We consider the embedding of piecewise-linear deep neural networks (ReLU networks) as surrogate models in mixed-integer linear programming (MILP) problems. A MILP formulation of ReLU networks has recently been applied by many authors to probe for various model properties subject to input bounds. The formulation is obtained by programming each ReLU operator with a binary variable and applying the big-M method. The efficiency of the formulation hinges on the tightness of the bounds defined by the big-M values. When ReLU networks are embedded in a larger optimization problem, the presence of output bounds can be exploited in bound tightening. To this end, we devise and study several bound tightening procedures that consider both input and output bounds. Our numerical results show that bound tightening may reduce solution times considerably, and that small-sized ReLU networks are suitable as surrogate models in mixed-integer linear programs.

A general modelling framework for optimization of multiphase flow networks with discrete decision variables is presented. The framework is expressed with the graph, and special attention is given to the convexity properties of the resulting programming formulation. Nonlinear pressure and temperature relations are modelled using multivariate splines and a special mixed-integer nonlinear programming (MINLP) formulation with spline constraints results. A global solution method is devised by combining the framework with a spline-compatible MINLP solver, recently presented in the literature. The solver is able to globally solve the nonconvex optimization problem. The new solution method is benchmarked with several local optimization methods on a set of three realistic subsea production optimization cases provided by the oil company BP.

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This paper describes a heuristic algorithm for finding good feasible solutions of convex mixed-integer nonlinear programs (MINLPs). The algorithm we propose is a modification of the feasibility pump heuristic, in which we aim at balancing the two goals of quickly obtaining a feasible solution and preserving quality of the solution with respect to the original objective. The effectiveness and merits of the proposed algorithm are assessed by evaluation of extensive computational results from a set of 146 convex MINLP test problems. We also show how a set of user-defined parameters may be selected to strike a balance between low computation time and high solution quality.