We study the p-dispersion problem with distance constraints (pDD), a variant of the well-known p-dispersion problem. In a pDD, the goal is to locate a set of facilities so as to maximize the minimum distance between any two of them, subject to additional constraints specifying minimum allowed distances. Two CP models for the pDD have recently been proposed. The first is a typical model that includes the global constraints Minimum and Element and explicitly represents the objective function, connecting it to the decision variables. However, as problem size grows, this model becomes increasingly inefficient. The second model adopts a simplistic approach that only uses binary constraints, essentially treating the pDD as a satisfaction problem. In this paper, after demonstrating the deficiencies of these models, we propose a new compact model that captures the problem through ternary constraints, instead of global or binary ones. We prove that, rather surprisingly, the pruning of the decision variables' domains achieved in our new model is equivalent to that achieved in the model with global constraints, resulting in the same search tree under the same variable and value ordering. Experiments demonstrate that our new model is by far superior to the existing ones, both in terms of solution quality and run times.

Abstract This paper reviews the literature on algorithms for solving bound-constrained mixed-integer derivative-free optimization problems and presents a systematic comparison of available implementations of these algorithms on a large collection of test problems. Thirteen derivative-free optimization solvers are compared using a test set of 267 problems. The testbed includes: (i) pure-integer and mixed-integer problems, and (ii) small, medium, and large problems covering a wide range of characteristics found in applications. We evaluate the solvers according to their ability to find a near-optimal solution, find the best solution among currently available solvers, and improve a given starting point. Computational results show that the ability of all these solvers to obtain good solutions diminishes with increasing problem size, but the solvers evaluated collectively found optimal solutions for 93% of the problems in our test set. The open-source solvers MISO and NOMAD were the best performers among all solvers tested. MISO outperformed all other solvers on large and binary problems, while NOMAD was the best performer on mixed-integer, non-binary discrete, small, and medium-sized problems.