Fixed charges of arbitrary sign: what survives and what fails
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For integer activities, conditioning on the support makes a fixed-charge objective affine. If every support-conditioned cell is integral, an optimal solution is a vertex of its cell for arbitrary fixed charges and marginal rates. A totally unimodular constraint matrix with integral data guarantees this condition. This surviving property is weaker than the classical conclusions. A negative fixed charge rewards opening an activity at its minimum positive level, which the vertex property does not constrain. We give unique-optimum instances that violate the fractional-period property in capacitated lot sizing and the positive-support forest properties in fixed-charge transportation and flow. A four-period lot-sizing instance further has unrestricted optimum \(-8\), while every schedule in the subplan state space used by a classical \(O(T^4)\) recursion costs at least \(-5\). Finally, the one-dimensional charge-if-positive function is piecewise concave in Zangwill’s sense exactly for a non-negative fixed charge. Thus integrality preserves a weak vertex statement across the sign change, while the stronger structures and an algorithmic state space built on them can fail.