Many problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important:[1]: Sec.2 optimization, violation, validity, separation, membership and emptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant.
In all problem descriptions, K denotes a compact and convex set in Rn.
^Grötschel, Martin; Lovász, László; Schrijver, Alexander (1993), Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, doi:10.1007/978-3-642-78240-4, ISBN 978-3-642-78242-8, MR 1261419
and 23 Related for: Algorithmic problems on convex sets information
Many problems in mathematical programming can be formulated as problemsonconvexsets or convex bodies. Six kinds of problems are particularly important:: Sec...
convex if it is possible to take convex combinations of points. Absorbing setAlgorithmicproblemsonconvexsets Bounded set (topological vector space) Brouwer...
numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities. Computing the convex hull...
this closure operator to finite sets of points. The algorithmicproblems of finding the convex hull of a finite set of points in the plane or other low-dimensional...
set must be found. They can include constrained problems and multimodal problems. An optimization problem can be represented in the following way: Given:...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer...
of a convex function. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which...
of the simplex algorithm. In the case in which Q is positive definite, the problem is a special case of the more general field of convex optimization....
points outside the feasible set. Convex feasible sets arise in many types of problems, including linear programming problems, and they are of particular...
greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy...
A convex polytope is a special case of a polytope, having the additional property that it is also a convexset contained in the n {\displaystyle n} -dimensional...
approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable...
sets, or when the sets are not convex, or that give faster convergence rates. Analysis of POCS and related methods attempt to show that the algorithm...
NPO(III)-problems are excluded from this class unless P=NP. Contains the set cover problem. NPO(V): :The class of NPO problems with polynomial-time algorithms...
vary, see "Dynamic problems". Yet another major class is the dynamic problems, in which the goal is to find an efficient algorithm for finding a solution...
research on specialized algorithms. A number of algorithms for other types of optimization problems work by solving linear programming problems as sub-problems...
their 49 city problem. While this paper did not give an algorithmic approach to TSP problems, the ideas that lay within it were indispensable to later...
(maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization...
the original on 2019-05-17. Gibbons, Alan (1985). Algorithmic Graph Theory. Cambridge University Press. Golumbic, Martin (1980). Algorithmic Graph Theory...
solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain...
Global Information