European day-ahead electricity markets clear prices by solving an optimization problem. The optimization includes integer variables — power plants are either on or off, they can't run at 37.4% of their minimum stable generation. The integers destroy convexity, and without convexity, classical pricing theory breaks down.
The consequence: paradoxical orders. Bids that should be accepted (they're cheaper than the clearing price) get rejected. Bids that should be rejected (they're more expensive) get accepted. The market routinely produces outcomes that violate basic economic logic, and the operators patch this with supplementary payments — side transfers that compensate the losers for the optimizer's mathematically necessary irrationality.
Strong duality — the theorem that guarantees the primal optimization and its dual produce the same answer — is generally impossible for mixed-integer linear programs. This is a foundational result in optimization theory. Wang, Hesamzadeh, and colleagues sidestep the impossibility by using Augmented Lagrangian Duality, which achieves strong duality for MILPs under specific structural conditions that electricity markets satisfy.
The result: incentive-compatible pricing without supplementary payments. The market clears rationally. No paradoxical orders. No side transfers. The impossibility theorem is circumvented not by relaxing the integers (which would misrepresent the physics of power plants) but by changing the duality framework — asking the same question through a different mathematical lens that happens to have an answer where the standard lens doesn't.
The through-claim: an impossibility result in mathematics is always relative to the framework that proves it. Changing the framework — not the problem, not the constraints, but the mathematical language used to analyze them — can transform impossible into solved. The paradox was in the method, not the market.