Optimization of Quadratic Forms and t-norm Forms on Interval Domain and Computational Complexity

Authors

Milan Hladik, Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, PragueFollow
Michal Čerńy, University of Economics, PragueFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

2-2018

Comments

Technical Report: UTEP-CS-18-14

To appear in Proceedings of the World Conference on Soft Computing, Baku, Azerbaijan, May 29-31, 2018.

Abstract

We consider the problem of maximization of a quadratic form over a box. We identify the NP-hardness boundary for sparse quadratic forms: the problem is polynomially solvable for O(log n) nonzero entries, but it is NP-hard if the number of nonzero entries is of the order n^ε for an arbitrarily small ε > 0. Then we inspect further polynomially solvable cases. We define a sunflower graph over the quadratic form and study efficiently solvable cases according to the shape of this graph (e.g. the case with small sunflower leaves or the case with a restricted number of negative entries). Finally, we define a generalized quadratic form, called t-norm form, where the quadratic terms are replaced by t-norms. We prove that the optimization problem remains NP-hard with an arbitrary Lipschitz continuous t-norm.