Hill Climbing: Fundamentals and Applications

By Fouad Sabry

What Is Hill Climbing

Hill climbing is a method of mathematical optimization that is used in numerical analysis. It is a member of the family of techniques known as local search. It is an iterative algorithm that begins with an arbitrary solution to a problem, and then seeks to discover a better answer by making incremental changes to the initial solution in order to see whether it leads to a better solution. If the change results in a better solution, then another incremental adjustment is made to the new solution, and so on and so forth, until there is no possibility of any further improvements being found.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Hill climbing

Chapter 2: Gradient descent

Chapter 3: Greedy algorithm

Chapter 4: Mean shift

Chapter 5: A* search algorithm

Chapter 6: Mathematical optimization

Chapter 7: Local search (optimization)

Chapter 8: Iterative method

Chapter 9: Travelling salesman problem

Chapter 10: Local optimum

(II) Answering the public top questions about hill climbing.

(III) Real world examples for the usage of hill climbing in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of hill climbing' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of hill climbing.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jul 1, 2023

Book preview

Chapter 8: Hill climbing

Hill climbing is a method of mathematical optimization that is used in numerical analysis. It is a member of the family of techniques known as local search. It is an iterative method that begins with an arbitrary solution to a problem, and then seeks to discover a better answer by making incremental changes to the initial solution in order to see whether it leads to a better solution. If the modification results in a better solution, then another incremental adjustment is made to the new solution, and so on and so forth, until there are no more improvements that can be identified.

The traveling salesman issue, for instance, lends itself well to the solution of hill climbing. An early solution that visits all of the cities can be found rather easily, but it is very unlikely to be particularly good in comparison to the ideal option. The algorithm begins with such a solution and then performs a series of incremental improvements to it, such as rearranging the order in which two cities are visited. In the end, it is probable that a path that is much shorter will be obtained.

Hill climbing will only identify local optima for issues that are not convex; for convex problems, it will find the best possible solutions (solutions that cannot be improved upon by any neighboring configurations), They may not always constitute the optimal answer out of all conceivable alternatives (the global optimum) (the search space).

Examples of algorithms that solve convex problems by hill-climbing include the simplex algorithm for linear programming and binary search.: 253 To attempt to avoid getting stuck in local optima, one could employ restarts (i.e.

repeated searches within the same area), or even more intricate plans that are built on iterations (like iterated local search), or by recollection (like reactive search optimization and tabu search), alternatively, on memoryless stochastic alterations (like simulated annealing).

Because of the algorithm's moderate level of complexity, it is often selected as a viable option for use in first optimization attempts. In artificial intelligence, it is put to extensive use for the purpose of progressing from an initial node to a desired destination state. In similar algorithms, a variety of alternatives are employed for both the initial nodes and the following nodes. It's possible that more complex algorithms, such simulated annealing or tabu search, may provide superior results more of the time, yet hill climbing can still be quite useful in some contexts. When the amount of time available to perform a search is limited, such as when dealing with real-time systems, hill climbing can frequently produce a better result than other algorithms can, so long as a small number of increments typically converges on a good solution. This is the case provided that the number of increments is kept to a minimum (the optimal solution or a close approximation). On the other hand, bubble sort may be interpreted as a hill climbing algorithm (every nearby element exchange reduces the number of disordered element pairs), however this strategy is not even remotely efficient for even tiny N, since the number of needed exchanges rises in a quadratic fashion.

Hill climbing is an anytime algorithm, which means that it may deliver a correct answer even if it is halted at any point before it completes.

Hill climbing attempts to maximize (or minimize) a target function f(\mathbf {x} ) , where \mathbf {x} is a vector of continuous and/or discrete values.

Each time during the cycle, hill climbing will adjust a single element in \mathbf {x} and determine whether the change improves the value of f(\mathbf {x} ) .

(It is important to note that this is not the same as gradient descent approaches, which adjust all of the values in \mathbf {x} at each iteration according to the gradient of the hill.) With hill climbing, any change that improves f(\mathbf {x} ) is accepted, and the process continues until no change can be found to improve the value of f(\mathbf {x} ) .

Then \mathbf {x} is said to be locally optimal.

In discrete vector spaces, each possible value for \mathbf {x} may be visualized as a vertex in a graph.

The graph will be followed from vertex to vertex as the hill is climbed, always locally increasing (or decreasing) the value of f(\mathbf {x} ) , until a local maximum (or local minimum) x_{m} is reached.

In straightforward hill climbing, the node that is closest to the top of the hill is selected, but in steepest ascent hill climbing, all of the successors are compared and the one that is most similar to the solution is selected. Both variants are invalid if the search space contains local maxima that are not solutions and there is no closer node. This might occur if there are local maxima in the search space. The steepest ascent technique is a kind of hill climbing that is analogous to the best-first search algorithm, which searches for all potential extensions of the present route rather than just one.

The neighbors are not taken into consideration while determining how to proceed in stochastic hill climbing. Instead, it picks a neighbor at random, and then, depending on how much progress that neighbor has made, it determines whether to go to that neighbor or to investigate another neighbor.

In each iteration, the coordinate descent algorithm does a line search along one coordinate direction beginning at the current position. Some implementations of the coordinate descent algorithm choose the next coordinate direction for each iteration at random.

The hill climbing algorithm is the foundation of the random-restart hill climbing meta-algorithm, which is constructed upon it.

Hill climbing with a shotgun is another name for it.

Hill climbing is done in an iterative fashion by it, each time with a random initial condition x_{0} .

The best x_{m} is kept: if a new run of hill climbing produces a better x_{m} than the stored state, It takes the place of the previously stored state.

In many contexts, the technique known as random-restart hill climbing proves to be remarkably successful. It has been shown that it is often more beneficial to spend CPU time exploring the area rather than cautiously optimizing from an initial state.

Hill climbing will not always lead to the discovery of the global maximum; rather, it may lead to the discovery of a local maximum. If the heuristic is convex, then there will be no occurrences of this issue. However, due to the fact that many functions are not convex, hill climbing does not always succeed in reaching a global maximum. Other types of local search algorithms, such as stochastic hill climbing, random walks, and simulated annealing, also make an attempt to solve this issue.

Hill climbers who maximize their performance in continuous expanses have a difficult dilemma once they encounter ridges. Hill climbers only alter one element in the vector at a time, therefore each step they take will move in the direction that is aligned with the axis. If the target function creates a narrow ridge that climbs in a direction that is not aligned with the axis, or if the goal is to minimize, if the target function creates a narrow alley that descends in a direction that is not aligned with the axis, then the only way for the hill climber to climb the ridge (or descend the alley) is by zigzagging. If the sides of the ridge (or the alley) are particularly steep, the hill climber may be compelled to take very little steps as it zigzags toward a better position on the ridge (or in the alley). As a consequence of this, it can take an unreasonably long amount of time for it to climb the ridge (or descend the alley).

On the other hand, gradient descent techniques are able to proceed in either direction, regardless of whether the ridge or alley is ascending or descending. Therefore, when the goal function can be differentiated, the gradient descent approach or the conjugate gradient method is chosen over hill climbing most of the time. However, hill climbers have the benefit of not needing the goal function to be differentiable; hence, hill climbers are likely to be chosen when the target function is difficult to differentiate.

Hill climbing might also provide you with the additional challenge of reaching a plateau at some point. When the search space is flat, or when it is sufficiently flat, the value that is returned by the target function becomes indistinguishable from the value that is returned for neighboring regions due