3 research outputs found
Using the primal-dual interior point algorithm within the branch-price-and-cut method
AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm
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
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