3,609 research outputs found
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
The Hidden Convexity of Spectral Clustering
In recent years, spectral clustering has become a standard method for data
analysis used in a broad range of applications. In this paper we propose a new
class of algorithms for multiway spectral clustering based on optimization of a
certain "contrast function" over the unit sphere. These algorithms, partly
inspired by certain Independent Component Analysis techniques, are simple, easy
to implement and efficient.
Geometrically, the proposed algorithms can be interpreted as hidden basis
recovery by means of function optimization. We give a complete characterization
of the contrast functions admissible for provable basis recovery. We show how
these conditions can be interpreted as a "hidden convexity" of our optimization
problem on the sphere; interestingly, we use efficient convex maximization
rather than the more common convex minimization. We also show encouraging
experimental results on real and simulated data.Comment: 22 page
Experimental study of energy-minimizing point configurations on spheres
In this paper we report on massive computer experiments aimed at finding
spherical point configurations that minimize potential energy. We present
experimental evidence for two new universal optima (consisting of 40 points in
10 dimensions and 64 points in 14 dimensions), as well as evidence that there
are no others with at most 64 points. We also describe several other new
polytopes, and we present new geometrical descriptions of some of the known
universal optima.Comment: 41 pages, 12 figures, to appear in Experimental Mathematic
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