71,247 research outputs found
A second-derivative trust-region SQP method with a "trust-region-free" predictor step
In (NAR 08/18 and 08/21, Oxford University Computing Laboratory, 2008) we introduced a second-derivative SQP method (S2QP) for solving nonlinear nonconvex optimization problems. We proved that the method is globally convergent and locally superlinearly convergent under standard assumptions. A critical component of the algorithm is the so-called predictor step, which is computed from a strictly convex quadratic program with a trust-region constraint. This step is essential for proving global convergence, but its propensity to identify the optimal active set is Paramount for recovering fast local convergence. Thus the global and local efficiency of the method is intimately coupled with the quality of the predictor step.\ud
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In this paper we study the effects of removing the trust-region constraint from the computation of the predictor step; this is reasonable since the resulting problem is still strictly convex and thus well-defined. Although this is an interesting theoretical question, our motivation is based on practicality. Our preliminary numerical experience with S2QP indicates that the trust-region constraint occasionally degrades the quality of the predictor step and diminishes its ability to correctly identify the optimal active set. Moreover, removal of the trust-region constraint allows for re-use of the predictor step over a sequence of failed iterations thus reducing computation. We show that the modified algorithm remains globally convergent and preserves local superlinear convergence provided a nonmonotone strategy is incorporated
A framework for proving the self-organization of dynamic systems
This paper aims at providing a rigorous definition of self- organization, one
of the most desired properties for dynamic systems (e.g., peer-to-peer systems,
sensor networks, cooperative robotics, or ad-hoc networks). We characterize
different classes of self-organization through liveness and safety properties
that both capture information re- garding the system entropy. We illustrate
these classes through study cases. The first ones are two representative P2P
overlays (CAN and Pas- try) and the others are specific implementations of
\Omega (the leader oracle) and one-shot query abstractions for dynamic
settings. Our study aims at understanding the limits and respective power of
existing self-organized protocols and lays the basis of designing robust
algorithm for dynamic systems
The Horcrux Protocol: A Method for Decentralized Biometric-based Self-sovereign Identity
Most user authentication methods and identity proving systems rely on a
centralized database. Such information storage presents a single point of
compromise from a security perspective. If this system is compromised it poses
a direct threat to users' digital identities. This paper proposes a
decentralized authentication method, called the Horcrux protocol, in which
there is no such single point of compromise. The protocol relies on
decentralized identifiers (DIDs) under development by the W3C Verifiable Claims
Community Group and the concept of self-sovereign identity. To accomplish this,
we propose specification and implementation of a decentralized biometric
credential storage option via blockchains using DIDs and DID documents within
the IEEE 2410-2017 Biometric Open Protocol Standard (BOPS)
The reliability horizon for semi-classical quantum gravity: Metric fluctuations are often more important than back-reaction
In this note I introduce the notion of the ``reliability horizon'' for
semi-classical quantum gravity. This reliability horizon is an attempt to
quantify the extent to which we should trust semi-classical quantum gravity,
and to get a better handle on just where the Planck regime resides. I point out
that the key obstruction to pushing semi-classical quantum gravity into the
Planck regime is often the existence of large metric fluctuations, rather than
a large back-reaction. There are many situations where the metric fluctuations
become large long before the back-reaction is significant. Issues of this type
are fundamental to any attempt at proving Hawking's chronology protection
conjecture from first principles, since I shall prove that the onset of
chronology violation is always hidden behind the reliability horizon.Comment: 6 pages; ReV_TeX 3.0; two-column format. Revisions: Central
definitions and results essentially unchanged. Discussion of the relationship
between this letter and the Kay-Radzikowski-Wald singularity theorems greatly
extended and clarified. Discussion of reliability horizon near curvature
singularities modified. Several references added. Minor typos fixed.
Technical TeX modification
Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions
Given a non-convex twice differentiable cost function f, we prove that the
set of initial conditions so that gradient descent converges to saddle points
where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue)
measure zero, even for cost functions f with non-isolated critical points,
answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016].
Moreover, this result extends to forward-invariant convex subspaces, allowing
for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce
an upper bound on the allowable step-size.Comment: 2 figure
Existence of regular neighborhoods for H-surfaces
In this paper, we study the global geometry of complete, constant mean
curvature hypersurfaces embedded in n-manifolds. More precisely, we give
conditions that imply properness of such surfaces and prove the existence of
fixed size one-sided regular neighborhoods for certain constant mean curvature
hypersurfaces in certain n-manifolds.Comment: 11 page
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
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