O(m) Bound on Number of Iterations in Sphere Methods for LP

Consider the linear program (LP): minimize z = cx, subject to Ax ≥ b, where A is an m × n matrix. Sphere methods (SMs) for solving this LP were introduced in Murty [5, 6], even though this name was not used there. Theorems in those papers claimed that a version of this method needs at most O(m) iterations to solve this LP, however Mirzaian [2] pointed out an error in the proofs of these theorems there. Here we prove the claim using the geometry of inspheres. Also the results in this paper provide a solution to the special case of the open problem 2 in page 441 of the book Murty [7] dealing only with inspheres encountered in the SM

Adjacency on the constrained assignment problem

AbstractLet Qc,r be the integer hull of the intersection of the assignment polytope with a given hyper-plane H = {x = (xij) ϵ Rn × n: ∑ni = 1 ∑nj = 1 cijxij = r}. We show that the problem of checking whether two given extreme points of Qc,r are nonadjacent on Qc,r is solvable in O(n5) time if c = (cij) is a 0–1 matrix, and that it is NP-Complete if c is a general integer matrix

A fundamental problem in linear inequalities with applications to the travelling salesman problem

We consider a system of m linearly independent equality constraints in n nonnegative variables: Ax = b, x ≧ 0. The fundamental problem that we discuss is the following: suppose we are given a set of r linearly independent column vectors of A , known as the special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special column vectors as basic column vectors and to find such a basis if one exists. Such an algorithm has several applications in the area of mathematical programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates that this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47908/1/10107_2005_Article_BF01584550.pd

On the number of solutions to the complementarity problem and spanning properties of complementary cones

The relationship between the number of solutions to the complementarity problem, w = Mz + q, w[ges]0, z[ges]0, wTz=0, the right-hand constant vector q and the matrix M are explored. The main results proved in this work are summarized below.The number of solutions to the complementarity problem is finite for all q [epsilon] Rn if and only if all the principal subdeterminants of M are nonzero. The necessary and sufficient condition for this solution to be unique for each q [epsilon] Rn is that all principal subdeterminants of M are strictly positive. When M[ges]0, there is at least one complementary feasible solution for each q [epsilon] Rn if and only if all the diagonal elements of M are strictly positive; and, in this case, the number of these solutions is an odd number whenever q is nondegenerate. If all principal subdeterminants of M are nonzero, then the number of complementary feasible solutions has the same parity (odd or even) for all q [epsilon] Rn which are nondegenerate. Also, if the number of complementary feasible solutions is a constant for each q [epsilon] Rn, then that constant is equal to one and M is a P-matrix.In the cartesian system of coordinates for Rn, an orthant is a convex cone generated by a set of n-column vectors in Rn, {A.1,...,A.n}, where for each j = 1 to n,A.j is either the jth column vector of the unit matrix of order n (denoted by I.j) or its negative - I.j. There are thus 2n orthants in Rn, and they partition the whole space. It is interesting to know what properties these orthants possess if we obtain them after replacing - I.j by some given column vector - M.j for j = 1 to n. Orthants obtained in this manner are called complementary cones, and their spanning properties are studied.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/34188/1/0000477.pd

A Geometric Problem in Simplicial Cones with Applications to Linear Complementarity Problems

We consider the following geometric question: suppose we are given a simplicial cone K in R^n. Can we find a point @) in the interior of K satisfying the property that the orthogonal projection of @) onto the linear hull of every face of K is in the relative interior of that fence? This question plays an important role in determining whether a certain class of linear complementarity problems (LCP 's) can be solved efficiently by a pivotal algorithm. The answer to this question is always in the affirmative if n=2, but not so for n=3. We establish some conditions for the answer to this question to be yes, and relate them to other well known properties of square matrices. e.g., world: simplicial cones, orthogonal projections, faces, linear complementarity problem, LCP, pivotal algorithms, P-matrices, symmetric positive definite matrices, 2-matrices, M-matrices

A feasible direction method for linear programming

We discuss a finite method of a feasible direction for linear programming problems. The method begins with a feasible basic vector for the problem, constructs a profitable direction to move using the updated column vectors of the nonbasic variables eligible to enter this basic vector. It then moves in this direction as far as possible, while retaining feasibility. This move in general takes it though the relative interior of a face of th set of a feasible solutions. The final point, , obtained at the end of this move will not in general be a basic solution. Using the method then constructs a basic feasible solution at which the objective value is better than, or the same as that at . The whole process repeats with the new basic feasible solution. We show that this method can be implemented using basis inverses. Initial computer runs of this method in comparison with the usual edge following primary simplex algorithms are very encouraging.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24731/1/0000153.pd

Computational behavior of a feasible direction method for linear programming

We discuss a finite method of feasible directions for linear programs. The method begins with a BFS (basic feasible solution) and constructs a profitable direction by combining the updated columns of several nonbasic variables eligible to enter. Moving in this direction as far as possible, while retaining feasibility, leads to a point which is not in general a basic solution of the original problem, but corresponds to a BFS of an augmented problem with a new column. So this is called an interior move or a column adding move. Next we can either carry another interior move, or a reduction process which starts with the present feasible solution and leads to a BFS of the original problem with the same or better objective value. We show that interior moves and reduction processes can be mixed in many ways leading to different methods, all of which can be implemented by maintaining the basis inverse or a factorization of it. Results of a computational experiment are presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27884/1/0000298.pd

A critical index algorithm for nearest point problems on simplicial cones

We consider the linear complementarity problem ( q, M ) in which M is a positive definite symmetric matrix of order n. This problem is equivalent to a nearest point problem [ Γ; b ] in which Γ = { A . 1 , ⋯, A. n } is a basis for R n , b is a given point in R n ; and it is required to find the nearest point in the simplicial cone Pos( Γ ) to b. We develop an algorithm for solving the linear complementarity problem ( q, M ) or the equivalent nearest point problem [ Γ; b ]. Computational experience in comparison with an existing algorithm is presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47910/1/10107_2005_Article_BF01583789.pd

Computational complexity of parametric linear programming

We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47907/1/10107_2005_Article_BF01581642.pd

Segments in enumerating faces

We introduce the concept of a segment of a degenerate convex polytope specified by a system of linear constraints, and explain its importance in developing algorithms for enumerating the faces. Using segments, we describe an algorithm that enumerates all the faces, in time polynomial in their number. The role of segments in the unsolved problem of enumerating the extreme points of a convex polytope specified by a degenerate system of linear constraints, in time polynomial in the number of extreme points, is discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47929/1/10107_2005_Article_BF01585927.pd