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Heuristic optimization seeks useful feasible solutions without necessarily providing a universal approximation ratio or a polynomial worst-case guarantee. Examples are local search, simulated annealing and tabu search. A claimed approximation algorithm needs a separate proved performance guarantee.

Local search repeatedly improves a feasible solution within a specified neighborhood until no improving move remains. For an allocation, neighbors can transfer an item or exchange items between bidders while preserving the partition. A local optimum need not be a global optimum; restarting from several initial solutions can explore different basins.

Represent a feasible allocation by a label for each item saying which bidder receives it. Every such label vector gives disjoint bundles covering all items, so the following heuristic optimization methods can search without violating allocation feasibility.

In local search, begin with any feasible partition and repeatedly make an improving move, such as transferring one item between bidders or exchanging two items. For a transfer $j:i\to r$, evaluate the changeStop when no allowed move improves the objective. Several starting partitions can avoid dependence on one local optimum, but no guarantee of a global optimum follows for general bids.

In simulated annealing, use the same feasible neighborhood but also accept a worsening move of loss $\Delta >0$ with probability $e^{-\Delta /T}$ at temperature $T$. Begin with a relatively high temperature, gradually cool, and retain the best allocation seen. Worsening moves can cross a barrier between local optima; a practical finite cooling schedule does not guarantee optimality.

In tabu search, retain a short memory of recent transfers or exchanges and temporarily prohibit their reversal. Select the best admissible move, possibly worsening the objective, to escape a local optimum. Permit an otherwise tabu move if it improves the best solution found, an aspiration rule. For the winner determination problem, bidder/item pairs or recent reverse transfers provide convenient tabu attributes. All three methods preserve the partition, but for arbitrary bids they are heuristics rather than algorithms with a proved constant approximation ratio.

Tabu search keeps a short-term memory of recently reversed moves or visited attributes and prohibits them temporarily, reducing cycling in local search. It can select the best admissible neighbor even when that neighbor worsens the objective; an aspiration rule permits a tabu move when it improves the best known solution. For allocations, transfers and exchanges maintain feasibility and recent reverse transfers can be tabu.