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:''This article is about the mathematical algorithm. For other meanings such as the branch of motorsport, please see Hillclimbing (disambiguation).''
'Hill climbing' is an optimization technique which belongs to the family of local search. It is a relatively simple technique to implement, making it a popular first choice. Although more advanced algorithms may give better results, there are situations where hill climbing works well.
Hill climbing can be used to solve problems that have many solutions but where some solutions are better than others. The algorithm is started with a (bad) solution to the problem, and sequentially makes small changes to the solution, each time improving it a little bit. At some point the algorithm arrives at a point where it cannot see any improvement anymore, at which point the algorithm terminates. Ideally, at that point a solution is found that is close to optimal, but it is not guaranteed that hill climbing will ever come close to the optimal solution.
An example of a problem that can be solved with hill climbing is the Traveling salesman problem. It is easy to find a solution that will visit all the cities, but this solution will probably be very bad compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. Eventually, a much better route is obtained.
Hill climbing is used widely in artificial intelligence fields, for reaching a goal state from a starting node. Choice of next node and starting node can be varied to give a list of related algorithms.

Contents

Mathematical Description

Variants

Problems

Local Maxima

Ridges

Plateau

Pseudocode

See also

References

Mathematical Description

Hill climbing attempts to maximize (or minimize) a function $f(x)$, where $x$ are discrete states. These states are typically represented by vertices in a graph, where edges in the graph encode nearness or similarity of a graph. Hill climbing will follow the graph from vertex to vertex, always locally increasing (or decreasing) the value of $f$, until a local maximum $x\_m$ is reached. Hill climbing can also operate on a continuous space: in that case, the algorithm is called gradient ascent (or gradient descent if the function is minimized).

Variants

In 'simple hill climbing', the first closer node is chosen, whereas in 'steepest ascent hill climbing' all successors are compared and the closest to the solution is chosen. Both forms fail if there is no closer node, which may happen if there are local maxima in the search space which are not solutions. Steepest ascent hill climbing is similar to best-first search, which tries all possible extensions of the current path in order instead of only one.
'Random-restart hill climbing' is a meta-algorithm built on top of the hill climbing algorithm. It is also known as 'Shotgun hill climbing'. Random-restart hill climbing simply runs an outer loop over hill-climbing, where each step of the outer loop chooses a random initial condition $x\_0$ to start hill climbing. The best $x\_m$ is kept: if a new run of hill climbing produces a better $x\_m$ than the stored state, it replaces the stored state.
Random-restart hill climbing is a surprisingly effective algorithm in many cases. It turns out that it is often better to spend CPU time exploring the space, rather than carefully optimizing from an initial condition.

Problems

Local Maxima

A problem with hill climbing is that it will find only local maxima. Unless the heuristic is convex, it will not necessarily reach a global maximum. Other local search algorithms try to overcome this problem such as stochastic hill climbing, random walks and simulated annealing.

Ridges

A ridge is a curve in the search place that leads to a maximum, but the orientation of the ridge compared to the available moves that are used to climb is such that each moves will lead to a smaller point. In other words, each point on a ridge looks to the algorithm like a local maximum, even though the point is part of a curve leading to a better optimum.

Plateau

Another problem with hill climbing is that of a plateau, which occurs when we get to a "flat" part of the search space, i.e. we have a path where the heuristics are all very close together. This kind of flatness can cause the algorithm to cease progress and wander aimlessly.

Pseudocode

Hill Climbing Algorithm

   bestEval = -INF;

   currentNode = startNode;

   bestNode = NULL;

   for MAX times

      if (EVAL(currentNode) > bestEval)

          bestEval = EVAL(currentNode);

          bestNode = currentNode;

      L = NEIGHBORS(currentNode);

      nextEval = -INF;

      for all x in L

         if (EVAL(x) > nextEval)

              currentNode = x;

              nextEval = EVAL(x);

   return currentNode;

Contrast genetic algorithm; random optimization.

See also

★ gradient descent

★ greedy algorithm

References

★ Stuart Russell, Peter Norvig, Chapter 4, Prentice-Hall, (1995), ISBN 0-13-103805-2