53,491 research outputs found
The Quadratic Cycle Cover Problem: special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of
node-disjoint cycles visiting all the nodes such that the total sum of
interaction costs between consecutive arcs is minimized. In this paper we study
the linearization problem for the quadratic cycle cover problem and related
lower bounds.
In particular, we derive various sufficient conditions for the quadratic cost
matrix to be linearizable, and use these conditions to compute bounds. We also
show how to use a sufficient condition for linearizability within an iterative
bounding procedure. In each step, our algorithm computes the best equivalent
representation of the quadratic cost matrix and its optimal linearizable matrix
with respect to the given sufficient condition for linearizability. Further, we
show that the classical Gilmore-Lawler type bound belongs to the family of
linearization based bounds, and therefore apply the above mentioned iterative
reformulation technique. We also prove that the linearization vectors resulting
from this iterative approach satisfy the constant value property.
The best among here introduced bounds outperform existing lower bounds when
taking both quality and efficiency into account
Proximal bundle method for contact shape optimization problem
From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction
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