Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I have a minimization problem of the form $$ \min_{n\in N} \sum_i A_i n_i +\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^2 $$ where $A_i\geq 0$ and $B_{ij}\geq 0$ for all $i,j$. This problem can be written as a convex quadratic integer program.

What are the standard algorithms to solve these problems numerically? Are they guaranteed to find a global solution? How fast are they?

  • $\begingroup$ This seems like a misnomer to me. According to Boyd and Vandenberghe, a Convex Optimization problem is an optimization problem whose domain is a convex set. But an integer programming problem is one where the solutions must be integers. The set of integers is not convex. $\endgroup$ – NicNic8 Jan 10 at 20:16
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    $\begingroup$ Yes, I know that using the word "convex" here is a bit tricky but several people in the literature use convex in this way for integer programs. $\endgroup$ – user_lambda Jan 10 at 20:17
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    $\begingroup$ Typically, a mixed integer nonlinear programming problem (MINLP) whose continuous relaxation is convex is referred to as a "Convex MINLP" This is terminology that's been around for at least 25 years... $\endgroup$ – Brian Borchers Jan 10 at 20:45
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    $\begingroup$ Your terminology is fine. You would think it might be possible to exploit the fact that it is unconstrained (that is, beyond the integrality constraints), but I'm not seeing it. $\endgroup$ – Michael Grant Jan 12 at 2:56

Typically, you'd use a branch and bound algorithm to solve this problem and obtain a globally optimal solution if the problem isn't unbounded. This does require worst-case exponential time.

Several commercial LP/QP packages have support for solving these MIQP problems.

  • $\begingroup$ Thanks. I was hoping that simpler algorithms might work given the nice properties of the problem but maybe that's not the case. $\endgroup$ – user_lambda Jan 10 at 21:14

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