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Infinite-Dimensional Relaxations of ...
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ProQuest Information and Learning Co.
Infinite-Dimensional Relaxations of Mixed-Integer Optimization Problems.
紀錄類型:
書目-語言資料,手稿 : Monograph/item
正題名/作者:
Infinite-Dimensional Relaxations of Mixed-Integer Optimization Problems./
作者:
Zhou, Yuan.
面頁冊數:
1 online resource (205 pages)
附註:
Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.
Contained By:
Dissertation Abstracts International79-01B(E).
標題:
Applied mathematics. -
電子資源:
click for full text (PQDT)
ISBN:
9780355152005
Infinite-Dimensional Relaxations of Mixed-Integer Optimization Problems.
Zhou, Yuan.
Infinite-Dimensional Relaxations of Mixed-Integer Optimization Problems.
- 1 online resource (205 pages)
Source: Dissertation Abstracts International, Volume: 79-01(E), Section: B.
Thesis (Ph.D.)
Includes bibliographical references
Optimization problems with integer variables form a class of mathematical models that are widely used in Operations Research and Mathematical Analytics. They provide a great modeling power, but it comes at a high price: Integer optimization problems are typically very hard to solve, both in theory and practice.
Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018
Mode of access: World Wide Web
ISBN: 9780355152005Subjects--Topical Terms:
1069907
Applied mathematics.
Index Terms--Genre/Form:
554714
Electronic books.
Infinite-Dimensional Relaxations of Mixed-Integer Optimization Problems.
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Optimization problems with integer variables form a class of mathematical models that are widely used in Operations Research and Mathematical Analytics. They provide a great modeling power, but it comes at a high price: Integer optimization problems are typically very hard to solve, both in theory and practice.
520
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A key ingredient in state-of-the-art solvers for integer optimization problems are cutting-plane algorithms. The need for next-generation cutting planes, prompted by ever-larger applications and models, has led to a recent trend, which revisits and extends foundational work in integer optimization by Ralph Gomory in the 1960s and Gomory--Johnson in the early 1970s, under the name of cut-generating functions. In this dissertation, we focus on the cut-generating functions in the single row Gomory--Johnson infinite group relaxation model for integer optimization problems.
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The proofs of cutting planes theorems in the literature were hand-written, and were dominated by tedious and error-prone case analysis. We ask how much of this process can be automated: In particular, can we use algorithms to discover and prove theorems about cutting planes?
520
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We develop a software for investigations with cut-generating functions, which implements a grid-free algorithmic extremality test. We conduct a computer-based search that leads to the discovery of new cut-generating functions, whose existence settles several open questions. Using a metaprogramming technique and semialgebraic computations, we provide computer-based proofs for old and new cutting-plane theorems.
520
$a
This dissertation also addresses the theoretical aspects of cut-generating functions. We improve the general theory of effective perturbations of minimal valid functions, which enables the grid-free algorithmic extremality test. We investigate the fine structure of a space of cut-generating functions, underlining the exceptional place that two-sided discontinuous functions take in the theory of the Gomory--Johnson functions. We establish a clear relation of three competing notions of facets in the infinite-dimensional Gomory--Johnson model, resolving a longstanding mystery.
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2018
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