mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

, the criss-cross algorithm is any of a family of

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s for

linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...

. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and

nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...

objective functions; there are criss-cross algorithms for

linear-fractional programming In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a rat ...

problems, quadratic-programming problems, and

linear complementarity problem In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968 ...

s. Like the

simplex algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are ...

of George B. Dantzig, the criss-cross algorithm is not a

polynomial-time algorithm In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations p ...

for linear programming. Both algorithms visit all 2^''D'' corners of a (perturbed)

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

in dimension ''D'', the Klee–Minty cube (after

Victor Klee Victor LaRue Klee, Jr. (September 18, 1925 – August 17, 2007) was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of ...

and George J. Minty), in the

worst case In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, b ...

. However, when it is started at a random corner, the criss-cross algorithm on average visits only ''D'' additional corners.The simplex algorithm takes on average ''D'' steps for a cube. : Thus, for the three-dimensional cube, the algorithm visits all 8 corners in the worst case and exactly 3 additional corners on average.

History

The criss-cross algorithm was published independently by Tamas Terlaky and by Zhe-Min Wang; related algorithms appeared in unpublished reports by other authors.

Comparison with the simplex algorithm for linear optimization

In linear programming, the criss-cross algorithm pivots between a sequence of bases but differs from the

. The simplex algorithm first finds a (primal-) feasible basis by solving a "''phase-one'' problem"; in "phase two", the simplex algorithm pivots between a sequence of basic ''feasible ''solutions so that the objective function is non-decreasing with each pivot, terminating with an optimal solution (also finally finding a "dual feasible" solution). The criss-cross algorithm is simpler than the simplex algorithm, because the criss-cross algorithm only has one phase. Its pivoting rules are similar to the least-index pivoting rule of Bland. Bland's rule uses only

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

s of coefficients rather than their (real-number) order when deciding eligible pivots. Bland's rule selects an entering variables by comparing values of reduced costs, using the real-number ordering of the eligible pivots. Unlike Bland's rule, the criss-cross algorithm is "purely combinatorial", selecting an entering variable and a leaving variable by considering only the signs of coefficients rather than their real-number ordering. The criss-cross algorithm has been applied to furnish constructive proofs of basic results in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, such as the lemma of Farkas. While most simplex variants are monotonic in the objective (strictly in the non-degenerate case), most variants of the criss-cross algorithm lack a monotone merit function which can be a disadvantage in practice.

Description

The criss-cross algorithm works on a standard pivot tableau (or on-the-fly calculated parts of a tableau, if implemented like the revised simplex method). In a general step, if the tableau is primal or dual infeasible, it selects one of the infeasible rows / columns as the pivot row / column using an index selection rule. An important property is that the selection is made on the union of the infeasible indices and the standard version of the algorithm does not distinguish column and row indices (that is, the column indices basic in the rows). If a row is selected then the algorithm uses the index selection rule to identify a position to a dual type pivot, while if a column is selected then it uses the index selection rule to find a row position and carries out a primal type pivot.

Computational complexity: Worst and average cases

The

time complexity In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations ...

of an

counts the number of

arithmetic operation Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Ar ...

s sufficient for the algorithm to solve the problem. For example,

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...

requires on the order of'' D''³ operations, and so it is said to have polynomial time-complexity, because its complexity is bounded by a

cubic polynomial In mathematics, a cubic function is a function (mathematics), function of the form f(x)=ax^3+bx^2+cx+d, that is, a polynomial function of degree three. In many texts, the ''coefficients'' , , , and are supposed to be real numbers, and the func ...

. There are examples of algorithms that do not have polynomial-time complexity. For example, a generalization of Gaussian elimination called Buchberger's algorithm has for its complexity an exponential function of the problem data (the degree of the polynomials and the number of variables of the

multivariate polynomial In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative intege ...

s). Because exponential functions eventually grow much faster than polynomial functions, an exponential complexity implies that an algorithm has slow performance on large problems. Several algorithms for linear programming— Khachiyan's ellipsoidal algorithm,

Karmarkar Narendra Krishna Karmarkar (born 1956) is an Indian mathematician. He developed Karmarkar's algorithm. He is listed as an ISI highly cited researcher. He invented one of the first probably Time complexity, polynomial time algorithms for linear ...

's projective algorithm, and central-path algorithms—have polynomial time-complexity (in the

and thus on average). The ellipsoidal and projective algorithms were published before the criss-cross algorithm. However, like the simplex algorithm of Dantzig, the criss-cross algorithm is ''not'' a polynomial-time algorithm for linear programming. Terlaky's criss-cross algorithm visits all the 2^''D'' corners of a (perturbed) cube in dimension ''D'', according to a paper of Roos; Roos's paper modifies the

Klee Paul Klee (; 18 December 1879 – 29 June 1940) was a Swiss-born German artist. His highly individual style was influenced by movements in art that included expressionism, cubism, and surrealism. Klee was a natural draftsman who experimented wi ...

–Minty construction of a

on which the simplex algorithm takes 2^''D'' steps. Like the simplex algorithm, the criss-cross algorithm visits all 8 corners of the three-dimensional cube in the worst case. When it is initialized at a random corner of the cube, the criss-cross algorithm visits only ''D'' additional corners, however, according to a 1994 paper by Fukuda and Namiki. Trivially, the simplex algorithm takes on average ''D'' steps for a cube. Like the simplex algorithm, the criss-cross algorithm visits exactly 3 additional corners of the three-dimensional cube on average.

Variants

The criss-cross algorithm has been extended to solve more general problems than linear programming problems.

Vertex enumeration

The criss-cross algorithm was used in an algorithm for enumerating all the vertices of a polytope, which was published by David Avis and Komei Fukuda in 1992. Avis and Fukuda presented an algorithm which finds the ''v'' vertices of a

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

defined by a nondegenerate system of ''n'' linear inequalities in ''D''

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

s (or, dually, the ''v''

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

s of the

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of ''n'' points in ''D'' dimensions, where each facet contains exactly ''D'' given points) in time O(''nDv'') and O(''nD'')

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

Oriented matroids

The criss-cross algorithm is often studied using the theory of oriented matroids (OMs), which is a

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

abstraction of linear-optimization theory.The theory of oriented matroids was initiated by R. Tyrrell Rockafellar. :

Rockafellar was influenced by the earlier studies of Albert W. Tucker and George J. Minty. Tucker and Minty had studied the sign patterns of the matrices arising through the pivoting operations of Dantzig's simplex algorithm.

Indeed, Bland's pivoting rule was based on his previous papers on oriented-matroid theory. However, Bland's rule exhibits cycling on some oriented-matroid linear-programming problems. The first purely combinatorial algorithm for linear programming was devised by Michael J. Todd. Todd's algorithm was developed not only for linear-programming in the setting of oriented matroids, but also for quadratic-programming problems and linear-complementarity problems. Todd's algorithm is complicated even to state, unfortunately, and its finite-convergence proofs are somewhat complicated. The criss-cross algorithm and its proof of finite termination can be simply stated and readily extend the setting of oriented matroids. The algorithm can be further simplified for ''linear feasibility problems'', that is for

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...

s with nonnegative variables; these problems can be formulated for oriented matroids. The criss-cross algorithm has been adapted for problems that are more complicated than linear programming: There are oriented-matroid variants also for the quadratic-programming problem and for the linear-complementarity problem.

Summary

The criss-cross algorithm is a simply stated algorithm for linear programming. It was the second fully combinatorial algorithm for linear programming. The partially combinatorial simplex algorithm of Bland cycles on some (nonrealizable) oriented matroids. The first fully combinatorial algorithm was published by Todd, and it is also like the simplex algorithm in that it preserves feasibility after the first feasible basis is generated; however, Todd's rule is complicated. The criss-cross algorithm is not a simplex-like algorithm, because it need not maintain feasibility. The criss-cross algorithm does not have polynomial time-complexity, however. Researchers have extended the criss-cross algorithm for many optimization-problems, including linear-fractional programming. The criss-cross algorithm can solve quadratic programming problems and linear complementarity problems, even in the setting of oriented matroids. Even when generalized, the criss-cross algorithm remains simply stated.

Notes

References

* * * * * * * * * * * *

External links

Komei Fukuda (ETH Zentrum, Zurich)
wit

Tamás Terlaky (Lehigh University)
wit
publications
{{Optimization algorithms, convex, state=collapsed Linear programming Oriented matroids Combinatorial optimization Optimization algorithms and methods Combinatorial algorithms Geometric algorithms Exchange algorithms