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Optimization Over Symmetric Cones Under Uncertainty1.09 MB
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Dissertation
Optimization Over Symmetric Cones Under Uncertainty
Doctor of Philosophy (PhD), Washington State University
01/2011
Handle:
https://hdl.handle.net/2376/3489
Abstract
We introduce and study two-stage stochastic symmetric programs (SSPs) with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a logarithmic barrier decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proceeds by showing that the log barrier associated with the recourse function of SSPs behaves as a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao's results [Math. Program., Ser. A, 90:507-536, 2001] for two-stage stochastic linear programs, and Mehrotra and Ozevin's results [SIAM J. of Optimization, 18(1): 206-222, 2007] for two-stage stochastic semidefinite programs (SSDPs). We also present another class of polynomial-time decomposition algorithms for SSPs based on the volumetric barrier. While this extends the work of Ariyawansa and Zhu [Mathematics of computation, 80: 1639-1661, 2011] for SSDPs, our analysis is based on utilizing the advantage of the special algebraic structure associated with the symmetric cone not utilized in [Mathematics of computation, 80: 1639-1661, 2011]. As a consequence, we are able to significantly simplify the proofs of central results. We then describe four applications leading to the SSP problem where, in particular, the underlying symmetric cones are second-order cones and rotated quadratic cones.
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Details
- Title
- Optimization Over Symmetric Cones Under Uncertainty
- Creators
- Baha' Alzalg
- Contributors
- K A Ariyawansa (Advisor)Robert Mifflin (Committee Member)David S Watkins (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Number of pages
- 167
- Identifiers
- 99900581457001842
- Language
- English
- Resource Type
- Dissertation