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Gabor Pataki, UNC-Chapel Hill, Bad semidefinite programs, linear algebra, and short proofs

March 13, 2018 | 3:00 pm - 4:00 pm EDT

Semidefinite programs (SDPs) — optimization problems with linear constraints, linear objective, and semidefinite matrix variables —  are some of the most useful, versatile, and pervasive optimization problems to emerge in the last 30 years. They find applications in combinatorial optimization, machine learning, and statistics, to name just a few areas.

Unfortunately, SDPs often behave pathologically: the optimal values of the primal and dual problems may differ and  may not be attained. Such SDPs are  theoretically interesting and  numerically often difficult, or impossible to solve. Yet, the pathological SDPs in the literature look strikingly similar, and our recent paper (Pataki: Bad semidefinite programs: they all look the same, SIAM J. Opt 2017)) explained why: it characterized pathological semidefinite systems by certain {\em excluded matrices}, which are easy to spot in all published examples.

Here we give short and elementary proofs of these results using mostly techniques from elementary linear algebra. Our main tool is a standard (canonical) form of semidefinite systems, from which their pathological behavior is easy to verify. The standard form is constructed in a surprisingly simple manner, using mostly elementary row operations inherited from Gaussian elimination.

Details

Date:
March 13, 2018
Time:
3:00 pm - 4:00 pm EDT
Event Category:
Website:
https://arxiv.org/abs/1709.02423

Venue

SAS 4201