648 research outputs found
An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow
A novel trust region method for solving linearly constrained nonlinear
programs is presented. The proposed technique is amenable to a distributed
implementation, as its salient ingredient is an alternating projected gradient
sweep in place of the Cauchy point computation. It is proven that the algorithm
yields a sequence that globally converges to a critical point. As a result of
some changes to the standard trust region method, namely a proximal
regularisation of the trust region subproblem, it is shown that the local
convergence rate is linear with an arbitrarily small ratio. Thus, convergence
is locally almost superlinear, under standard regularity assumptions. The
proposed method is successfully applied to compute local solutions to
alternating current optimal power flow problems in transmission and
distribution networks. Moreover, the new mechanism for computing a Cauchy point
compares favourably against the standard projected search as for its activity
detection properties
Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows
In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to
the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems
A branch, price, and cut approach to solving the maximum weighted independent set problem
The maximum weight-independent set problem (MWISP) is one of the most
well-known and well-studied NP-hard problems in the field of combinatorial
optimization.
In the first part of the dissertation, I explore efficient branch-and-price (B&P)
approaches to solve MWISP exactly. B&P is a useful integer-programming tool for
solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint
decompositions of the underlying graph. MWISPâÂÂs on the resulting subgraphs are less
challenging, on average, to solve. I use the B&P framework to solve MWISP on the
original graph G using these specially constructed subproblems to generate columns. I
demonstrate that vertex-disjoint partitioning scheme gives an effective approach for
relatively sparse graphs. I also show that the edge-disjoint approach is less effective than
the vertex-disjoint scheme because the associated DWD reformulation of the latter
entails a slow rate of convergence.
In the second part of the dissertation, I address convergence properties associated
with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the
edge-disjoint B&P scheme and show that these methods improve the rate of
convergence.
In the third part of the dissertation, I focus on identifying new cut-generation
methods within the B&P framework. Such methods have not been explored in the
literature. I present two new methodologies for generating generic cutting planes within
the B&P framework. These techniques are not limited to MWISP and can be used in
general applications of B&P. The first methodology generates cuts by identifying faces
(facets) of subproblem polytopes and lifting associated inequalities; the second
methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully
demonstrate the feasibility of both approaches and present preliminary computational
tests of each
SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0)
SDPNAL+ is a {\sc Matlab} software package that implements an augmented
Lagrangian based method to solve large scale semidefinite programming problems
with bound constraints. The implementation was initially based on a majorized
semismooth Newton-CG augmented Lagrangian method, here we designed it within an
inexact symmetric Gauss-Seidel based semi-proximal ADMM/ALM (alternating
direction method of multipliers/augmented Lagrangian method) framework for the
purpose of deriving simpler stopping conditions and closing the gap between the
practical implementation of the algorithm and the theoretical algorithm. The
basic code is written in {\sc Matlab}, but some subroutines in C language are
incorporated via Mex files. We also design a convenient interface for users to
input their SDP models into the solver. Numerous problems arising from
combinatorial optimization and binary integer quadratic programming problems
have been tested to evaluate the performance of the solver. Extensive numerical
experiments conducted in [Yang, Sun, and Toh, Mathematical Programming
Computation, 7 (2015), pp. 331--366] show that the proposed method is quite
efficient and robust, in that it is able to solve 98.9\% of the 745 test
instances of SDP problems arising from various applications to the accuracy of
in the relative KKT residual
Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows
In this paper, we develop a simultaneous column-and-row generation algorithm for a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints. These constraints are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on the fly within an efficient solution method. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm and then introduce the associated set of pricing subproblems in detail. The proposed methodology is demonstrated on numerical examples for the multi-stage cutting stock and the quadratic set covering problems
The Recoverable Robust Tail Assignment Problem
This is the author accepted manuscript. The final version is available from Institute for Operations Research and the Management Sciences (INFORMS) via the DOI in this record Schedule disruptions are commonplace in the airline industry with many flight-delaying events
occurring each day. Recently there has been a focus on introducing robustness into airline planning
stages to reduce the effect of these disruptions. We propose a recoverable robustness technique as
an alternative to robust optimisation to reduce the effect of disruptions and the cost of recovery. We
formulate the recoverable robust tail assignment problem (RRTAP) as a stochastic program, solved
using column generation in the master and subproblems of the Benders decomposition. We implement a two-phase algorithm for the Benders decomposition incorporating the Magnanti-Wong [21]
enhancement techniques. The RRTAP includes costs due to flight delays, cancellation, and passenger
rerouting, and the recovery stage includes cancellation, delay, and swapping options. To highlight
the benefits of simultaneously solving planning and recovery problems in the RRTAP we compare
our tail assignment solution with the tail assignment generated using a connection cost function
presented in Gr¨onkvist [15]. Using airline data we demonstrate that by developing a better tail assignment plan via the RRTAP framework, one can reduce recovery costs in the event of a disruption.Australian Research Council Centre of Excellence for MathematicsMASCOS
On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations
The strengthening of linear relaxations and bounds of mixed integer linear
programs has been an active research topic for decades. Enumeration-based
methods for integer programming like linear programming-based branch-and-bound
exploit strong dual bounds to fathom unpromising regions of the feasible space.
In this paper, we consider the strengthening of linear programs via a composite
of Dantzig-Wolfe and Fenchel decompositions. We provide geometric
interpretations of these two classical methods. Motivated by these geometric
interpretations, we introduce a novel approach for solving Fenchel sub-problems
and introduce a novel decomposition combining Dantzig-Wolfe and Fenchel
decompositions in an original manner. We carry out an extensive computational
campaign assessing the performance of the novel decomposition on the
unsplittable flow problem. Very promising results are obtained when the new
approach is compared to classical decomposition methods
An investigation of new methods for estimating parameter sensitivities
The method proposed for estimating sensitivity derivatives is based on the Recursive Quadratic Programming (RQP) method and in conjunction a differencing formula to produce estimates of the sensitivities. This method is compared to existing methods and is shown to be very competitive in terms of the number of function evaluations required. In terms of accuracy, the method is shown to be equivalent to a modified version of the Kuhn-Tucker method, where the Hessian of the Lagrangian is estimated using the BFS method employed by the RQP algorithm. Initial testing on a test set with known sensitivities demonstrates that the method can accurately calculate the parameter sensitivity
Accelerating Benders' Decomposition: Algorithmic Enhancements and Model Selection Criteria
Not AvailableThis research was supported by the U.S. Department of Transportation under Contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)
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