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18j-simplex - Algorithms Non-Lecture J Linear Programming...

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Algorithms Non-Lecture J: Linear Programming Algorithms Simplicibus itaque verbis gaudet Mathematica Veritas, cum etiam per se simplex sit Veritatis oratio. [And thus Mathematical Truth prefers simple words, because the language of Truth is itself simple.] — Tycho Brahe (quoting Seneca (quoting Euripides)) Epistolarum astronomicarum liber primus (1596) When a jar is broken, the space that was inside Merges into the space outside. In the same way, my mind has merged in God; To me, there appears no duality. — Sankara, Viveka-Chudamani (c. 700), translator unknown J Linear Programming Algorithms ? In this lecture, we’ll see a few algorithms for actually solving linear programming problems. The most famous of these, the simplex method , was proposed by George Dantzig in 1947. Although most variants of the simplex algorithm performs well in practice, no simplex variant is known to run in sub-exponential time in the worst case. However, if the dimension of the problem is considered a constant, there are several linear programming algorithms that run in linear time. I’ll describe a particularly simple randomized algorithm due to Raimund Seidel. My approach to describing these algorithms will rely much more heavily on geometric intuition than the usual linear-algebraic formalism. This works better for me, but your mileage may vary. For a more traditional description of the simplex algorithm, see Robert Vanderbei’s excellent textbook Linear Programming: Foundations and Extensions [ Springer, 2001 ] , which can be freely downloaded (but not legally printed) from the author’s website. J.1 Bases, Feasibility, and Local Optimality Consider the canonical linear program max { c · x | Ax b , x 0 } , where A is an n × d constraint matrix, b is an n -dimensional coefficient vector, and c is a d -dimensional objective vector. We will interpret this linear program geometrically as looking for the lowest point in a convex polyhedron in IR d , described as the intersection of n + d halfspaces. As in the last lecture, we will consider only non-degenerate linear programs: Every subset of d constraint hyperplanes intersects in a single point; at most d constraint hyperplanes pass through any point; and objective vector is linearly independent from any d - 1 constraint vectors. A basis is a subset of d constraints, which by our non-degeneracy assumption must be linearly independent. The location of a basis is the unique point x that satisfies all d constraints with equality; geometrically, x is the unique intersection point of the d hyperplanes. The value of a basis is c · x , where x is the location of the basis. There are precisely n + d d bases. Geometrically, the set of constraint hyperplanes defines a decomposition of IR d into convex polyhedra; this cell decomposition is called the arrangement of the hyperplanes. Every subset of d hyperplanes (that is, every basis) defines a vertex of this arrangement (the location of the basis). I will use the words ‘vertex’ and ‘basis’ interchangeably.

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