Digital Circuit Optimization via Geometric Programming
Operations Research
A Superior Representation Method for Piecewise Linear Functions
INFORMS Journal on Computing
A review of recent advances in global optimization
Journal of Global Optimization
A Note on “A Superior Representation Method for Piecewise Linear Functions”
INFORMS Journal on Computing
Modeling disjunctive constraints with a logarithmic number of binary variables and constraints
Mathematical Programming: Series A and B
A simple recipe for concise mixed 0-1 linearizations
Operations Research Letters
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This paper presents an approach for representing functions of discrete variables, and their products, using logarithmic numbers of binary variables. Given a univariate function whose domain consists of n distinct values, it begins by employing a base-2 expansion to express the function in terms of the ceiling of log2n binary and n continuous variables, using linear restrictions to equate the functional values with the possible binary realizations. The representation of the product of such a function with a nonnegative variable is handled via an appropriate scaling of the linear restrictions. Products of m functions are treated in an inductive manner from i = 2 to m, where each step i uses such a scaling to express the product of function i and a nonnegative variable denoting a translated version of the product of functions 1 through i-1 as a newly defined variable. The resulting representations, both in terms of one function and many, are important for reformulating general discrete variables as binary, and also for linearizing mixed-integer generalized geometric and discrete nonlinear programs, where it is desired to economize on the number of binary variables. The approach provides insight into, improves upon, and subsumes related linearization methods for products of functions of discrete variables.