7,947 research outputs found
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
End-to-End Decision Focused Learning using Learned Solvers
Achieving fusion of deep learning with combinatorial algorithms promises transformativechanges to AI. Creating an impact in a real-world setting requires AI techniques to span a pipeline from data, to predictive models, to decisions. Aligning these components together requires careful consideration, as having these components trained separately does not account for the end goal of the model. This work surveys general frameworks for melding these components, we focus on the integration of optimization methods with machine learning architectures. We address some challenges and limitations associated with these methods and propose a novel approach to address some of the bottlenecks that arise
Regret Models and Preprocessing Techniques for Combinatorial Optimization under Uncertainty
Ph.DDOCTOR OF PHILOSOPH
A Computational Comparison of Optimization Methods for the Golomb Ruler Problem
The Golomb ruler problem is defined as follows: Given a positive integer n,
locate n marks on a ruler such that the distance between any two distinct pair
of marks are different from each other and the total length of the ruler is
minimized. The Golomb ruler problem has applications in information theory,
astronomy and communications, and it can be seen as a challenge for
combinatorial optimization algorithms. Although constructing high quality
rulers is well-studied, proving optimality is a far more challenging task. In
this paper, we provide a computational comparison of different optimization
paradigms, each using a different model (linear integer, constraint programming
and quadratic integer) to certify that a given Golomb ruler is optimal. We
propose several enhancements to improve the computational performance of each
method by exploring bound tightening, valid inequalities, cutting planes and
branching strategies. We conclude that a certain quadratic integer programming
model solved through a Benders decomposition and strengthened by two types of
valid inequalities performs the best in terms of solution time for small-sized
Golomb ruler problem instances. On the other hand, a constraint programming
model improved by range reduction and a particular branching strategy could
have more potential to solve larger size instances due to its promising
parallelization features
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
Inverse polynomial optimization
We consider the inverse optimization problem associated with the polynomial
program f^*=\min \{f(x): x\in K\}y\in
K\tilde{f}fy\tilde{f}Kd\tilde{f}\Vert f-\tilde{f}\Vert\ell_1\ell_2\ell_\infty\tilde{f}_df(\y)f^*\ell_1\tilde{f}$ takes a
simple and explicit canonical form. Some variations are also discussed.Comment: 25 pages; to appear in Math. Oper. Res; Rapport LAAS no. 1114
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