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Concave and Convex Functions
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
Geometrically, part (vii) is about a convex slice that a vertical hyperplane cuts out of the hypograph. Recall that a function is concave iff its hypograph is a convex set, and a hypograph is convex if every hyperplane intersecting it produces a slice that is a convex set.
Projection onto the exponential cone: a univariate root-finding problem
Published in Optimization Methods and Software, 2023
From a mathematical modelling perspective, the expressive abilities of the exponential cone is vast. For instance, given the simple relationship between and above, it should come as no surprise that both the exponential epigraph and logarithmic hypograph sets can be represented using the exponential cone, namely, More broadly, it can be used to represent convex compositions of exponentials, logarithms, entropy functions, product logarithms (such as Lambert W), softmax and softplus known from neural networks, and generalized posynomials known from geometric programming [4,13]. These constructions are usually made in conjunction with quadratic cones, but note that this is purely for the sake of simplicity and computational performance. In principle, the exponential cone is powerful enough to represent semidefiniteness of a symmetric matrix variable (which is an orthogonal transformation of a 3-dimensional quadratic cone) [3], and thus single-handedly represent all the above.
Weak subgradient method for solving nonsmooth nonconvex optimization problems
Published in Optimization, 2021
Gulcin Dinc Yalcin, Refail Kasimbeyli
Hence, we conclude that Geometrically this means that the hypograph of the function where is a supporting cone to the epigraph of f at the point Figure 1 illustrates the graph of the spiral function f and the supporting cone at the point
Minimal convex majorants of functions and Demyanov–Rubinov exhaustive super(sub)differentials
Published in Optimization, 2019
Let X be a real vector space. A concave function is a maximal concave minorant of an u-proper function if and only if its hypograph is a convex component of the hypograph of the function f.