Quadratic programming: Difference between revisions

Revision as of 21:33, 16 December 2023

Revision as of 21:39, 16 December 2023

Line 123:

For [[positive-definite matrix|positive definite]] {{mvar|Q}}, when the problem is convex, the [[ellipsoid method]] solves the problem in (weakly) [[polynomial time]].<ref>{{cite journal| last=Kozlov | first=M. K. |author2=S. P. Tarasov | author3-link=Leonid Khachiyan |author3=Leonid G. Khachiyan | year=1979 | title=[Polynomial solvability of convex quadratic programming] | journal=[[Doklady Akademii Nauk SSSR]] | volume=248 | pages=1049–1051}} Translated in: {{cite journal| journal=Soviet Mathematics - Doklady | volume=20 | pages=1108–1111}}</ref>

If, on the other hand, {{mvar|Q}} is indefinite, (so the problem is non-convex) then the problem is [[NP-hard]].<ref>{{cite journal | last = Sahni | first = S. | title = Computationally related problems | journal = SIAM Journal on Computing | volume = 3 | issue = 4 | pages = 262–279 | year = 1974 | doi=10.1137/0203021| url = http://www.cise.ufl.edu/~sahni/papers/comp.pdf | citeseerx = 10.1.1.145.8685 }}</ref> A heresimple way to see this is to consider the non-convex quadratic constraint ''x_i''² = ''x_i''. This constraint is equivalent to requiring that ''x_i'' is in {0,1}, that is, ''x_i'' is a binary integer variable. Therefore, such constraints can be severalused stationaryto pointsmodel andany local[[Integer minimaprogramming|integer forprogram]] with binary variables, which is known to be NP-hard.

Moreover, these non-convex problems might have several stationary points and local minima. In fact, even if {{mvar|Q}} has only one negative [[eigenvalue]], the problem is (strongly) [[NP-hard]].<ref>{{cite journal | title = Quadratic programming with one negative eigenvalue is (strongly) NP-hard | first1 = Panos M. | last1 = Pardalos | first2 = Stephen A. | last2 = Vavasis | journal = Journal of Global Optimization | volume = 1 | issue = 1 | year = 1991 | pages = 15–22 | doi=10.1007/bf00120662| s2cid = 12602885 }}</ref>

== Integer constraints ==