A separation oracle (also called a cutting-plane oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization algorithm. Separation oracles are used as input to ellipsoid methods.[1]: 87, 96, 98 

Definition

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Let K be a convex and compact set in Rn. A strong separation oracle for K is an oracle (black box) that, given a vector y in Rn, returns one of the following:[1]: 48 

  • Assert that y is in K.
  • Find a hyperplane that separates y from K: a vector a in Rn, such that   for all x in K.

A strong separation oracle is completely accurate, and thus may be hard to construct. For practical reasons, a weaker version is considered, which allows for small errors in the boundary of K and the inequalities. Given a small error tolerance d>0, we say that:

  • A vector y is d-near K if its Euclidean distance from K is at most d;
  • A vector y is d-deep in K if it is in K, and its Euclidean distance from any point in outside K is at least d.

The weak version also considers rational numbers, which have a representation of finite length, rather than arbitrary real numbers. A weak separation oracle for K is an oracle that, given a vector y in Qn and a rational number d>0, returns one of the following::[1]: 51 

  • Assert that y is d-near K;
  • Find a vector a in Qn, normalized such that its maximum element is 1, such that   for all x that are d-deep in K.

Implementation

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A special case of a convex set is a set represented by linear inequalities:  . Such a set is called a convex polytope. A strong separation oracle for a convex polytope can be implemented, but its run-time depends on the input format.

Representation by inequalities

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If the matrix A and the vector b are given as input, so that  , then a strong separation oracle can be implemented as follows.[2] Given a point y, compute  :

  • If the outcome is at most  , then y is in K by definition;
  • Otherwise, there is at least one row   of A, such that   is larger than the corresponding value in  ; this row   gives us the separating hyperplane, as   for all x in K.

This oracle runs in polynomial time as long as the number of constraints is polynomial.

Representation by vertices

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Suppose the set of vertices of K is given as an input, so that   the convex hull of its vertices. Then, deciding whether y is in K requires to check whether y is a convex combination of the input vectors, that is, whether there exist coefficients z1,...,zk such that: [1]: 49 

  •  ;
  •   for all i in 1,...,k.

This is a linear program with k variables and n equality constraints (one for each element of y). If y is not in K, then the above program has no solution, and the separation oracle needs to find a vector c such that

  •   for all i in 1,...,k.

Note that the two above representations can be very different in size: it is possible that a polytope can be represented by a small number of inequalities, but has exponentially many vertices (for example, an n-dimensional cube). Conversely, it is possible that a polytope has a small number of vertices, but requires exponentially many inequalities (for example, the convex hull of the 2n vectors of the form (0,...,±1,...,0).

Problem-specific representation

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In some linear optimization problems, even though the number of constraints is exponential, one can still write a custom separation oracle that works in polynomial time. Some examples are:

  • The minimum-cost arborescence problem: given a weighted directed graph and a vertex r in it, find a subgraph of minimum cost that contains a directed path from r to any other vertex. The problem can be presented as an LP with a constraint for each subset of vertices, which is an exponential number of constraints. However, a separation oracle can be implemented using n-1 applications of the minimum cut procedure.[3]
  • The maximum independent set problem. It can be approximated by an LP with a constraint for every odd-length cycle. While there are exponentially-many such cycles, a separation oracle that works in polynomial time can be implemented by just finding an odd cycle of minimum length, which can be done in polynomial time.[3]
  • The dual of the configuration linear program for the bin packing problem. It can be approximated by an LP with a constraint for each feasible configuration. While there are exponentially-many such cycles, a separation oracle that works in pseudopolynomial time can be implemented by solving a knapsack problem. This is used by the Karmarkar-Karp bin packing algorithms.

Non-linear sets

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Let f be a convex function on Rn. The set   is a convex set in Rn+1. Given an evaluation oracle for f (a black box that returns the value of f for every given point), one can easily check whether a vector (y, t) is in K. In order to get a separation oracle, we need also an oracle to evaluate the subgradient of f.[1]: 49  Suppose some vector (y, s) is not in K, so f(y) > s. Let g be the subgradient of f at y (g is a vector in Rn). Denote  .Then,  , and for all (x, t) in K:  . By definition of a subgradient:   for all x in Rn. Therefore,  , so   , and c represents a separating hyperplane.

Usage

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A strong separation oracle can be given as an input to the ellipsoid method for solving a linear program. Consider the linear program  . The ellipsoid method maintains an ellipsoid that initially contains the entire feasible domain  . At each iteration t, it takes the center   of the current ellipsoid, and sends it to the separation oracle:

  • If the oracle says that   is feasible (that is, contained in the set  ), then we do an "optimality cut" at  : we cut from the ellipsoid all points x for which  . These points are definitely not optimal.
  • If the oracle says that   is infeasible, then it typically returns a specific constraint that is violated by  , that is, a row   in the matrix A, such that  . Since   for all feasible x, this implies that   for all feasible x. Then, we do a "feasibility cut" at  : we cut from the ellipsoid all points y for which  . These points are definitely not feasible.

After making a cut, we construct a new, smaller ellipsoid, that contains the remaining region. It can be shown that this process converges to an approximate solution, in time polynomial in the required accuracy.

Converting a weak oracle to a strong oracle

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Given a weak separation oracle for a polyhedron, it is possible to construct a strong separation oracle by a careful method of rounding, or by diophantine approximations.[1]: 159 

See also

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References

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  1. ^ a b c d e f 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
  2. ^ "MIT 6.854 Spring 2016 Lecture 12: From Separation to Optimization and Back; Ellipsoid Method - YouTube". www.youtube.com. Retrieved 2021-01-03.
  3. ^ a b Vempala, Santosh (2016). "Separation oracle" (PDF).