3,954 research outputs found
Compact relaxations for polynomial programming problems
Reduced RLT constraints are a special class of Reformulation- Linearization Technique (RLT) constraints. They apply to nonconvex (both continuous and mixed-integer) quadratic programming problems subject to systems of linear equality constraints. We present an extension to the general case of polynomial programming problems and discuss the derived convex relaxation. We then show how to perform rRLT constraint generation so as to reduce the number of inequality constraints in the relaxation, thereby making it more compact and faster to solve. We present some computational results validating our approach
Semidefinite relaxations for semi-infinite polynomial programming
This paper studies how to solve semi-infinite polynomial programming (SIPP)
problems by semidefinite relaxation method. We first introduce two SDP
relaxation methods for solving polynomial optimization problems with finitely
many constraints. Then we propose an exchange algorithm with SDP relaxations to
solve SIPP problems with compact index set. At last, we extend the proposed
method to SIPP problems with noncompact index set via homogenization. Numerical
results show that the algorithm is efficient in practice.Comment: 23 pages, 4 figure
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even for
problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended relaxations
that we call product relaxations and is motivated by the study of the
Sherali-Adams (SA) hierarchy. Then we show that for every approximate
relaxation of a polytope P, there is a product relaxation that has the same
size and is at least as strong. We provide a methodology for proving lower
bounds on the size of approximate product relaxations by lower bounding the
chromatic number of an underlying hypergraph, whose vertices correspond to
gap-inducing vectors.
We extend the definition of product relaxations and our methodology to mixed
integer sets. However in this case we are able to show that mixed product
relaxations are at least as powerful as a special family of extended
formulations. As an application of our method we show an exponential lower
bound on the size of approximate mixed product formulations for the metric
capacitated facility location problem, a problem which seems to be intractable
for linear programming as far as constant-gap compact formulations are
concerned
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