10,053 research outputs found
Local convergence of random graph colorings
Let be a random graph whose average degree is below the
-colorability threshold. If we sample a -coloring of
uniformly at random, what can we say about the correlations between the colors
assigned to vertices that are far apart? According to a prediction from
statistical physics, for average degrees below the so-called {\em condensation
threshold} , the colors assigned to far away vertices are
asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences
2007]. We prove this conjecture for exceeding a certain constant .
More generally, we investigate the joint distribution of the -colorings that
induces locally on the bounded-depth neighborhoods of any fixed number
of vertices. In addition, we point out an implication on the reconstruction
problem
A second derivative SQP method: local convergence
In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact â„“1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud
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Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud
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Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set
A Method to Guarantee Local Convergence for Sequential Quadratic Programming with Poor Hessian Approximation
Sequential Quadratic Programming (SQP) is a powerful class of algorithms for
solving nonlinear optimization problems. Local convergence of SQP algorithms is
guaranteed when the Hessian approximation used in each Quadratic Programming
subproblem is close to the true Hessian. However, a good Hessian approximation
can be expensive to compute. Low cost Hessian approximations only guarantee
local convergence under some assumptions, which are not always satisfied in
practice. To address this problem, this paper proposes a simple method to
guarantee local convergence for SQP with poor Hessian approximation. The
effectiveness of the proposed algorithm is demonstrated in a numerical example
Local convergence of random graph colorings
Let G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring \SIGMA of G(n,m) uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart?
According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} \dc, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007].
We prove this conjecture for k exceeding a certain constant k_0. More generally, we investigate the joint distribution of the k-colorings that \SIGMA induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem
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