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Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone. The class of conic optimization problems includes some of the most well known classes of convex optimization problems, namely linear and semidefinite programming.
A second-order cone program (SOCP) ... SCS (Splitting Conic Solver) is a numerical optimization package for solving large-scale convex quadratic cone problems.
The problem is named after its historical application by Allied forces in World War II to the estimation of the monthly rate of German tank production from very limited data. This exploited the manufacturing practice of assigning and attaching ascending sequences of serial numbers to tank components (chassis, gearbox, engine, wheels), with some ...
In mathematics, specifically in order theory and functional analysis, if is a cone at the origin in a topological vector space such that and if is the neighborhood filter at the origin, then is called normal if = [], where []:= {[]:} and where for any subset , []:= (+) is the -saturatation of . [1]
A subset of a vector space over an ordered field is a cone (or sometimes called a linear cone) if for each in and positive scalar in , the product is in . [2] Note that some authors define cone with the scalar ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). [3]
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration:
Fleming (1962) gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in R 3. De Giorgi (1965) showed that if there is no non-planar area-minimizing cone in R n−1 then the analogue of Bernstein's theorem is true for graphs in R n, which in particular implies that it is true in R 4.
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