Invited Session Tue.2.H 2036

Tuesday, 13:15 - 14:45 h, Room: H 2036

Cluster 4: Conic programming [...]

Advances in convex optimization


Chair: Javier Pena



Tuesday, 13:15 - 13:40 h, Room: H 2036, Talk 1

Luis Zuluaga
Positive polynomials on unbounded domains

Coauthors: Javier F. Peña, Juan C. Vera


Certificates of non-negativity are fundamental tools in optimization. A "certificate'' is generally understood as an expression that makes the non-negativity of the function in question evident. Recently, sum-of-squares certificates of non-negativity for polynomials have been used to obtain powerful numerical techniques for solving polynomial optimization problems; in particular, for mixed integer programs, and non-convex binary programs. We present a new certificate of non-negativity for polynomials over the intersection of a closed set S and the zero set of a given polynomial h(x). The certificate is written in terms of the set of non-negative polynomials over S and the ideal generated by h(x). Our certificate of non-negativity yields a copositive programming reformulation for a very general class of polynomial optimization problems.



Tuesday, 13:45 - 14:10 h, Room: H 2036, Talk 2

Martin Lotz
Conditioning of the convex feasibility problem and sparse recovery

Coauthor: Dennis Amelunxen


The problem of whether certain simple or sparse solutions to linear systems of equations can be found or approximated efficiently can often be cast in terms of a convex feasibility problem. In particular, condition numbers introduced for the complexity analysis of conic optimization problems play an important role in the analysis of such problems. We present results and geometric methods from the probabilistic analysis of condition numbers for optimization problems, and indicate how this analysis can be used to obtain sparse and simple recovery thresholds for problems with noise.



Tuesday, 14:15 - 14:40 h, Room: H 2036, Talk 3

Javier Pena
A smooth primal-dual perceptron-von Neumann algorithm

Coauthor: Negar Soheili


We propose an elementary algorithm for solving a system of linear
inequalities ATy > 0 or its alternative Ax ≥ 0, x ≠ 0.
Our algorithm is a smooth version of the perceptron and von Neumann's
algorithms. Our algorithm retains the simplicity of these algorithms
but has a significantly improved convergence rate.


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