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An algorithm parameter space, commonly written $\Theta$, is the set of allowed settings for a parameterized algorithm. It is the search space of algorithm configuration, rather than the set of problem instances that the target algorithm must solve. ParamILS explicitly accounts for parameters whose relevance depends on other settings.^[ref]

For an original finite example, let a mode be either $A$ or $B$. Mode $A$ has an extra setting $h\in \{0,1,2\}$, while mode $B$ ignores $h$. A rectangular encoding has $2\times 3=6$ tuples, but only four distinct effective configurations:The star means that the conditional algorithm parameter is inactive. Treating the three encodings of $B$ as three distinct experiments spends effort revisiting the same behavior. If $(A,2)$ is a forbidden parameter combination, the valid effective space has three members.

The representation also determines what a nearby configuration means. A search that changes one active parameter can move among the $A$ settings or change mode; changing an inactive value alone does not change behavior. Thus the number of encoded tuples, number of meaningful settings and neighborhood used by local search are separate design choices.

Continuous settings require declared ranges and a search method that supports them; replacing a range with a finite grid changes the problem being searched. Whatever the representation, a returned setting must be valid in $\Theta$ before its measured cost can be compared with other candidates.

Consider the chosen finite minimization problemOnly adjacent integers are neighbors. At each interior state propose left or right with probability $1/2$ each; at an endpoint propose its sole neighbor with probability $1/2$ and itself with probability $1/2$. This proposal distribution is symmetric, including at the endpoints. The local minimum at $1$ is relative to these allowed moves; $4$ is the unique global minimum.

At $1$, improving-only local search rejects both neighbors because $E(0)=2$ and $E(2)=4$ exceed $E(1)=1$. For simulated annealing, choose $T_{\ast }=1/\ln 2$, using the natural logarithm. Conditional on proposing $2$ from $1$, the acceptance probability isThe one-step transition probability is instead $P_{T_{\ast }}(1,2)=(1/2)(1/8)=1/16$. Both subsequent rightward proposals are downhill, so the conditional probability of the precise next three states being $2,3,4$, given $X_0=1$ and fixed $T_{\ast }$, isThis is one possible escape route, not the probability of eventually reaching $4$. Rejections, excursions toward $0$, and repeated attempts give many other sample paths. The calculation establishes possibility, without claiming a general performance advantage from a selected sample path.

For comparison, halving $T_{\ast }$ makes the uphill acceptance probability $1/64$; doubling it makes that probability $1/(2\sqrt{2})$. Cooling makes it harder to leave this local minimum, even though the stationary distribution increasingly favors the global minimum. That tension is the central scheduling problem.

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.

A local search method that retains one current candidate and repeatedly chooses an improving neighbor of its objective. For a conflict-minimization constraint satisfaction problem, a candidate can assign all variables within their constraint domains while violating some inter-variable constraints; the objective counts those violations. Zero conflicts certifies a solution, while a positive-conflict local minimum need not certify impossibility. Plateaus and local minima motivate sideways moves, randomized choices and restarting. Prescribed variables must remain fixed in every initialization and move.

Iterated local search is a metaheuristic that alternates perturbation, local search and acceptance decisions. The local search improves candidates within a declared optimization neighborhood; perturbation helps reach other locally optimal solutions.^[ref]

For minimization, start with $s=\mathcal{L}(s_0)$ and generate $q=\mathcal{L}(P(s))$, where $\mathcal{L}$ performs local search and $P$ perturbs its input. Accepting only $f(q)<f(s)$ or accepting every candidate are two possible rules.^[ref]

For an original complete example, let feasible solutions be the integers $0$ through $6$, with neighbors differing by one and objective valuesStrictly improving local search starting at $0$ moves to $1$ and stops: its two neighbors have costs $4$ and $3$, both worse than $2$. Perturb the solution from $1$ to $4$, allowing a temporary rise to cost $4$. Local improvement then follows $4\to 5\to 6$, reaching cost $0$, which is the global minimum by inspection of the complete list. The perturbation escaped the first local minimum without requiring the local improvement rule itself to cross an uphill edge.

Perturbation and acceptance must work together: weak changes can return to the same local minimum. The authors’ travelling salesman problem comparison used ten runs of 120 seconds on a Pentium 266 MHz PC, illustrating why evaluation must specify the embedded search and computational budget.^[ref]

An iterated local search acceptance rule chooses the next current solution after a perturbation has been followed by local search. Let the current locally improved solution be $s$ and the new locally improved candidate be $q$, with a minimized objective function $f$. The decision concerns these search states, not just the potentially poor intermediate output of the local-search perturbation.^[ref]

One rule accepts only improvement:The paper calls this policy Better. Its current costs cannot increase, which follows immediately from the two cases. That property does not establish that the search will reach a global minimum. The RW rule instead retains the latest candidate regardless of cost, allowing movement among locally improved solutions.^[ref]

For an original numerical comparison, suppose $f(s)=10$. A perturbation temporarily raises the cost to18, and subsequent local search lowers it to12. Better rejects the candidate with cost12, while RW accepts it; neither decision is made by comparing10 with the intermediate18. If the same procedure instead reaches cost8, both accept.

A simulated annealing-type option accepts a worse candidate with probability $\exp [-(f(q)-f(s))/T]$, with $T>0$. At cost difference2 and $T=2$, this probability is $e^{-1}\simeq 0.368$. The temperature is a parameter of the rule, not a guarantee that a particular finite run escapes every trap.^[ref]

A permissive current-state rule can retain a candidate worse than an earlier result. Keeping a separate best-so-far solution preserves that result while allowing exploration. Report both the current-state acceptance policy and the best-result bookkeeping when comparing search runs.

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.

A local-search perturbation deliberately changes the current solution before local search resumes. In iterated local search, the perturbation and improvement form a composite step: $s\mapsto \mathcal{L}(P(s))$, where $P$ changes the candidate and $\mathcal{L}$ applies the chosen local improvement procedure. The perturbation need not improve the objective function; a later iterated local search acceptance rule decides which improved candidate becomes current.^[ref]

A fully specified example shows why a weak perturbation can be ineffective. Let feasible solutions be the integers $0$ through $5$, with neighbors differing by one, and costsUse strictly improving, best-neighbor local search. Starting at5 is locally optimal: its only neighbor4 has cost3, above the current cost1. A one-step kick to4 is immediately undone by local improvement, which returns to5.

A different kick from5 to2 temporarily raises the cost to4, but the subsequent improving path is $2\to 1\to 0$, with costs $4\to 2\to 0$. The resulting candidate beats the old stopping point. These values are an original constructed example; they show how changing the perturbation can change which solution the same local search reaches.

Perturbation strength therefore concerns more than accepting a single worse local move. Too little change may repeatedly return to the same solution, while a nearly independent redraw can discard useful structure from the current candidate. Feasibility must also be handled explicitly: either the perturbation stays within the feasible set or the method specifies how an infeasible intermediate is repaired. Problem-specific moves and acceptance decisions must be evaluated together.^[ref]

Olivier C. Martin coauthored the 2001 survey Iterated Local Search by Helena R. Lourenço and others with Helena Ramalhinho Dias Lourenço and Thomas Stützle. Its title page identifies him with Université Paris-Sud, Orsay. The survey describes searching over outputs of an embedded optimizer, rather than treating every feasible solution as an equally likely search state.^[ref]

An original finite example distinguishes uniformly chosen starting states from uniformly chosen local minima. Let a deterministic local search operator $\mathcal{L}$ map six possible starts as follows:Here $a,b,c$ are its three possible locally optimized outputs. A uniform random start reaches them with probabilities $1/2$, $1/3$ and $1/6$, not $1/3$ each. This follows simply by counting the starts mapped to each output. It is a chosen abstract optimizer map, not a claim about a particular benchmark landscape.

In iterated local search, a local-search perturbation followed by $\mathcal{L}$ induces another transition mechanism on those outputs. Its probabilities depend on both components. Consequently a chain of locally optimized candidates is not automatically a uniform sample of local minima, nor does it automatically have a prescribed equilibrium distribution. The optimizer and transition rule must be specified before making such sampling claims.

An optimization neighborhood assigns to each feasible solution $x$ a set $N(x)$ of candidates reachable by the chosen elementary moves. It need not be a neighborhood in a previously chosen geometric metric, nor must reachability be symmetric. For minimization, a solution is locally optimal under these moves whenChanging $N$ can change that conclusion while leaving the objective function unchanged. Specifying the embedded local search is therefore part of specifying an iterated local search method.^[ref]

For a complete binary string example, let the feasible set contain all eight strings of length three. Assign $f(000)=0$, $f(111)=-1$, and $f(x)=1$ for the other six strings. With one-bit flips,so $000$ is a strict local optimum. Allowing flips of one or two bits adds $110,101,011$, but all six neighbors still cost $1$. Allowing three-bit flips finally reaches $111$, so $000$ ceases to be locally optimal. The best value in the entire feasible set is $-1$ throughout; the move rule changes what the local procedure can see.

For length $n$, the one-bit-flip neighborhood has $n$ members, while including all two-bit flips adds $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ members. A larger neighborhood may expose an improving move but cost more to examine. The corresponding directed graph has feasible solutions as vertices and a directed edge from $x$ to each member of $N(x)$. If this graph cannot reach a region from the initial solution, even repeated local moves cannot explore that region without a different move or a restart.

For the three-colourability problem, represent a candidate by a complete colour assignment $c:V\to \{r,g,b\}$, initially chosen at random. Give it the conflict objective functionA neighbor changes the colour of one vertex to either other colour. For hill climbing, maximize $-H$, or equivalently repeatedly choose a recolouring that reduces $H$. One can inspect all $2|V|$ neighbors, or focus on conflicted vertices and evaluate their alternative colours using only incident edges. Stop successfully as soon as $H=0$, which is exactly a proper graph colouring with three colours.

If no improving move exists at positive $H$, the method is stuck in a local minimum. Random restarts, limited sideways moves or other exploratory local search choices can reduce this failure mode, but do not turn a finite unsuccessful run into a proof of impossibility. On a graph with no proper graph colouring using three colours, every complete colouring has $H>0$, so this procedure cannot return a valid colouring. Conversely, a positive-conflict stopping point also occurs on some graphs admitting a proper graph colouring with three colours. Failure of hill climbing alone does not certify non-colourability. For a concrete trap, use a bipartite graph with three vertices on each side and every cross-edge, and give each side one vertex of each colour. There are three conflicts. Recolouring any single vertex leaves it with exactly one same-colour neighbor, so no single change improves $H$, even though colouring each whole side differently solves the instance.

A useful local search state is any occupied-square subset $S\subseteq Q$, allowing temporary attacks. Let $C(S)$ count unordered attacking pairs and $N=n^2$. MaximizeACTIONS can offer insertion at an empty square, deletion of an occupied square, and relocation of one knight to an empty square. RESULT is respectively $S\cup \{q\}$, $S\setminus \{p\}$, and $(S\setminus \{p\})\cup \{q\}$. Insertions and deletions alone connect the whole state space; relocation is a useful additional proposal.

Every conflicting state has $F(S)\le N-(N+1)<0$, whereas the empty board has score zero. Every conflict-free state has score exactly its knight count. Hence every global maximizer is a maximum-size nonattacking placement. If a knight participates in $r\ge 1$ conflicts, deleting it changes the score by $-1+(N+1)r>0$, so a positive-conflict local stopping point cannot persist when improving deletions are considered.

From a random board, choose score-improving moves, with random restarts or occasional non-improving proposals to escape poor conflict-free local optima. A maximal placement is not necessarily a maximum placement: accepting only legal insertions can get stuck too early. This objective function defines the intended optimization precisely; a finite heuristic run still gives no general global-optimality certificate.

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.

Thomas Stützle researches computational optimization, including local search and algorithm configuration. His author-maintained IRIDIA homepage describes the irace configuration software and its use of racing algorithms.^[ref] He also coauthored Iterated Local Search by Helena R. Lourenço and others with Helena Ramalhinho Dias Lourenço and Olivier C. Martin.^[ref]

The modular iterated local search framework keeps the embedded local search distinct from a local-search perturbation and the subsequent acceptance decision. That separation makes a solver's settings part of the experimental description rather than incidental implementation detail.

An original budget comparison illustrates a configuration pitfall. Suppose setting A takes $1\mathrm{ms}$ per objective evaluation and setting B takes $2\mathrm{ms}$, with all other costs neglected. In a $100\mathrm{ms}$ budget they can perform $100$ and $50$ evaluations respectively. Comparing both after $100$ evaluations instead gives B twice as much time. Neither evaluation count nor elapsed time is inherently the wrong budget; the comparison must state which resource is constrained. These are hypothetical costs, not irace timings.

This is why algorithm configuration and algorithm selection are related but distinct: one tunes a parameterized solver, while the other chooses among candidates for an instance. A useful evaluation records the tuned parameters, computational budget and held-out instances alongside solution quality.