Violator Spaces: Structure and Algorithms

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006; author spelling fixe

Computational complexity of LCPs associated with positive definite symmetric matrices

Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of order n for n ≥ 2—he has shown that to solve the problem of order n , either of the two methods goes through 2 n pivot steps before termination.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47905/1/10107_2005_Article_BF01588254.pd

Finite reduction and Morse index estimates for mechanical systems

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover, the Hessian of the reduced function preserves all the relevant information of the original one, by Schur's complement, which spontaneously appears in this context. Finally, the results are straightforwardly extended to the case of a Dirichlet problem on a bounded domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential Equations and Application

Fermi's golden rule and exponential decay as a RG fixed point

We discuss the decay of unstable states into a quasicontinuum using models of the effective Hamiltonian type. The goal is to show that exponential decay and the golden rule are exact in a suitable scaling limit, and that there is an associated renormalization group (RG) with these properties as a fixed point. The method is inspired by a limit theorem for infinitely divisible distributions in probability theory, where there is a RG with a Cauchy distribution, i.e. a Lorentz line shape, as a fixed point. Our method of solving for the spectrum is well known; it does not involve a perturbation expansion in the interaction, and needs no assumption of a weak interaction. We use random matrices for the interaction, and show that the ensemble fluctuations vanish in the scaling limit. Thus the limit is the same for every model in the ensemble with probability one.Comment: 20 pages, 1 figur

Generic properties of the complementarity problem

Given f : R + n → R n , the complementarity problem is to find a solution to x ≥ 0, f(x) ≥ 0, and 〈 x, f(x) 〉 = 0. Under the condition that f is continuously differentiable, we prove that for a generic set of such an f , the problem has a discrete solution set. Also, under a set of generic nondegeneracy conditions and a condition that implies existence, we prove that the problem has an odd number of solutions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47915/1/10107_2005_Article_BF01584674.pd

NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.Comment: 20 page

A finite characterization of K -matrices in dimensions less than four

The class of real n × n matrices M , known as K -matrices, for which the linear complementarity problem w − Mz = q, w ≥ 0, z ≥ 0, w T z =0 has a solution whenever w − Mz =q, w ≥ 0, z ≥ 0 has a solution is characterized for dimensions n <4. The characterization is finite and ‘practical’. Several necessary conditions, sufficient conditions, and counterexamples pertaining to K -matrices are also given. A finite characterization of completely K -matrices ( K -matrices all of whose principal submatrices are also K -matrices) is proved for dimensions <4.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47913/1/10107_2005_Article_BF01589438.pd

CP-rays in simplicial cones

An interior point of a triangle is called CP-point if its orthogonal projection on the line containing each side lies in the relative interior of that side. In classical mathematics, interest in the concept of regularity of a triangle is mainly centered on the property of every interior point of the triangle being a CP-point. We generalize the concept of regularity using this property, and extend this work to simplicial cones in ℝ n , and derive necessary and sufficient conditions for this property to hold in them. These conditions highlight the geometric properties of Z-matrices. We show that these concepts have important ramifications in algorithmic studies of the linear complementarity problem. We relate our results to other well known properties of square matrices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47921/1/10107_2005_Article_BF01582265.pd