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Distances to Lattice Points and Integer Linear Programming

 

I received my PhD in mathematics from Cardiff University at the start of 2022 under the supervision of Professor Iskander Aliev. My PhD thesis entitled "distances to lattice points in rational polyhedra" is available below. My current research lies within the field of integer optimisation. This mathematical discipline looks at optimisation or feasibility problems where some or all of the variables are restricted to integers. Because integer optimisation is concerned with discrete values, it is unsurprising that it is widely used to model problems in many real world scenarios including operational research tasks such as scheduling and manufacturing.

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I currently focus on 'integer linear programming' (ILP) problems. These problems have the property that both the constraints and objective functions are linear. It is well-known that in general, finding a solution to an ILP is NP-complete. Despite this, one can obtain an approximation within polynomial time by solving the problem's related 'linear program' (LP). One of my aims is to under certain assumptions refine current worst-case upper bounds for the distance between an optimal solution to the LP to some nearby feasible integer solution, which solves the ILP. Further, I am actively interested in how the right-hand side of an ILP impacts on the feasibility of the problem and the sparsity of the underlying solutions. 

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For those who are interested, my current Erdös Number is 3 (meaning I am a coauthor of a coauthor of a coauthor of Erdös).  In particular, the connection to Erdös is A. Williams => I. Aliev => P. Gruber => P. Erdös.

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Publications & Preprints

  • Insights to Weighted Sum Sampling Approaches for Multi-Criteria Decision Making Problems

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A. Williams and Y. Cai (2024)​

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[arXiv.org]

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  • Markov Chain Monte Carlo Methods for Dynamic Student Assessment

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A. Williams (2024)

 

[EdArXiv.org]

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  • Identifying Job Satisfaction Parameters among the Employees in Higher Educational Institutions: A Mathematical Model​

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M. Bhatia and A. Williams (2024)

 

 [WSEAS Transactions on Business and Economics] ; [arXiv.org]

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  • Considering a Classical Upper Bound on the Frobenius Number

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A. Williams and D. Haijima (2023)

 

[arXiv.org]

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  • Selection of Criteria Using MCDM Techniques - An Application in Renewable Energy​​
     

M. Bhatia and A. Williams (2023)

 

[American Journal of Operations Management and Information Systems] ; [arXiv.org]

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  • Target Setting: A Case Study Looking at How Greater Collaboration Can Impact the Effectiveness of Target Setting​
     

A. Williams (2022)

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[WSEAS Transactions on Business and Economics]
 

  • Finding an Optimal Proximity Bound in a Very Special Scenario 

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A. Williams (2022)

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[WSEAS Transactions on Mathematics]

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  • Distances to Lattice Points in Rational Polyhedra (PhD Thesis)


A. Williams (2022)

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[Online Research @ Cardiff]

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  • Intelligent Security Control Based on the New Criterion of Edwards and Montgomery Curves, Isogenous  of These Curves Supersingularity

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R. Skuratovskii, A. Williams and Y. Osadchyy (2021)

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[Proceedings of International Conference on Advanced Computing Applications]​
 

  • Distance-Sparsity Transference for Vertices of Corner Polyhedra

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I. Aliev, M. Celaya, M. Henk and A. Williams (2021)

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[SIAM Journal on Optimization] ; [arXiv.org]​

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  • Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band

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R. Skuratovskii and A. Williams (2020)

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[Rendiconti del Circolo Matematico di Palermo Series 2 (Springer)] 

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  • Some Approach to Key Exchange Protocol Based on Non-Commutative Groups

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R. Skuratovskii and A. Williams (2020)

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[International Journal of Mathematical Models and Methods in Applied Sciences] ; [arXiv.org]

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  • A Solution of the Inverse Problem to Doubling of Twisted Edwards Curve Point over Finite Field

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R. Skuratovskii and A. Williams (2019)

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[Wydawnictwo Naukowe Akademii Techniczno-Humanistycznej w Bielsku-Białej]

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  • Minimal Generating Set and a Structure of the Wreath Product of Groups, and the Fundamental Group of the Orbit Morse Function

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R. Skuratovskii and A. Williams (2019)

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[Вісник Донецького національного університету. Серія А: Природничі науки] ; [arXiv.org]

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Conferences & Workshops

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Contributed Talk:

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  • International Meeting on Numerical Semigroups (IMNS) | Universidad de Cádiz (2024)

  • 33rd European Conference on Operational Research | Technical University of Denmark (2024)

  • 75th British Mathematical Colloquium (Number Theory Workshop) | University of Manchester (2024)

  • Sharing Best Practices in Teaching (LSE ECR Network's Peer Learning Café) | LSE (2023)

  • Seminar on Combinatorics, Games and Optimisation | LSE (2023)

  • 6th European Conference on Electrical Engineering and Computer Science | Bern, Switzerland (2022)

  • PhD Seminar on Combinatorics, Games and Optimisation | LSE (2022)

  • World Conference on Information Systems for Business Management | Bangkok, Thailand (2022)

  • 7th International Conference on Mathematics and Computers in Sciences and Industry | Athens, Greece (2022)

  • British Early Career Mathematicians' Colloquium | University of Birmingham (2021)

  • Young Researchers in Mathematics | University of Bristol (2021)

  • International Conference of Young Mathematicians | Online Kiev, Ukraine (2021)

  • British Early Career Mathematicians' Colloquium | University of Birmingham (2020)

  • Algebraic and Geometric Methods of Analysis | Odessa, Ukraine (2020)

  • Intellectual Systems and Information Technologies | Odessa, Ukraine (2019)

  • British Combinatorial Conference | University of Birmingham (2019)

  • SIAM-UKIE National Student Chapter Conference | University of Manchester (2019)

  • International Conference of Young Mathematicians | Institute of Mathematics NAS of Ukraine (2019)

  • Welsh Mathematics Colloquium | Gregynog Hall (2019)

  • British Young Mathematicians Colloquium | University of Birmingham (2019)

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Contributed Poster:

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  • ​SIAM-UKIE National Student Chapter Conference | University of Manchester (2019)

  • SIAM-IMA Student Chapter Conference | Oxford University (2019)

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Attended:

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  • Induction Course for New Lecturers in the Mathematical Sciences | University of Cambridge (2022)

  • NATCOR Multi Criteria Decision Making | University of Portsmouth (2022)

  • IPCO and Summer School | London School of Economics (2020)

  • SCIP Online Workshop 2020 | University of Exeter (2020)

  • Mixed Integer Programming Workshop 2020 | Online Edition (2020)

  • LMS Computer Science Colloquium “Mathematics of Security” | London (2019)

  • NATCOR Combinatorial Optimisation | University of Southampton (2019)

  • NATCOR Stochastic Modelling | Lancaster University (2019)

  • Computational Optimisation in Action | Edinburgh University (2018) 

  • NATCOR Convex Optimisation | Edinburgh University (2018) 

  • Young Researchers in Mathematics | Southampton University (2018)

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Society Membership

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  • Society for Industrial and Applied Mathematics (Discrete Mathematics, Linear Algebra, Optimisation)

  • South Wales Operational Research Discussion Society (SWORDS)

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Other Responsibilities 

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  • Editorial Board Member of the American Journal of Operations Management and information Systems

  • Joint organiser of the Combinatorics, Games and Optimisation (CGO) Seminar | London School of Economics

  • Member of the inaugural Early Career Research (ECR) Steering Group | London School of Economics

  • Undergraduate Academic Mentor | London School of Economics 

  • International Journal for Mathematics Teaching and Learning | Reviewer

  • Canadian Journal of Science, Mathematics and Technology Education (Springer) | Reviewer

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