15,726 research outputs found
Hessian barrier algorithms for linearly constrained optimization problems
In this paper, we propose an interior-point method for linearly constrained
optimization problems (possibly nonconvex). The method - which we call the
Hessian barrier algorithm (HBA) - combines a forward Euler discretization of
Hessian Riemannian gradient flows with an Armijo backtracking step-size policy.
In this way, HBA can be seen as an alternative to mirror descent (MD), and
contains as special cases the affine scaling algorithm, regularized Newton
processes, and several other iterative solution methods. Our main result is
that, modulo a non-degeneracy condition, the algorithm converges to the
problem's set of critical points; hence, in the convex case, the algorithm
converges globally to the problem's minimum set. In the case of linearly
constrained quadratic programs (not necessarily convex), we also show that the
method's convergence rate is for some
that depends only on the choice of kernel function (i.e., not on the problem's
primitives). These theoretical results are validated by numerical experiments
in standard non-convex test functions and large-scale traffic assignment
problems.Comment: 27 pages, 6 figure
Speeding up SOR Solvers for Constraint-based GUIs with a Warm-Start Strategy
Many computer programs have graphical user interfaces (GUIs), which need good
layout to make efficient use of the available screen real estate. Most GUIs do
not have a fixed layout, but are resizable and able to adapt themselves.
Constraints are a powerful tool for specifying adaptable GUI layouts: they are
used to specify a layout in a general form, and a constraint solver is used to
find a satisfying concrete layout, e.g.\ for a specific GUI size. The
constraint solver has to calculate a new layout every time a GUI is resized or
changed, so it needs to be efficient to ensure a good user experience. One
approach for constraint solvers is based on the Gauss-Seidel algorithm and
successive over-relaxation (SOR).
Our observation is that a solution after resizing or changing is similar in
structure to a previous solution. Thus, our hypothesis is that we can increase
the computational performance of an SOR-based constraint solver if we reuse the
solution of a previous layout to warm-start the solving of a new layout. In
this paper we report on experiments to test this hypothesis experimentally for
three common use cases: big-step resizing, small-step resizing and constraint
change. In our experiments, we measured the solving time for randomly generated
GUI layout specifications of various sizes. For all three cases we found that
the performance is improved if an existing solution is used as a starting
solution for a new layout
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