360 research outputs found
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
Solving variational inequalities and cone complementarity problems in nonsmooth dynamics using the alternating direction method of multipliers
This work presents a numerical method for the solution of variational inequalities arising in nonsmooth flexible multibody problems that involve set-valued forces. For the special case of hard frictional contacts, the method solves a second order cone complementarity problem. We ground our algorithm on the Alternating Direction Method of Multipliers (ADMM), an efficient and robust optimization method that draws on few computational primitives. In order to improve computational performance, we reformulated the original ADMM scheme in order to exploit the sparsity of constraint jacobians and we added optimizations such as warm starting and adaptive step scaling. The proposed method can be used in scenarios that pose major difficulties to other methods available in literature for complementarity in contact dynamics, namely when using very stiff finite elements and when simulating articulated mechanisms with odd mass ratios. The method can have applications in the fields of robotics, vehicle dynamics, virtual reality, and multiphysics simulation in general
Newton-type Alternating Minimization Algorithm for Convex Optimization
We propose NAMA (Newton-type Alternating Minimization Algorithm) for solving
structured nonsmooth convex optimization problems where the sum of two
functions is to be minimized, one being strongly convex and the other composed
with a linear mapping. The proposed algorithm is a line-search method over a
continuous, real-valued, exact penalty function for the corresponding dual
problem, which is computed by evaluating the augmented Lagrangian at the primal
points obtained by alternating minimizations. As a consequence, NAMA relies on
exactly the same computations as the classical alternating minimization
algorithm (AMA), also known as the dual proximal gradient method. Under
standard assumptions the proposed algorithm possesses strong convergence
properties, while under mild additional assumptions the asymptotic convergence
is superlinear, provided that the search directions are chosen according to
quasi-Newton formulas. Due to its simplicity, the proposed method is well
suited for embedded applications and large-scale problems. Experiments show
that using limited-memory directions in NAMA greatly improves the convergence
speed over AMA and its accelerated variant
Some recent advances in projection-type methods for variational inequalities
AbstractProjection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods
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