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Circuits play a fundamental role in polyhedral theory and linear programming.For instance, circuits are used as step directions in various augmentationschemes for solving linear programs or to leave degenerate vertices whilerunning the simplex method. However, there are significant challenges whenimplementing these approaches: The set of circuits of a polyhedron may be ofexponential size and is highly sensitive to the representation of thepolyhedron.  In this paper, we provide a universal framework for enumerating the set ofcircuits and optimizing over sets of circuits of a polyhedron in anyrepresentation - we propose a polyhedral model in which the circuits of theoriginal polyhedron are encoded as extreme rays or vertices. Many methods inthe literature and software assume that a polyhedron is in standard form; ourframework is a direct generalization. We demonstrate its value by showing thatthe conversion of a general representation to standard form may introduceexponentially many new circuits.  We then discuss the main advantages of the generalized polyhedral model. Itenables the direct enumeration of useful subsets of circuits such as strictlyfeasible circuits or sign-compatible circuits, as well as optimization overthese sets. In particular, this leads to the efficient computation of asteepest-descent circuit, which can be used in an augmentation scheme forsolving linear programs or the construction of sign-compatible circuit walkswith additional

A Polyhedral Model for Enumeration and Optimization over the Set of Circuits