---
OA_place: publisher
_id: '21957'
abstract:
- lang: eng
  text: "This thesis investigates algorithmic certification and approximation methods
    for degenerate semidefinite programs (SDPs) and the singular roots of polynomial
    systems. In the first part, we present a hybrid symbolic-numeric algorithm for
    certifying the feasibility of weakly feasible, degenerate SDPs. By reformulating
    linear matrix inequalities (LMIs) into a structured polynomial system via facial
    reduction and incidence varieties, we guarantee the existence of an isolated exact
    solution. This algebraic reduction enables the certification of maximum-rank numerical
    approximations using methods from algebraic geometry.\r\n\r\nIn the second part,
    we address the severe ill-conditioning and loss of quadratic convergence that
    plague standard path-tracking methods near isolated singular roots. To overcome
    this, we propose tracking algorithms that achieve superlinear convergence without
    the computational bloat characteristic of classical deflation techniques. By modeling
    the solution path as a generalized fractional Puiseux series, our approach combines
    an explicitly derived algebraic predictor with a localized hyperplane desingularization
    phase during the corrector step. Furthermore, we introduce a continuous path-limit
    method and an extension of the geometric sequence rule to directly extract exact
    fractional exponents. This bypasses traditional heuristic trial-and-error methods
    and explicitly accommodates sparse series expansions. Numerical experiments confirm
    that our method significantly reduces the cumulative number of matrix inversions
    while achieving high-accuracy root approximations, even for heavily degenerate
    systems exhibiting higher coranks."
acknowledgement: 'Funding: Vienna Graduate School on Computational Optimization (FWF),
  grant DOI: 10.55776/W1260.'
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Jeferson
  full_name: Zapata, Jeferson
  id: 00223538-AF8F-11E9-A4C7-F729E6697425
  last_name: Zapata
citation:
  ama: 'Zapata J. Overcoming degeneracy and singularity: Techniques for semidefinite
    programs and homotopy continuation endgames. 2026. doi:<a href="https://doi.org/10.15479/AT-ISTA-21957">10.15479/AT-ISTA-21957</a>'
  apa: 'Zapata, J. (2026). <i>Overcoming degeneracy and singularity: Techniques for
    semidefinite programs and homotopy continuation endgames</i>. Institute of Science
    and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-21957">https://doi.org/10.15479/AT-ISTA-21957</a>'
  chicago: 'Zapata, Jeferson. “Overcoming Degeneracy and Singularity: Techniques for
    Semidefinite Programs and Homotopy Continuation Endgames.” Institute of Science
    and Technology Austria, 2026. <a href="https://doi.org/10.15479/AT-ISTA-21957">https://doi.org/10.15479/AT-ISTA-21957</a>.'
  ieee: 'J. Zapata, “Overcoming degeneracy and singularity: Techniques for semidefinite
    programs and homotopy continuation endgames,” Institute of Science and Technology
    Austria, 2026.'
  ista: 'Zapata J. 2026. Overcoming degeneracy and singularity: Techniques for semidefinite
    programs and homotopy continuation endgames. Institute of Science and Technology
    Austria.'
  mla: 'Zapata, Jeferson. <i>Overcoming Degeneracy and Singularity: Techniques for
    Semidefinite Programs and Homotopy Continuation Endgames</i>. Institute of Science
    and Technology Austria, 2026, doi:<a href="https://doi.org/10.15479/AT-ISTA-21957">10.15479/AT-ISTA-21957</a>.'
  short: 'J. Zapata, Overcoming Degeneracy and Singularity: Techniques for Semidefinite
    Programs and Homotopy Continuation Endgames, Institute of Science and Technology
    Austria, 2026.'
corr_author: '1'
das_tickbox: '1'
date_created: 2026-06-08T13:29:52Z
date_published: 2026-06-09T00:00:00Z
date_updated: 2026-07-27T14:30:42Z
day: '09'
ddc:
- '500'
degree_awarded: PhD
department:
- _id: GradSch
- _id: VlKo
doi: 10.15479/AT-ISTA-21957
doi_confirm: '1'
file:
- access_level: closed
  checksum: b11a959e99d3dcf61040282b5c837141
  content_type: application/zip
  creator: jzapata
  date_created: 2026-06-08T13:20:02Z
  date_updated: 2026-06-08T13:20:02Z
  file_id: '21958'
  file_name: istaustriathesis_JZapata.zip
  file_size: 40811933
  relation: source_file
- access_level: open_access
  checksum: edf1e5899b2e31505cd1aa3fe8bd4b7f
  content_type: application/pdf
  creator: jzapata
  date_created: 2026-06-10T13:33:25Z
  date_updated: 2026-06-10T13:33:25Z
  file_id: '21992'
  file_name: 4_Final_Thesis_JZapata_REX.pdf
  file_size: 2207892
  relation: main_file
  success: 1
file_date_updated: 2026-06-10T13:33:25Z
has_accepted_license: '1'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: '89'
project:
- _id: 9B9290DE-BA93-11EA-9121-9846C619BF3A
  grant_number: W1260-N35
  name: Vienna Graduate School on Computational Optimization
publication_identifier:
  isbn:
  - 978-3-99078-079-4
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '21144'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Vladimir
  full_name: Kolmogorov, Vladimir
  id: 3D50B0BA-F248-11E8-B48F-1D18A9856A87
  last_name: Kolmogorov
title: 'Overcoming degeneracy and singularity: Techniques for semidefinite programs
  and homotopy continuation endgames'
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2026'
...