3 that it is still NP-hard to find an optimal solution to the quadratic program (QP).

Plugging constraints (2) into the objective function, the quadratic program (QP) can be rewritten as

Moreover, a convex quadratic program of the form min [c.

j], this leads exactly to the integer quadratic program (IQP) that has been introduced in Section 2.

In contrast to the case without nontrivial release dates, we cannot directly prove that this quadratic program is convex.

We show how an arbitrary optimal solution to the quadratic program can be iteratively turned into one satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i and k.

Observe that this defines a new feasible solution to the quadratic program where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has been decreased to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and all other [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], have remained unchanged.

j] in the objective function by the right-hand side of (19), we get the following convex quadratic programming relaxation, which we denote by (CQP) since it generalizes the convex quadratic program developed in Section 2:

However, in contrast to the time-indexed LP relaxation, the construction of the convex quadratic program (CQP) contains more insights into the structure of an optimal schedule.

2, the following convex quadratic program, which we denote by (CQP'p), is a relaxation of the preemptive problem R|[r.

To get the improved bound in the absence of release dates, notice that the objective function (4) of the quadratic program (QP) is equal to the sum of the right-hand sides of (26) and (27).

We start with a reformulation of the integer quadratic program (IQP) from Section 2 in variables [x.