Kuhn-Tucker Optimality Ne...
Proof of Theorem 1.1. Define a positive linear form F (u) = G(u) , then, we obG(g) viously have F (g) = 1. From Lemma 2.1, if we take as a convex function ϕ(x) = xp for p ≥ 1,...Pečarić,Matrix inequalities for convex functions - Mond, E - 1997 () ...
objective / cost function inequality constraints equality constraints feasible optimal value optimal point active inactive 【Why Convex Optimization?】 Contains various types of problems, e.g., many ML and OR tasks. Repeatability: different runs give the same results. Some convex problems can be solve...
2.1 smoothness of function \(g(\mathbf {u})\) besides the local restricted strong convexity, we can also prove the smoothness of \(g(\mathbf {u})\) , which is built in the following theorem. theorem 2 let \({\hat{l}}=2\vert \nabla f(\mathbf {v}\mathbf {v}^t)\vert _2+l...
Besides, Boolean function integer(x′(Ω)) returns a value of true if vector x′(Ω) is composed entirely of integers, and false, otherwise. Initially we put: (5.2.5)Ω={0≤xj≤Mj:j=1,2,…,k},i.e.,dj(Ω)=0,gj(Ω)=Mjj=1,2,…,k. Most of the remarks made for the ...
We give an analysis of the gradient method with steplengths satisfying the Armijo and Wolfe inexact line search conditions on the nonsmooth convex function f(x) = a|x^{(1)}| + \\sum_{i=2}^{n} x^{(i)} f(x) = a|x^{(1)}| + \\sum_{i=2}^{n} x^{(i)} . We show ...
1. Introduction Let f : I ⊆ ℝ → ℝ be a convex function defined on the interval I of real numbers and a
摘要: This paper studies the subdifferential of the composition of a nonde- creasing convex function with a mapping taking values in an ordered topological vector space. Necessary and sufficient conditions are derived for general convex optimisation problems....
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For the function composition f(g(x)) the compact notation f∘g∘x is applied. The set of invertible second order tensors with positive determinant is denoted by ≔GL+(3)≔{X∈R3×3|detX>0}, the special orthogonal group in R3 by ≔SO(3)≔{X∈R3×3|XTX=1,detX=1}....