59 research outputs found
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization
The purpose of this note is to survey a methodology to solve systems of
polynomial equations and inequalities. The techniques we discuss use the
algebra of multivariate polynomials with coefficients over a field to create
large-scale linear algebra or semidefinite programming relaxations of many
kinds of feasibility or optimization questions. We are particularly interested
in problems arising in combinatorial optimization.Comment: 28 pages, survey pape
A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions
This paper considers the problem of minimizing the ordered weighted average
(or ordered median) function of finitely many rational functions over compact
semi-algebraic sets. Ordered weighted averages of rational functions are not,
in general, neither rational functions nor the supremum of rational functions
so that current results available for the minimization of rational functions
cannot be applied to handle these problems. We prove that the problem can be
transformed into a new problem embedded in a higher dimension space where it
admits a convenient representation. This reformulation admits a hierarchy of
SDP relaxations that approximates, up to any degree of accuracy, the optimal
value of those problems. We apply this general framework to a broad family of
continuous location problems showing that some difficult problems (convex and
non-convex) that up to date could only be solved on the plane and with
Euclidean distance, can be reasonably solved with different -norms and
in any finite dimension space. We illustrate this methodology with some
extensive computational results on location problems in the plane and the
3-dimension space.Comment: 27 pages, 1 figure, 7 table
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Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics
The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session
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