Consider the following program: \begin{align} \max_x ~& x^TQx \\ \mbox{s.t.} ~& Ax \geq b \end{align} where $Q$ is a symmetric (possibly indefinite) matrix and the inequality is element-wise and constrains feasible solutions to a convex polytope.
This is NP-hard to solve, but what are known approximation results?
A relevant result is given by (Kough 1979). It is shown that this program can be optimized using Benders' decomposition to within $\epsilon$ of the optimum. However, the paper does not seem to clearly specify what this means, or the complexity of the procedure.
I believe the $\epsilon$-approximation is in the usual sense employed in the field of mathematical programming, that is, if $OPT$ is the optimal value of the program, $ALG$ is the result of the above procedure and $MIN$ is the minimal value attainable by a feasible solution, $$ \frac{ALG-MIN}{OPT-MIN} \geq (1-\epsilon). $$ Or something of the sort.
Questions:
- Is the mentioned procedure a polynomial-time algorithm?
- Are there known polynomial-time algorithms yielding approximations to the above program in the traditional sense, i.e. $ALG \geq \alpha OPT $ for some $\alpha < 1$, constant or not.
Update: I don't think it's possible to get an $\alpha OPT$ approximation for general quadratic programming for a very simple reason: in negative definite instances, the optimum is negative.
Still, it would be interesting to know if approximations are possible when the quadratic form is nonegative in the feasible set. Also, it would be nice to know the complexity of the Bender's decomposition method.
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Kough, Paul F. "The indefinite quadratic programming problem." Operations Research 27.3 (1979): 516-533.
