Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions

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Mixed-integer nonlinear optimization*† (2013, methodology): “In our development, we assume the piecewise linear function f̂ is continuous, although many of these formulations can be extended to lower semicontinuous piecewise linear functions (Vielma et al., 2010).”
Optimization Models for Machine Learning: A Survey (2019, background): “Piecewise affine forms emerge as candidate models when the fitting function is known to be discontinuous [92], separable [80], or approximate to complex nonlinear expressions [79, 181, 207].”
Mixed Integer Linear Programming Formulation Techniques (2015, methodology): “3) reduces to a standard MIP formulation for piecewise linear functions that is denoted the incremental model in [155].”
Identification of critical combination of vulnerable links in transportation networks – a global optimisation approach (2016, methodology): “Therefore, following the Log model (Vielma, Ahmed, and Nemhauser 2010), we have (8)–(10) to approximate the single-variable non-linear term la: ∑ v∈V(P) αva · xva = xa, ∑ v∈V(P) αva · (xva/ya)4 = l̃a, (8) αva ≥ 0 ∀v ∈ V(P),∑ v∈V(P) αva = 1, (9) ∑ v∈Lk αva ≤ λka, ∑ v∈Rk αva ≤ 1 − λka, λka ∈ {0, 1}…”
Modeling disjunctive constraints with a logarithmic number of binary variables and constraints (2008, methodology): “However, preliminary tests in [45] showed that the computational effectiveness of”