Branch-and-bound for biobjective mixed-integer linear programming

We present a generic branch-and-bound method for finding all the Pareto solutions of a biobjective mixed integer program. Our main contribution is new algorithms for obtaining dual bounds at a node, for checking node fathoming, presolve and duality gap measurement. Our various procedures are implemented and empirically validated on instances from literature and a new set of hard instances. We also perform comparisons against the triangle splitting method of Boland et al. [\emph{INFORMS Journal on Computing}, \textbf{27} (4), 2015], which is a objective space search algorithm as opposed to our variable space search algorithm. On each of the literature instances, our branch-and-bound is able to compute the entire Pareto set in significantly lesser time. Most of the instances of the harder problem set were not solved by either algorithm in a reasonable time limit, but our algorithm performs better on average on the instances that were solved.Comment: 35 pages, 12 figures. Original preprint at Optimization Online, October 201

Multilinear Formulations for Computing a Nash Equilibrium of Multi-Player Games

We present multilinear and mixed-integer multilinear programs to find a Nash equilibrium in multi-player noncooperative games. We compare the formulations to common algorithms in Gambit, and conclude that a multilinear feasibility program finds a Nash equilibrium faster than any of the methods we compare it to, including the quantal response equilibrium method, which is recommended for large games. Hence, the multilinear feasibility program is an alternative method to find a Nash equilibrium in multi-player games, and outperforms many common algorithms. The mixed-integer formulations are generalisations of known mixed-integer programs for two-player games, however unlike two-player games, these mixed-integer programs do not give better performance than existing algorithms

A mean-risk mixed integer nonlinear program for transportation network protection

This paper focuses on transportation network protection to hedge against extreme events such as earthquakes. Traditional two-stage stochastic programming has been widely adopted to obtain solutions under a risk-neutral preference through the use of expectations in the recourse function. In reality, decision makers hold different risk preferences. We develop a mean-risk two-stage stochastic programming model that allows for greater flexibility in handling risk preferences when allocating limited resources. In particular, the first stage minimizes the retrofitting cost by making strategic retrofit decisions whereas the second stage minimizes the travel cost. The conditional value-at-risk (CVaR) is included as the risk measure for the total system cost. The two-stage model is equivalent to a nonconvex mixed integer nonlinear program (MINLP). To solve this model using the Generalized Benders Decomposition (GBD) method, we derive a convex reformulation of the second-stage problem to overcome algorithmic challenges embedded in the non-convexity, nonlinearity, and non-separability of first- and second-stage variables. The model is used for developing retrofit strategies for networked highway bridges, which is one of the research areas that can significantly benefit from mean-risk models. We first justify the model using a hypothetical nine-node network. Then we evaluate our decomposition algorithm by applying the model to the Sioux Falls network, which is a large-scale benchmark network in the transportation research community. The effects of the chosen risk measure and critical parameters on optimal solutions are empirically explored

Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in $\R^2$ and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch-and-bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to $10^7$ points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB