Volume 53, Number 1, January–March 2019
|Page(s)||81 - 109|
|Published online||31 January 2019|
Undecidability and hardness in mixed-integer nonlinear programming
CNRS LIX, École Polytechnique,
* Corresponding author: firstname.lastname@example.org
Accepted: 4 May 2018
We survey two aspects of mixed-integer nonlinear programming which have attracted less attention (so far) than solution methods, solvers and applications: namely, whether the class of these problems can be solved algorithmically, and, for the subclasses which can, whether they are hard to solve. We start by reviewing the problem of representing a solution, which is linked to the correct abstract computational model to consider. We then cast some traditional logic results in the light of mixed-integer nonlinear programming, and come to the conclusion that it is not a solvable class: instead, its formal sentences belong to two different theories, one of which is decidable while the other is not. Lastly, we give a tutorial on computational complexity and survey some interesting hardness results in nonconvex quadratic and nonlinear programming.
Mathematics Subject Classification: 90C11 / 90C26
Key words: Undecidability / hardness / mathematical programming
© EDP Sciences, ROADEF, SMAI 2019
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