Orbitopal fixing

The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant, since this kind of symmetry unnecessarily blows up the search tree.We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the search tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been introduced in [14]. It does, however, not explicitly add inequalities to the model. Instead, it uses certain fixing rules for variables. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem.     

Algorithms for highly symmetric linear and integer programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.     

Solving large Steiner Triple Covering Problems

Computing the 1-width of the incidence matrix of a Steiner Triple System gives rise to highly symmetric and computationally challenging set covering problems. The largest instance solved so far corresponds to a Steiner Tripe System of order 81. We present optimal solutions for systems of orders 135 and 243. These are computed by a tailored implementation of constraint orbital branching, a method designed to exploit symmetry in integer programs.     

Experimental Results on Using General Disjunctions in Branch-and-Bound for General-Integer Linear Programs

Typical implementations of branch-and-bound for integer linear programs choose to branch on single variables. In this paper we explore the use of general disjunctions for branching when solving linear programs with general-integer variables. We give computational results that show that the size of the enumeration tree can be greatly reduced by branching on such disjunctions rather than on single variables.     

Active-constraint variable ordering for faster feasibility of mixed integer linear programs

The selection of the branching variable can greatly affect the speed of the branch and bound solution of a mixed-integer or integer linear program. Traditional approaches to branching variable selection rely on estimating the effect of the candidate variables on the objective function. We present a new approach that relies on estimating the impact of the candidate variables on the active constraints in the current LP relaxation. We apply this method to the problem of finding the first feasible solution as quickly as possible. Empirical experiments demonstrate a significant improvement compared to a state-of-the art commercial MIP solver.     

Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs with binary variables in the first stage and general integer variables in the second stage. We develop decomposition algorithms akin to the $L$ -shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is flexible in that it allows several modes of implementation, all of which lead to finitely convergent algorithms. We illustrate our algorithms using examples from the literature. We report computational results using the stochastic server location problem instances which suggest that our decomposition-based approach scales better with increases in the number of scenarios than a state-of-the art solver which was used to solve the deterministic equivalent formulation.     

Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices

We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate the problem of rounding a continuous solution to a mixed integer solution. A worst case complexity of a Lenstra-type algorithm is established using an approximate log-barrier center to obtain an ellipsoidal rounding of the feasible set. The results for the mixed integer convex programming also establish a complexity result for the mixed integer second order cone programming and mixed integer semidefinite programming feasibility problems as a special case. Our results motivate an alternative reformulation technique and a branching heuristic using a generalized (e.g., ellipsoidal) norm reduced basis available at the root node.     

On Compact Formulations for Integer Programs Solved by Column Generation

Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.     

Mixed-integer quadratic programming

This paper considers mixed-integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables, and the constraints are linear in the variables of both types. The generalized Benders' decomposition is a suitable approach for solving such programs. However, the program does not become more tractable if this method is used, since Benders' cuts are quadratic in the integer variables. A new equivalent formulation that renders the program tractable is developed, under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables. Benders' cuts that are derived from the new formulation are linear in the integer variables, and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems. The new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm.The author wishes to thank two anonymous referees for their helpful comments and suggestions for revising the paper.     

A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints

We present a branch-and-bound algorithm for discretely-constrained mathematical programs with equilibrium constraints (DC-MPEC). This is a class of bilevel programs with an integer program in the upper-level and a complementarity problem in the lower-level. The algorithm builds on the work by Gabriel et al. (Journal of the Operational Research Society 61(9):1404–1419, 2010) and uses Benders decomposition to form a master problem and a subproblem. The new dynamic partition scheme that we present ensures that the algorithm converges to the global optimum. Partitioning is done to overcome the non-convexity of the Benders subproblem. In addition Lagrangean relaxation provides bounds that enable fathoming in the branching tree and warm-starting the Benders algorithm. Numerical tests show significantly reduced solution times compared to the original algorithm. When the lower level problem is stochastic our algorithm can easily be further decomposed using scenario decomposition. This is demonstrated on a realistic case.     

Information-theoretic approaches to branching in search

Deciding what question to branch on at each node is a key element of search algorithms. In this paper, we describe a collection of techniques for branching decisions that are motivated from an information-theoretic perspective. The idea is to drive the search to reduce the uncertainty (entropy) in the current subproblem. We present four families of methods for branch question selection in mixed integer programming that use this idea. In the first, a variable to branch on is selected based on lookahead. This method performs comparably to strong branching on MIPLIB, and better than strong branching on hard real-world procurement optimization instances on which CPLEX’s default strong branching outperforms CPLEX’s default branching strategy. The second family combines this idea with strong branching. The third family does not use lookahead, but instead exploits the tie between indicator variables and the variables they govern. This significantly outperforms the state-of-the-art branching strategies on both combinatorial procurement problems and facility location problems. The fourth family concerns branching using carefully constructed linear inequality constraints over sets of variables.     

The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs

We consider a class of non-linear mixed integer programs with n integer variables and k continuous variables. Solving instances from this class to optimality is an NP-hard problem. We show that for the cases with k=1 and k=2, every optimal solution is integral. In contrast to this, for every k≥3 there exist instances where every optimal solution takes non-integral values. Received: August 2001 / Accepted: January 2002?Published online March 27, 2002     

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

Binary integer programs with two variables per inequality

Several recent papers have shown that some properties of the maximum weight stable set problem hold true in the more general setting of binary integer programs with two variables per inequality. In this paper, we show that in fact the two problems are equivalent by using the transitive closure of the binary integer program and (possibly) reducing the number of variables by fixing, complementing, or identifying them. We use this equivalence to prove two conjectures made by Johnson and Padberg regarding the perfection of bidirected graphs.     

SICOpt: Solution Approach for Nonlinear Integer Stochastic Programming Problems

We propose a novel solution approach for the class of two-stage nonlinear integer stochastic programming models. These problems are characterized by large scale dimensions, as the number of constraints and variables depend on the number of realizations (scenarios) used to capture the underlying distributions of the random data. In addition, the integrality constraints on the decision variables make the solution process even much more difficult preventing the application of general purpose solvers. The proposed solution approach integrates the branch-and-bound framework with the interior point method. The main advantage of this choice is the effective exploitation of the specific structure exhibited by the different subproblems at each node of the search tree. A specifically designed warm start procedure and an early branching technique improve the overall efficiency. Our contribution is well founded from a theoretical point of view and is characterized by good computational efficiency, without any loss in terms of effectiveness. Some preliminary numerical results, obtained by solving a challenging real-life problem, prove the robustness and the efficiency of the proposed approach.     

Discrete and geometric Branch and Bound algorithms for?medical image registration

Aiming at the development of an exact solution method for registration problems, we present two different Branch & Bound algorithms for a mixed integer programming formulation of the problem. The first B&B algorithm branches on binary assignment variables and makes use of an optimality condition that is derived from a graph matching formulation. The second, geometric B&B algorithm applies a geometric branching strategy on continuous transformation variables. The two approaches are compared for synthetic test examples as well as for 2-dimensional medical data. The results show that medium sized problem instances can be solved to global optimality in a reasonable amount of time.     

Point-to-point shortest paths on dynamic time-dependent road networks

This a summary of the author’s PhD thesis supervised by Leo Liberti, Philippe Baptiste and Daniel Krob and defended on 18 June 2009 at Ecole Polytechnique, Palaiseau, France. The thesis is written in English and is available at . The computation of point-to-point shortest paths in road networks has many practical applications that require very fast solution times, meaning that Dijkstra’s algorithm is not a viable option. In this work we develop an efficient algorithm to find the shortest route between two nodes of a large-scale, time-dependent graph, where we also allow the time-dependent arc cost functions to be updated at regular intervals. Furthermore, we propose a mathematical programming formulation for the shortest paths problem on time-dependent networks, that gives rise to integer programs. Within the context of solving Mixed-Integer Linear Programs through a Branch-and-Bound algorithm, we propose a new strategy for branching, mixing branching on single variables with branching on general hyperplanes. Finally, we introduce an effective heuristic for nonconvex Mixed-Integer Nonlinear Programss which combines VNS, Local Branching, NLP local search and Branch-and-Bound.     

Aggregation of constraints in integer programming

Constraint aggregation provides a method of formulating equivalent integer programs with a smaller number of constraints. This approach was widely researched in the seventies but its use was discounted due to large coefficients in the equivalent problem. We provide a method that yields numerically smaller constraint coefficients. This method has enabled us to investigate other computational issues relating to the use of constraint aggregation in solving integer programming problems, more thoroughly than has previously been possible.     

An interior-point Benders based branch-and-cut algorithm for mixed integer programs

We present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). We show that the ACCPM based Benders cuts are both pareto-optimal and valid for any node of the branch-and-bound tree. The valid cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances.