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Modeling tools are separate pieces of software that let the user specify an optimization in higher-level syntax. They manage all transformations to and from the user's high-level model and the solver's input/output format. The table below shows a mix of modeling tools (such as CVXPY and Convex.jl) and solvers (such as CVXOPT and MOSEK).
For a convex function , the sublevel sets {: <} and {: ()} with are convex sets. A function that satisfies this property is called a quasiconvex function and may fail to be a convex function. Consequently, the set of global minimisers of a convex function f {\displaystyle f} is a convex set: argmin f {\displaystyle {\operatorname {argmin} }\,f ...
The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. Given a set X, a convexity over X is a collection 𝒞 of subsets of X satisfying the following axioms: [9] [10] [23] The empty set and X are in 𝒞. The intersection of any collection from 𝒞 is in 𝒞.
is a convex set. [2] The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis . Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures.
A set of convex-shaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a convex set.
Bauer's maximum principle is the following theorem in mathematical optimization: . Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at some extreme point of that set.
The classes of s-convex measures form a nested increasing family as s decreases to −∞" . or, equivalently {} {}.Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class.
For every proper convex function : [,], there exist some and such that ()for every .. The sum of two proper convex functions is convex, but not necessarily proper. [4] For instance if the sets and are non-empty convex sets in the vector space, then the characteristic functions and are proper convex functions, but if = then + is identically equal to +.