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Fast computation of zeros of polynomial systems with bounded degree under finite-precision

- Briquel, Irenee, Cucker, Felipe, Peña, Javier, Roshchina, Vera

**Authors:**Briquel, Irenee , Cucker, Felipe , Peña, Javier , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Mathematics of Computation Vol. 83, no. 287 (2014), p. 1279-1317**Full Text:**false**Reviewed:****Description:**A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite precision. In this paper we describe a finite-precision version of this algorithm. Our main result shows that this version works within the same time bounds and requires a precision which, on the average, amounts to a polynomial amount of bits in the mantissa of the intervening floating-point numbers. © 2013 American Mathematical Society.

A complementarity partition theorem for multifold conic systems

- Peña, Javier, Roshchina, Vera

**Authors:**Peña, Javier , Roshchina, Vera**Date:**2012**Type:**Text , Journal article**Relation:**Mathematical Programming Vol.142 , no.1-2 (2012), p.579-589**Full Text:**false**Reviewed:****Description:**Consider a homogeneous multifold convex conic system {Mathematical expression}and its alternative system {Mathematical expression}, where K 1,..., K r are regular closed convex cones. We show that there is a canonical partition of the index set {1,..., r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman-Tucker Theorem for linear programming. © 2012 Springer and Mathematical Optimization Society.

Some preconditioners for systems of linear inequalities

- Peña, Javier, Roshchina, Vera, Soheili, Negar

**Authors:**Peña, Javier , Roshchina, Vera , Soheili, Negar**Date:**2014**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 8, no. 7 (2014), p. 2145-2152**Full Text:**false**Reviewed:****Description:**We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities. © 2014, Springer-Verlag Berlin Heidelberg.

Solving second-order conic systems with variable precision

- Cucker, Felipe, Peña, Javier, Roshchina, Vera

**Authors:**Cucker, Felipe , Peña, Javier , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Mathematical Programming Vol. 150, no. 2 (2014), p. 217-250**Full Text:**false**Reviewed:****Description:**We describe and analyze an interior-point method to decide feasibility problems of second-order conic systems. A main feature of our algorithm is that arithmetic operations are performed with finite precision. Bounds for both the number of arithmetic operations and the finest precision required are exhibited. © 2014, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

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