A linear program always attains its minimum. If the feasible region is nonempty and the objective is bounded below, there exists a point in the feasible region where the objective achieves its infimum. This is a theorem — not an assumption, not a convenience, but a mathematical fact that follows from the polyhedral geometry of linear constraints.
Second-order cone programs (SOCPs) are not linear programs. They include constraints like ||x|| ≤ t — the point (x, t) must lie inside or on the boundary of a cone. SOCPs are convex, efficiently solvable, and widely used in engineering, finance, and signal processing. The implicit assumption is that they behave like linear programs: if the problem is feasible and bounded, the minimum exists.
It does not always exist (arXiv:2510.00318). The paper constructs a specific SOCP that is feasible, bounded, and satisfies every classical solvability condition — yet has no optimal solution. The infimum is a finite number that no feasible point achieves. You can get arbitrarily close but never arrive.
The mechanism is the non-polyhedral boundary of the cone. In a linear program, the feasible region has flat faces, and the optimum sits at a vertex or along an edge — a point you can write down. In an SOCP, the feasible region has curved boundaries. The sequence of improving points can converge toward the boundary along a curve that asymptotically approaches but never reaches the infimum. The boundary is closed — the cone includes its surface — but the intersection of multiple conic constraints can create a limiting point that is feasible for each constraint individually but infeasible for their intersection.
This is not pathological in the sense of being artificial. The example is robust — it persists under small perturbations of the data. Any solver applied to this problem will converge to a sequence of decreasing objective values that never terminates, not because the solver is broken but because there is no solution to find.
The minimum is real. The minimizer is not.