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DTSTART:20260308T070000
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DTSTART;TZID=America/Toronto:20260806T130000
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URL:https://uwaterloo.ca/combinatorics-and-optimization/events/phd-defense-
 leo-jung-preprocessing-hard-optimization
SUMMARY:PhD Defense - Leo Jung - Preprocessing for Hard Optimization Proble
 ms\nAcross Structurally Diverse Models
CLASS:PUBLIC
DESCRIPTION:SPEAKER:\n\nLeo Jung\n\nLOCATION:\n MC 5029\n\nABSTRACT: \n\nD
 ifficulties in solving large-scale optimization problems often arise\nfrom
  structural pathologies such as ill-conditioning\, Hadamard\nill-posedness
 \, and degeneracy\, particularly due to the failure of\nconstraint qualifi
 cations. While standard algorithms often struggle to\naddress these issues
 \, preprocessing based on structural analysis\noffers an effective strateg
 y for overcoming such challenges. This\nthesis investigates several prepro
 cessing methods targeting various\nsources of these difficulties. \nIn Par
 t I\, we study a nonclassical\, average condition number of linear\nsystem
 s\, the $\\omega$-condition number. Our results demonstrate\nseveral advan
 tages of the $\\omega$-condition number over the classical\n$\\kappa$-cond
 ition number. First\, $\\omega$ provides a more accurate\nmeasure of the c
 onditioning of linear systems by more faithfully\ncapturing the effects of
  perturbations observed in practice. Second\,\n$\\omega$ exhibits superior
  numerical stability compared to $\\kappa$.\nThird\, when used in precondi
 tioner design\, $\\omega$ more effectively\npromotes eigenvalue clustering
 \, which is crucial for the efficiency of\niterative solvers. Finally\, th
 e analytical simplicity of $\\omega$\nenables the derivation of explicit o
 ptimality conditions\, allowing for\nclosed-form expressions of optimal pr
 econditioners under various\nframeworks\, including low rank updates of th
 e generalized Jacobian for\nsemismooth Newton methods and diagonal or bloc
 k-diagonal scaling. For\ndiagonal preconditioning\, we further include a c
 omparison between two\ndistinct notions of conditioning. \nIn Part II\, we
  first answer in the affirmative a long-standing open\nquestion of whether
  the smooth stress function admits local nonglobal\nminimizers. This quart
 ic nonconvex objective function arises in the\nexact recovery of a Euclide
 an distance matrix (EDM) of a given\nembedding dimension. By eliminating t
 he Hadamard ill-posedness caused\nby translation and rotation invariance\,
  we stabilize Newton's method\nand avoid singular Hessians. \\\\ \nWe then
  consider the single-element error correction problem as a case\nstudy. We
  first show that the standard nearest EDM formulation based\non minimizing
  the smooth stress function fails to recover the correct\nEDM in this sett
 ing. We then introduce divide-and-conquer strategies\nbased on facial redu
 ction. Our approach efficiently recovers the\ncorrect EDM with high accura
 cy\, and we further provide criteria\ncharacterizing the existence of mult
 iple solutions. \nIn Part III\, we relate FR to the analysis of the conver
 gence behaviour\nof a semismooth Newton method for projection onto a spect
 rahedron\,\ni.e.\, the intersection of a linear manifold and the semidefin
 ite cone.\nIn this process\, we derive an explicit formula for the project
 ion onto\na face of the semidefinite cone obtained via regularization and\
 nanalyze pathologies that arise in the absence of strict feasibility.\nWe 
 further show that ill-conditioning of the Jacobian near optimality\ncharac
 terizes the degeneracy of the projection point. \\\\ \nAs an application\,
  we consider a simplified Wasserstein barycenter\nproblem\, a well-known N
 P-hard problem. We compute the Wasserstein\nbarycenter by exploiting the s
 tructure of the linear constraints to\nobtain a facially reduced doubly no
 nnegative (DNN) relaxation. This\nreduction provides a natural splitting f
 or applying the symmetric\nalternating direction method of multipliers (sA
 DMM). The resulting\nalgorithm exploits structure in the subproblems to co
 mpute strong\nupper and lower bounds. In most of the instances\, we achiev
 e the small\ngap between these bounds\, which means that the original prob
 lem is\nsolved.
DTSTAMP:20260805T042147Z
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