464 research outputs found
A Lagrangian penalty function method for monotone variational inequalities
A Lagrange-type penalty function method is proposed for a class of variational inequalities. The penalty function may have both positive and negative values. Each penalized subproblem is required to be solved only approximately. A condition under which a Lagrangian penalty function is exact, and an estimate for the penalty coefficient are given
Exact Penalty Functions with Multidimensional Penalty Parameter and Adaptive Penalty Updates
We present a general theory of exact penalty functions with vectorial
(multidimensional) penalty parameter for optimization problems in infinite
dimensional spaces. In comparison with the scalar case, the use of vectorial
penalty parameters provides much more flexibility, allows one to adaptively and
independently take into account the violation of each constraint during an
optimization process, and often leads to a better overall performance of an
optimization method using an exact penalty function. We obtain sufficient
conditions for the local and global exactness of penalty functions with
vectorial penalty parameters and study convergence of global exact penalty
methods with several different penalty updating strategies. In particular, we
present a new algorithmic approach to an analysis of the global exactness of
penalty functions, which contains a novel characterisation of the global
exactness property in terms of behaviour of sequences generated by certain
optimization methods.Comment: In the second version, a number of small mistakes found in the paper
was correcte
Scalable Semidefinite Relaxation for Maximum A Posterior Estimation
Maximum a posteriori (MAP) inference over discrete Markov random fields is a
fundamental task spanning a wide spectrum of real-world applications, which is
known to be NP-hard for general graphs. In this paper, we propose a novel
semidefinite relaxation formulation (referred to as SDR) to estimate the MAP
assignment. Algorithmically, we develop an accelerated variant of the
alternating direction method of multipliers (referred to as SDPAD-LR) that can
effectively exploit the special structure of the new relaxation. Encouragingly,
the proposed procedure allows solving SDR for large-scale problems, e.g.,
problems on a grid graph comprising hundreds of thousands of variables with
multiple states per node. Compared with prior SDP solvers, SDPAD-LR is capable
of attaining comparable accuracy while exhibiting remarkably improved
scalability, in contrast to the commonly held belief that semidefinite
relaxation can only been applied on small-scale MRF problems. We have evaluated
the performance of SDR on various benchmark datasets including OPENGM2 and PIC
in terms of both the quality of the solutions and computation time.
Experimental results demonstrate that for a broad class of problems, SDPAD-LR
outperforms state-of-the-art algorithms in producing better MAP assignment in
an efficient manner.Comment: accepted to International Conference on Machine Learning (ICML 2014
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