Would anyone mind helping me solve this problem
$$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$
where $x$ is a vector whose entries are positive integers and $Q$ is positive definite.
2026-03-30 11:10:57.1774869057
Quadratic integer Programming
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These kinds of (mixed) integer quadratic programming (MIQP) problems are typically solved by branch and bound algorithms. One widely used commercial code is IBM's CPLEX. You might also be interested in Sven Leyffer's MIQPBB code.
More general codes for mixed integer nonlinear programming (MINLP) can also be used for this problem. See for example the BonMin code.