15,582 research outputs found
Singular Continuation: Generating Piece-wise Linear Approximations to Pareto Sets via Global Analysis
We propose a strategy for approximating Pareto optimal sets based on the
global analysis framework proposed by Smale (Dynamical systems, New York, 1973,
pp. 531-544). The method highlights and exploits the underlying manifold
structure of the Pareto sets, approximating Pareto optima by means of
simplicial complexes. The method distinguishes the hierarchy between singular
set, Pareto critical set and stable Pareto critical set, and can handle the
problem of superposition of local Pareto fronts, occurring in the general
nonconvex case. Furthermore, a quadratic convergence result in a suitable
set-wise sense is proven and tested in a number of numerical examples.Comment: 29 pages, 12 figure
Efficient Numerical Methods to Solve Sparse Linear Equations with Application to PageRank
In this paper, we propose three methods to solve the PageRank problem for the
transition matrices with both row and column sparsity. Our methods reduce the
PageRank problem to the convex optimization problem over the simplex. The first
algorithm is based on the gradient descent in L1 norm instead of the Euclidean
one. The second algorithm extends the Frank-Wolfe to support sparse gradient
updates. The third algorithm stands for the mirror descent algorithm with a
randomized projection. We proof converges rates for these methods for sparse
problems as well as numerical experiments support their effectiveness.Comment: 26 page
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
A simplex-like search method for bi-objective optimization
We describe a new algorithm for bi-objective optimization, similar to the Nelder Mead simplex
algorithm, widely used for single objective optimization. For diferentiable bi-objective functions on
a continuous search space, internal Pareto optima occur where the two gradient vectors point in
opposite directions. So such optima may be located by minimizing the cosine of the angle between
these vectors. This requires a complex rather than a simplex, so we term the technique the \cosine
seeking complex". An extra beneft of this approach is that a successful search identifes the direction
of the effcient curve of Pareto points, expediting further searches. Results are presented for some
standard test functions. The method presented is quite complicated and space considerations here
preclude complete details. We hope to publish a fuller description in another place
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