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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 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.
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.
Graduated optimization is commonly used in image processing for locating objects within a larger image. This problem can be made to be more convex by blurring the images. . Thus, objects can be found by first searching the most-blurred image, then starting at that point and searching within a less-blurred image, and continuing in this manner until the object is located with precision in the ...
Bregman divergences between functions include total squared error, relative entropy, and squared bias; see the references by Frigyik et al. below for definitions and properties. Similarly Bregman divergences have also been defined over sets, through a submodular set function which is known as the discrete analog of a convex function.
Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i {\displaystyle m_{i}} modified to ensure the monotonicity of the resulting Hermite spline.
The policy is characterized by two numbers s and S, , such that when the inventory level falls below level s, an order is issued for a quantity that brings the inventory up to level S, and nothing is ordered otherwise. Gallego and Sethi [2] have generalized the concept of K-convexity to higher dimensional Euclidean spaces.