1,491 research outputs found
Inexact Block Coordinate Descent Algorithms for Nonsmooth Nonconvex Optimization
In this paper, we propose an inexact block coordinate descent algorithm for
large-scale nonsmooth nonconvex optimization problems. At each iteration, a
particular block variable is selected and updated by inexactly solving the
original optimization problem with respect to that block variable. More
precisely, a local approximation of the original optimization problem is
solved. The proposed algorithm has several attractive features, namely, i) high
flexibility, as the approximation function only needs to be strictly convex and
it does not have to be a global upper bound of the original function; ii) fast
convergence, as the approximation function can be designed to exploit the
problem structure at hand and the stepsize is calculated by the line search;
iii) low complexity, as the approximation subproblems are much easier to solve
and the line search scheme is carried out over a properly constructed
differentiable function; iv) guaranteed convergence of a subsequence to a
stationary point, even when the objective function does not have a Lipschitz
continuous gradient. Interestingly, when the approximation subproblem is solved
by a descent algorithm, convergence of a subsequence to a stationary point is
still guaranteed even if the approximation subproblem is solved inexactly by
terminating the descent algorithm after a finite number of iterations. These
features make the proposed algorithm suitable for large-scale problems where
the dimension exceeds the memory and/or the processing capability of the
existing hardware. These features are also illustrated by several applications
in signal processing and machine learning, for instance, network anomaly
detection and phase retrieval
Proximal bundle method for contact shape optimization problem
From the mathematical point of view, the contact shape optimization is a problem of nonlinear optimization with a specific structure, which can be exploited in its solution. In this paper, we show how to overcome the difficulties related to the nonsmooth cost function by using the proximal bundle methods. We describe all steps of the solution, including linearization, construction of a descent direction, line search, stopping criterion, etc. To illustrate the performance of the presented algorithm, we solve a shape optimization problem associated with the discretized two-dimensional contact problem with Coulomb's friction
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