Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series

Counting and computing regions of DD-decomposition: algebro-geometric approach

New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families of polynomial and matrices is given. For the case of two parametric family more sharp estimate is proven. Theoretic results are supported by various numerical simulations that show higher precision of presented methods with respect to traditional ones. The presented methods are inherently global and could be applied for studying $D$-decomposition for the space of parameters as a whole instead of some prescribed regions. For symbolic computations the Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure

Bounding the rational LS-category of certain spaces via the Hilbert function

AbstractLet (ΛV,d) be the Sullivan model of a pure elliptic space S. We give an algorithm, based on the computation of the Hilbert function of certain ideal, that provides a lower and an upper bound for the rational L.–S. category of S,cat0(ΛV,d). When S has null Euler homotopy characteristic, both bounds are sharp

Computing the Sullivan Milnor–Moore S.S. and the rational LS category of certain spaces

AbstractLet (ΛV,d) be the Sullivan model of an elliptic space S and (ΛV,dσ) be the associated pure model. We give an algorithm, based on Groebner basis computations, that computes the stage lσ=l0(ΛV,dσ) at which the (Sullivan version of the) Milnor–Moore spectral sequence of (ΛV,dσ) collapses. When (d−dσ)V⊂Λ>lσV we call S a Ginsburg space. We show that the rational LS category of any Ginsburg space S, cat0(ΛV,d), coincides with that of the associated pure space cat0(ΛV,dσ). A previous algorithm due to the author computes cat0(ΛV,dσ). So we obtain an algorithm that determines whether a space is Ginsburg and which in this case computes its rational LS category

Note on Integer Factoring Methods IV

This note continues the theoretical development of deterministic integer factorization algorithms based on systems of polynomials equations. The main result establishes a new deterministic time complexity bench mark in integer factorization.Comment: 20 Pages, New Versio

Alexander Invariants of Complex Hyperplane Arrangements

Let A be an arrangement of complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism from a finitely generated free group to the pure braid group. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.Comment: 26 pages; LaTeX2e with amscd, amssymb package

A straightening algorithm for row-convex tableaux

We produce a new basis for the Schur and Weyl modules associated to a row-convex shape, D. The basis is indexed by new class of "straight" tableaux which we introduce by weakening the usual requirements for standard tableaux. Spanning is proved via a new straightening algorithm for expanding elements of the representation into this basis. For skew shapes, this algorithm specializes to the classical straightening law. The new straight basis is used to produce bases for flagged Schur and Weyl modules, to provide Groebner and sagbi bases for the homogeneous coordinate rings of some configuration varieties and to produce a flagged branching rule for row-convex representations. Systematic use of supersymmetric letterplace techniques enables the representation theoretic results to be applied to representations of the general linear Lie superalgebra as well as to the general linear group.Comment: 31 pages, latex2e, submitted to J. Algebr