433 research outputs found
Modified Cholesky Factorizations In Interior-Point Algorithms For Linear Programming
. We investigate a modified Cholesky algorithm similar to those used in current interior-point codes for linear programming. Cholesky-based interior-point codes are popular for three reasons: their implementation requires only minimal changes to standard sparse Cholesky codes (allowing us to take full advantage of software written by specialists in that area); they tend to be more efficient than competing approaches that use different factorizations; and they perform robustly on most practical problems, yielding good interior-point steps even when the coefficient matrix is ill conditioned. We explain the surprisingly good performance of the Cholesky-based approach by using analytical tools from matrix perturbation theory and error analysis, illustrating our results with computational experiments. Finally, we point out the limitations of this approach. Key words. Interior-point algorithms and software, Cholesky factorization, Matrix perturbations, Error analysis. 1. Introduction. Most ..
Low-Rank Modifications of Riccati Factorizations for Model Predictive Control
In Model Predictive Control (MPC) the control input is computed by solving a
constrained finite-time optimal control (CFTOC) problem at each sample in the
control loop. The main computational effort is often spent on computing the
search directions, which in MPC corresponds to solving unconstrained
finite-time optimal control (UFTOC) problems. This is commonly performed using
Riccati recursions or generic sparsity exploiting algorithms. In this work the
focus is efficient search direction computations for active-set (AS) type
methods. The system of equations to be solved at each AS iteration is changed
only by a low-rank modification of the previous one, and exploiting this
structured change is important for the performance of AS type solvers. In this
paper, theory for how to exploit these low-rank changes by modifying the
Riccati factorization between AS iterations in a structured way is presented. A
numerical evaluation of the proposed algorithm shows that the computation time
can be significantly reduced by modifying, instead of re-computing, the Riccati
factorization. This speed-up can be important for AS type solvers used for
linear, nonlinear and hybrid MPC
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
Controlling the level of sparsity in MPC
In optimization routines used for on-line Model Predictive Control (MPC),
linear systems of equations are usually solved in each iteration. This is true
both for Active Set (AS) methods as well as for Interior Point (IP) methods,
and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main
computational effort is spent while solving these linear systems of equations,
and hence, it is of greatest interest to solve them efficiently. Classically,
the optimization problem has been formulated in either of two different ways.
One of them leading to a sparse linear system of equations involving relatively
many variables to solve in each iteration and the other one leading to a dense
linear system of equations involving relatively few variables. In this work, it
is shown that it is possible not only to consider these two distinct choices of
formulations. Instead it is shown that it is possible to create an entire
family of formulations with different levels of sparsity and number of
variables, and that this extra degree of freedom can be exploited to get even
better performance with the software and hardware at hand. This result also
provides a better answer to an often discussed question in MPC; should the
sparse or dense formulation be used. In this work, it is shown that the answer
to this question is that often none of these classical choices is the best
choice, and that a better choice with a different level of sparsity actually
can be found
A new interior-point approach for large two-stage stochastic problems
Two-stage stochastic models give rise to very large optimization problems. Several approaches havebeen devised for efficiently solving them, including interior-point methods (IPMs). However, usingIPMs, the linking columns associated to first-stage decisions cause excessive fill-in for the solutionof the normal equations. This downside is usually alleviated if variable splitting is applied to first-stage variables. This work presents a specialized IPM that applies variable splitting and exploits thestructure of the deterministic equivalent of the stochastic problem. The specialized IPM combinesCholesky factorizations and preconditioned conjugate gradients for solving the normal equations.This specialized IPM outperforms other approaches when the number of first-stage variables is largeenough. This paper provides computational results for two stochastic problems: (1) a supply chainsystem and (2) capacity expansion in an electric system. Both linear and convex quadratic formu-lations were used, obtaining instances of up to 38 million variables and six million constraints. Thecomputational results show that our procedure is more efficient than alternative state-of-the-art IPMimplementations (e.g., CPLEX) and other specialized solvers for stochastic optimizationPeer ReviewedPreprin
- …