Optimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques. Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixedinteger nonlinear programming (MINLP) optimization problems. In this work, we survey the various techniques used for domain reduction of NLP and MINLP optimization problems. We also present a computational analysis of the impact of these techniques on the performance of various widely available global solvers on a collection of 1740 test problems.
Whereas much research in the area of optimization is directed toward developing algorithms for optimization of feasible models, the diagnosis of infeasible models has not received as much attention. Identification of irreducible infeasible sets (IISs) can facilitate the process of correcting infeasible models. Several filtering algorithms have been proposed for IIS identification but efficient implementations are available only for linear programs. We propose a novel approach for IIS identification that is applicable to linear programs (LPs), nonlinear programs (NLPs), mixed-integer linear programs (MIPs), and mixed-integer nonlinear programs (MINLPs). The approach makes use of a deletion presolve procedure that exploits bounds tightening techniques to reduce the model to an infeasible set (IS) in a computationally efficient manner. The IS is subsequently reduced to an IIS by applying one of the currently available exact filtering algorithms for IIS identification. We implement the proposed deletion presolve along with four filtering algorithms for IIS identification within the global solver BARON. The effectiveness and usefulness of the proposed approach is demonstrated through computational experiments on a test set of 790 infeasible LPs, NLPs, MIPs, and MINLPs. Deletion presolve rapidly eliminates a large fraction of the problem constraints and speeds up the filtering algorithms by over forty times on average. Speedups of as high as 1,000 times are observed for some problems, while, for 40% of the test problems, the deletion presolve itself reduces the original model to an IIS.
Air Liquide operates several industrial gas pipeline networks around the world, connecting air separation plants to customers of industrial gases. The operation of such a network of plants, pipelines, and customers is complicated due to fluctuating electricity prices and customer demands. We describe a complex industrial problem for real-time optimization of network operations in the presence of these challenges. We then summarize a concerted modeling and algorithmic effort toward global optimization of this model. The resulting advances include development of a regression-based fully-deterministic nonconvex optimization model, a tool for diagnosing infeasibilities during model development, reformulations and scaling to make the model more amenable for optimization, and development of strengthened relaxations for its efficient solution. We provide details on the development of these tools and techniques that facilitated the solution of this model in a reasonable computational time with the global solver BARON.
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