Exact solution approaches for the discrete α\alpha-neighbor pp-center problem

The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances

A scaleable projection‐based branch‐and‐cut algorithm for the p‐center problem

The p-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose p of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds. We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation. Finally, we also present a computational study on instances from the literature with up to more than 700,000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

International audienceBilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

An outer approximation algorithm for multi-objective mixed-integer linear and non-linear programming

In this paper, we present the first outer approximation algorithm for multi-objective mixed-integer linear programming problems with any number of objectives. The algorithm also works for certain classes of non-linear programming problems. It produces the non-dominated extreme points as well as the facets of the convex hull of these points. The algorithm relies on an oracle which solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the non-dominated extreme points in the case of multiobjective mixed-integer programming (MOMILP). Thus, for MOMILP problems for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay. From a practical perspective, the algorithm starts from a valid lower bound set for the non-dominated extreme points and iteratively improves it. Therefore it can be used in multi-objective branch-and-bound algorithms and still provide a valid bound set at any stage, even if interrupted before converging. Moreover, the oracle produces Pareto optimal solutions, which makes the algorithm also attractive from the primal side in a multi-objective branch-and-bound context. Finally, the oracle can also be called with any relaxation of the primal problem, and the obtained points and facets still provide a valid lower bound set. A computational study on a set of benchmark instances from the literature and new non-linear multi-objective instances is provided.Comment: 21 page

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.Comment: arXiv admin note: substantial text overlap with arXiv:2111.0682