21,943 research outputs found
An Efficient Interior-Point Decomposition Algorithm for Parallel Solution of Large-Scale Nonlinear Problems with Significant Variable Coupling
In this dissertation we develop multiple algorithms for efficient parallel solution of structured nonlinear programming problems by decomposition of the linear augmented system solved at each iteration of a nonlinear interior-point approach. In particular, we address large-scale, block-structured problems with a significant number of complicating, or coupling variables. This structure arises in many important problem classes including multi-scenario optimization, parameter estimation, two-stage stochastic programming, optimal control and power network problems. The structure of these problems induces a block-angular structure in the augmented system, and parallel solution is possible using a Schur-complement decomposition. Three major variants are implemented: a serial, full-space interior-point method, serial and parallel versions of an explicit Schur-complement decomposition, and serial and parallel versions of an implicit PCG-based Schur-complement decomposition. All of these algorithms have been implemented in C++ in an extensible software framework for nonlinear optimization.
The explicit Schur-complement decomposition is typically effective for problems with a few hundred coupling variables. We demonstrate the performance of our implementation on an important problem in optimal power grid operation, the contingency-constrained AC optimal power ow problem. In this dissertation, we present a rectangular IV formulation for the contingency-constrained ACOPF problem and demonstrate that the explicit Schur-complement decomposition can dramatically reduce solution times for a problem with a large number of contingency scenarios. Moreover, a comparison of the explicit Schur-complement decomposition implementation and the Progressive Hedging approach provided by Pyomo is provided, showing that the internal decomposition approach is computationally favorable to the external approach. However, the explicit Schur-complement decomposition approach is not appropriate for problems with a large number of coupling variables because of the high computational cost associated with forming and solving the dense Schur-complement.
We show that this bottleneck can be overcome by solving the Schur-complement equations implicitly using a quasi-Newton preconditioned conjugate gradient method.
This new algorithm avoids explicit formation and factorization of the Schur-complement.
The computational efficiency of the serial and parallel versions of this algorithm are compared with the serial full-space approach, and the serial and parallel explicit
Schur-complement approach on a set of quadratic parameter estimation problems and nonlinear optimization problems. These results show that the PCG implicit Schur-complement approach dramatically reduces the computational expense for problems with many coupling variables
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
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