We develop explicit, piecewise-linear formulations of functions f(x):ℝ n ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.
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机译：我们开发了函数f（x）：ℝ n ↦ℝ，n≤3的显式分段线性公式，它们在顶点的正交网格上定义。如果可以轻松有效地将涉及多维分段线性函数的混合整数线性优化问题（MILP）求解为全局最优值，则可以将非解析函数用作大型优化问题的目标函数或约束函数。固定网格点之间的线性插值还可以用于近似通用非线性函数，从而使我们能够使用混合整数线性优化方法来近似解决问题。为此，我们开发了分段线性函数的两种不同的显式公式，并讨论了将这些公式集成到优化问题中的结果。
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