Abstract. Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers.An example is given where many state-of-the-art MILP solvers fail.It is then shown how, using directed rounding and interval arithmetic, cheap pre-and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.

Abstract. Results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables. OverviewAs the recent survey Neumaier [24] of complete solution techniques in global optimization documents, there are now about a dozen solvers for constrained global optimization that claim to solve global optimization and/or constraint satisfaction problems to global optimality by performing a complete search.Within the COCONUT project [30,31], we evaluated many of the existing software packages for global optimization and constraint satisfaction problems. This is the first time that different constrained global optimization and constraint satisfaction algorithms are compared on a systematic basis and with a test set that allows to derive statistically significant conclusions. We tested the global solvers BARON, GlobSol, ICOS, LGO, LINGO, OQNLP, Premium Solver, the local solver MINOS, and a basic combination strategy COCOS implemented in the COCONUT platform. 1The testing process turned out to be extremely time-consuming, due to various reasons not initially anticipated. A lot of effort went into creating appropriate interfaces, making the comparison fair and reliable, and making it possible to process a large number of test examples in a semiautomatic fashion.In a recent paper about testing local optimization software, Dolan & Moré [6,7] write: We realize that testing optimization software is a notoriously difficult problem and that there may be objections to the testing presented in this report. For example, performance of a particular solver may improve significantly if non-default options are given. Another objection is that we only use one starting point per problem and that the performance of a solver may be sensitive to the choice of starting point. We also have used the default stopping criteria of the solvers. This choice may bias results but should not affect comparisons that rely on large time differences. In spite of these objections, we feel that it is essential that we provide some indication of the performance of optimization solvers on interesting problems.These difficulties are also present with our benchmarking studies. Section 2 describes our testing methodology. We use a large test set of over 1000 problems from various collections. Our main performance criterium is currently how often the attainment of the global optimum, or the infeasibility of a problem, is correctly or incorrectly claimed (within some time limit). All solvers are tested with the default options suggested by the providers of the codes, with the request to stop at a time limit or after the solver believed that first global solution was obtained.These are very high standards, much more demanding than what had been done by anyone before. Thorough comparisons are indeed very rare, due to the difficulty of performing extensive and meaningful testing. Indeed, we know of only two comparative studies [17,22] in global ...

Abstract. A benchmarking suite describing over 1000 optimization problems and constraint satisfaction problems covering problems from different traditions is described, annotated with best known solutions, and accompanied by recommended benchmarking protocols for comparing test results.

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