Ellipsoidal Approach to Box-Constrained Quadratic Problems

Lower Bound Improvement and Forcing Rule for Quadratic Binary Programming

“…If x * = (x u , x l ) T is an optimal solution of the problem (1), then x * is a Pareto optimal solution to the problem (10).…”

“…If there exists an optimal solution x * to the problem (1) such that x * i = y u i (i ∈ I u ), then x * j = y l j for all j / ∈ I u .…”

“…The MAX CUT problem has long been known to be NP-hard even if the problem is unweighted, that is, w i j = 1 for all (i, j) ∈ E. This fact is another confirmation of NP-hardness of the quadratic binary programming problem (1). Some other aspects of the complexity of problem (1) are addressed in [19], where it is shown that: (a) the problem of checking if the quadratic binary programming problem has a unique global solution is NP-hard; (b) the quadratic binary programming problem with a unique global solution remains NP-hard; (c) the problem of finding a discrete local minimum for problem (2) with two components fixed is also NP-hard.…”

“…In addition, some global optimality conditions also have appeared for the boolean quadratic programming problem [1,3].…”

“…HUANG, PARDALOS AND PROKOPYEV (i) IfL 0, i.e.,L is positive semidefinite, and there exists a globally optimal solution x * to (1) such that x * i = x u i , ∀i ∈ I u , then x is also a globally optimal solution to (1); (ii) IfŪ 0, i.e.,Ū is positive semidefinite, and there exists a globally optimal solution x * to (1) such that x * i = x l i , ∀i / ∈ I u , then x is also a globally optimal solution to (1).…”