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80103 Numerical and Computational Mathematics
20802 Computation Theory and Mathematics
10101 Pure Mathematics
10906 Electrical and Electronic Engineering
1Arithmetic operations
1Billiard invisibility
1Billiards
1Computer science
1Condition numbers
1Conic feasibility system
1Conic system
1Cyclic projections
1De Pierro conjecture
1Demyanov converter
1Demyanov–Ryabova conjecture
1Difference of convex (delta-convex, DC) functions
1Differences of sets
1Directional derivatives
1Error bounds

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A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections

- Cominetti, Roberto, Roshchina, Vera, Williamson, Andrew

**Authors:**Cominetti, Roberto , Roshchina, Vera , Williamson, Andrew**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Vol. 68, no. 1 (2019), p. 3-12**Full Text:****Reviewed:****Description:**The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge. © 2018 Informa UK Limited, trading as Taylor & Francis Group.

**Authors:**Cominetti, Roberto , Roshchina, Vera , Williamson, Andrew**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Vol. 68, no. 1 (2019), p. 3-12**Full Text:****Reviewed:****Description:**The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by the ε-under-relaxed cyclic projection method converge when ε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets in R3 for which the ε-under-relaxed cycles do not converge. © 2018 Informa UK Limited, trading as Taylor & Francis Group.

The Demyanov–Ryabova conjecture is false

**Authors:**Roshchina, Vera**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 1 (2019), p. 227-234**Full Text:****Reviewed:****Description:**It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.

**Authors:**Roshchina, Vera**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 1 (2019), p. 227-234**Full Text:****Reviewed:****Description:**It was conjectured by Demyanov and Ryabova (Discrete Contin Dyn Syst 31(4):1273–1292, 2011) that the minimal cycle in the sequence obtained via repeated application of the Demyanov converter to a finite family of polytopes is at most two. We construct a counterexample for which the minimal cycle has length 4.

On local coincidence of a convex set and its tangent cone

- Meng, Kaiwen, Roshchina, Vera, Yang, Xiaoqi

**Authors:**Meng, Kaiwen , Roshchina, Vera , Yang, Xiaoqi**Date:**2015**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 164, no. 1 (2015), p. 123-137**Full Text:**false**Reviewed:****Description:**In this paper, we introduce the exact tangent approximation property for a convex set and provide its characterizations, including the nonzero extent of a convex set. We obtain necessary and sufficient conditions for the closedness of the positive hull of a convex set via a limit set defined by truncated upper level sets of the gauge function. We also apply the exact tangent approximation property to study the existence of a global error bound for a proper, lower semicontinuous and positively homogeneous function.

Bodies with mirror surface invisible from two points

- Plakhov, Andrew, Roshchina, Vera

**Authors:**Plakhov, Andrew , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Nonlinearity Vol. 27, no. 6 (June 2014), p. 1193-1203**Full Text:**false**Reviewed:****Description:**We consider a setting where a bounded set with a piecewise smooth boundary in Euclidean space is identified with a body with a mirror surface, and the billiard in the complement of the set is identified with the dynamics of light rays outside the body in the framework of geometric optics. We show that in this setting it is possible to construct a body invisible from two points. Â© 2014 IOP Publishing Ltd & London Mathematical Society.

Directed subdifferentiable functions and the directed subdifferential without Delta-convex structure

- Baier, Robert, Farkhi, Elza, Roshchina, Vera

**Authors:**Baier, Robert , Farkhi, Elza , Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 391-414**Full Text:**false**Reviewed:****Description:**We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the delta-convex structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions. © 2013 Springer Science+Business Media New York.

Facially exposed cones are not always nice

**Authors:**Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 257-268**Full Text:****Reviewed:****Description:**We address the conjecture proposed by GÃ¡bor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case; however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice.

**Authors:**Roshchina, Vera**Date:**2014**Type:**Text , Journal article**Relation:**SIAM Journal on Optimization Vol. 24, no. 1 (2014), p. 257-268**Full Text:****Reviewed:****Description:**We address the conjecture proposed by GÃ¡bor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case; however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice.

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.

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.

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.

Fractal bodies invisible in 2 and 3 directions

- Plakhov, Alexander, Roshchina, Vera

**Authors:**Plakhov, Alexander , Roshchina, Vera**Date:**2013**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series A Vol. 33, no. 4 (2013), p. 1615-1631**Full Text:****Reviewed:****Description:**We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 12], where two-dimensional bodies invisible in one direction and threedimensional bodies invisible in one and two orthogonal directions were constructed.**Description:**2003010679

**Authors:**Plakhov, Alexander , Roshchina, Vera**Date:**2013**Type:**Text , Journal article**Relation:**Discrete and Continuous Dynamical Systems - Series A Vol. 33, no. 4 (2013), p. 1615-1631**Full Text:****Reviewed:****Description:**We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1, 12], where two-dimensional bodies invisible in one direction and threedimensional bodies invisible in one and two orthogonal directions were constructed.**Description:**2003010679

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.

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