630 research outputs found
Design and implementation of a modular interior-point solver for linear optimization
This paper introduces the algorithmic design and implementation of Tulip, an
open-source interior-point solver for linear optimization. It implements a
regularized homogeneous interior-point algorithm with multiple centrality
corrections, and therefore handles unbounded and infeasible problems. The
solver is written in Julia, thus allowing for a flexible and efficient
implementation: Tulip's algorithmic framework is fully disentangled from linear
algebra implementations and from a model's arithmetic. In particular, this
allows to seamlessly integrate specialized routines for structured problems.
Extensive computational results are reported. We find that Tulip is competitive
with open-source interior-point solvers on the H. Mittelmann's benchmark of
barrier linear programming solvers. Furthermore, we design specialized linear
algebra routines for structured master problems in the context of Dantzig-Wolfe
decomposition. These routines yield a tenfold speedup on large and dense
instances that arise in power systems operation and two-stage stochastic
programming, thereby outperforming state-of-the-art commercial interior point
method solvers. Finally, we illustrate Tulip's ability to use different levels
of arithmetic precision by solving problems in extended precision
Constraint interface preconditioning for topology optimization problems
The discretization of constrained nonlinear optimization problems arising in
the field of topology optimization yields algebraic systems which are
challenging to solve in practice, due to pathological ill-conditioning, strong
nonlinearity and size. In this work we propose a methodology which brings
together existing fast algorithms, namely, interior-point for the optimization
problem and a novel substructuring domain decomposition method for the ensuing
large-scale linear systems. The main contribution is the choice of interface
preconditioner which allows for the acceleration of the domain decomposition
method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com
Improving an interior-point approach for large block-angular problems by hybrid preconditioners
The computational time required by interior-point methods
is often domi-
nated by the solution of linear systems of equations. An efficient spec
ialized
interior-point algorithm for primal block-angular proble
ms has been used to
solve these systems by combining Cholesky factorizations for the
block con-
straints and a conjugate gradient based on a power series precon
ditioner for
the linking constraints. In some problems this power series prec
onditioner re-
sulted to be inefficient on the last interior-point iterations, wh
en the systems
became ill-conditioned. In this work this approach is combi
ned with a split-
ting preconditioner based on LU factorization, which is main
ly appropriate
for the last interior-point iterations. Computational result
s are provided for
three classes of problems: multicommodity flows (oriented and no
noriented),
minimum-distance controlled tabular adjustment for statistic
al data protec-
tion, and the minimum congestion problem. The results show that
, in most
cases, the hybrid preconditioner improves the performance an
d robustness of
the interior-point solver. In particular, for some block-ang
ular problems the
solution time is reduced by a factor of 10.Peer ReviewedPreprin
FATROP : A Fast Constrained Optimal Control Problem Solver for Robot Trajectory Optimization and Control
Trajectory optimization is a powerful tool for robot motion planning and
control. State-of-the-art general-purpose nonlinear programming solvers are
versatile, handle constraints in an effective way and provide a high numerical
robustness, but they are slow because they do not fully exploit the optimal
control problem structure at hand. Existing structure-exploiting solvers are
fast but they often lack techniques to deal with nonlinearity or rely on
penalty methods to enforce (equality or inequality) path constraints. This
works presents FATROP: a trajectory optimization solver that is fast and
benefits from the salient features of general-purpose nonlinear optimization
solvers. The speed-up is mainly achieved through the use of a specialized
linear solver, based on a Riccati recursion that is generalized to also support
stagewise equality constraints. To demonstrate the algorithm's potential, it is
benchmarked on a set of robot problems that are challenging from a numerical
perspective, including problems with a minimum-time objective and no-collision
constraints. The solver is shown to solve problems for trajectory generation of
a quadrotor, a robot manipulator and a truck-trailer problem in a few tens of
milliseconds. The algorithm's C++-code implementation accompanies this work as
open source software, released under the GNU Lesser General Public License
(LGPL). This software framework may encourage and enable the robotics community
to use trajectory optimization in more challenging applications
An interior-point and decomposition approach to multiple stage stochastic programming
There is no abstract of this repor
Structure-exploiting interior point methods for security constrained optimal power flow problems
The aim of this research is to demonstrate some more efficient approaches to solve the n-1 security constrained optimal power flow (SCOPF) problems by using structure-exploiting
primal-dual interior point methods (IPM).
Firstly, we consider a DC-SCOPF model, which is a linearized version of AC-SCOPF. One
new reformulation of the DC-SCOPF model is suggested, in which most matrices that need to
be factorized are constant. Consequently, most numerical factorizations and a large number of
back-solve operations only need to be performed once throughout the entire IPM process. In
the framework of the structure-exploiting IPM implementation, one of the major computational
efforts consists of forming the Schur complement matrix, which is very computationally expensive
if no further measure is applied. One remedy is to apply a preconditioned iterative method
to solve the corresponding linear systems which appear in assembling the Schur complement
matrix. We suggest two main schemes to pick a good and robust preconditioner for SCOPF
problems based on combining different âactiveâ contingency scenarios. The numerical results
show that our new approaches are much faster than the default structure-exploiting method in
OOPS, and also that it requires less memory.
The second part of this thesis goes to the standard AC-SCOPF problem, which is a nonlinear
and nonconvex optimization problem. We present a new contingency generation algorithm:
it starts with solving the basic OPF problem, which is a much smaller problem of the same
structure, and then generates contingency scenarios dynamically when needed. Some theoretical
analysis of this algorithm is shown for the linear case, while the numerical results are exciting,
as this new algorithm works for both AC and DC cases. It can find all the active scenarios and
significantly reduce the number of scenarios one needs to contain in the model. As a result, it
speeds up the solving process and may require less IPM iterations.
Also, some heuristic algorithms are designed and presented to predict the active contingencies
for the standard AC-SCOPF, based on the use of AC-OPF or DC-SCOPF. We test our
heuristic algorithms on the modified IEEE 24-bus system, and also present their corresponding
numerical results in the thesis
Advances in Interior Point Methods for Large-Scale Linear Programming
This research studies two computational techniques that improve the practical performance of existing implementations of interior point methods for linear
programming. Both are based on the concept of symmetric neighbourhood as the
driving tool for the analysis of the good performance of some practical algorithms.
The symmetric neighbourhood adds explicit upper bounds on the complementarity pairs, besides the lower bound already present in the common Nâ1 neighbourhood. This allows the algorithm to keep under control the spread among
complementarity pairs and reduce it with the barrier parameter Ό. We show that
a long-step feasible algorithm based on this neighbourhood is globally convergent
and converges in O(nL) iterations.
The use of the symmetric neighbourhood and the recent theoretical under-
standing of the behaviour of Mehrotraâs corrector direction motivate the introduction of a weighting mechanism that can be applied to any corrector direction,
whether originating from Mehrotraâs predictorâcorrector algorithm or as part of
the multiple centrality correctors technique. Such modification in the way a correction is applied aims to ensure that any computed search direction can positively
contribute to a successful iteration by increasing the overall stepsize, thus avoid-
ing the case that a corrector is rejected. The usefulness of the weighting strategy is
documented through complete numerical experiments on various sets of publicly
available test problems. The implementation within the hopdm interior point
code shows remarkable time savings for large-scale linear programming problems.
The second technique develops an efficient way of constructing a starting point
for structured large-scale stochastic linear programs. We generate a computation-
ally viable warm-start point by solving to low accuracy a stochastic problem of
much smaller dimension. The reduced problem is the deterministic equivalent
program corresponding to an event tree composed of a restricted number of scenarios. The solution to the reduced problem is then expanded to the size of the
problem instance, and used to initialise the interior point algorithm. We present
theoretical conditions that the warm-start iterate has to satisfy in order to be
successful. We implemented this technique in both the hopdm and the oops
frameworks, and its performance is verified through a series of tests on problem
instances coming from various stochastic programming sources
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