Elementary reformulation and succinct certificates in conic linear programming Public Deposited

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  • March 19, 2019
Creator
  • Liu, Minghui
    • Affiliation: College of Arts and Sciences, Department of Statistics and Operations Research
Abstract
  • The first part of this thesis deals with infeasibility in semidefinite programs (SDPs). In SDP, unlike in linear programming, Farkas’ lemma may fail to prove infeasibility. We obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any equality constrained semidefinite system using only elementary row operations, and rotations. When a system is infeasible, the reformulated system is trivially infeasible. When a system is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. The second part is about simple and exact duals, and certificates of infeasibility and weak infeasibility in conic linear programming that do not rely on any constraint qualifi- cation and retain most of the simplicity of the Lagrange dual. Some of our infeasibility certificates generalize the row echelon form of a linear system of equations, as they consist of a small, trivially infeasible subsystem. The “easy” proofs – as sufficiency of a certificate to prove infeasibility – are elementary. We also derived some fundamental geometric corollaries: 1) an exact characterization of when the linear image of a closed convex cone is closed, 2) an exact characterization of nice cones, and 3) bounds on the number of constraints that can be dropped from, or added to a (weakly) infeasible conic LP while keeping it (weakly) infeasible. Using our infeasibility certificates we generated a public domain library of infeasible and weakly infeasible SDPs. The status of our instances is easy to verify by inspection in exact arithmetic, but they turn out to be challenging for commercial and research codes.
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  • In Copyright
Advisor
  • Budhiraja, Amarjit
  • Bhamidi, Shankar
  • Pataki, Gabor
  • Provan, Scott
  • Lu, Shu
Degree
  • Doctor of Philosophy
Degree granting institution
  • University of North Carolina at Chapel Hill Graduate School
Graduation year
  • 2015
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  • Chapel Hill, NC
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