James Renegar

A polynomial-time algorithm, based on Newton's method, for linear programming

A new interior method for linear programming is presented and a polynomial time bound for it is proven. The proof is substantially different from those given for the ellipsoid algorithm and for Karmarkars algorithm. Also, the algorithm is conceptually simpler than either of those algorithms.

On the computational complexity and geometry of the first-order theory of the reals. Par I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals

This series of papers presents a complete development and complexity analysis of a decision method, and a quantifier elimination method, for the first order theory of the reals. The complexity upper bounds which are established are the best presently available, both for sequential and parallel computation, and both for the bit model of computation and the real number model of computation; except for the bounds pertaining to the sequential decision method in the bit model of computation, all bounds represent significant improvements over previously established bounds.

Linear programming, complexity theory and elementary functional analysis

We propose analyzing interior-point methods using notions of problem-instance size which are direct generalizations of the condition number of a matrix. The notions pertain to linear programming quite generally; the underlying vector spaces are not required to be finite-dimensional and, more importantly, the cones defining nonnegativity are not required to be polyhedral. Thus, for example, the notions are appropriate in the context of semi-definite programming. We prove various theorems to demonstrate how the notions can be used in analyzing interior-point methods. These theorems assume little more than that the interiors of the cones (defining nonnegativity) are the domains of self-concordant barrier functions.

On the worst-case arithmetic complexity of approximating zeros of polynomials

Abstract Let Pd(R) denote the set of degree d complex polynomials with all zeros ζ satisfying |ζ| ≤ R. For d ≥ 2 fixed, we show that with respect to a certain model of computation, the worst-case computational complexity of obtaining an e-approximation either to one, or to each, zero of arbitrary f ∈ Pd(R) is Θ(log log(R/e)), that is, we prove both upper and lower bounds. A new algorithm, based on Newtons method, is introduced for proving the upper bound.

Hyperbolic Programs, and Their Derivative Relaxations

We study the algebraic and facial structures of hyperbolic programs, and examine natural relaxations of hyperbolic programs, the relaxations themselves being hyperbolic programs.

On the computational complexity and geometry of the first-order theory of the reals. Part III: quantifier elimination

This series of papers presents a complete development and complexity analysis of a decision method, and a quantifier elimination method, for the first order theory of the reals. The complexity upper bounds which are established are the best presently available, both for sequential and parallel computation, and both for the bit model of computation and the real number model of computation; except for the bounds pertaining to the sequential decision method in the bit model of computation, all bounds represent significant improvements over previously established bounds.

On the worst-case arithmetic complexity of approximating zeros of systems of polynomials

Let

A faster PSPACE algorithm for deciding the existential theory of the reals

d_1 , \cdots ,d_n

Computing approximate solutions for convex conic systems of constraints

be positive integers. Let

On the computational complexity and geometry of the first-order theory of the reals. Part II: The general decision problem. Preliminaries for quantifier elimination

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