Local Search and Optimization

Presented by Collin Kanaley

Local Search Algorithms and Optimization

Problems

Local Search Algorithms

-Local search algorithms are useful when the path to the goal does not matter; for example, in the eight-queens problem, what matters is the configuration of the queens, not the order in which they are added to the board.

-This class of problems includes many important applications such as integrated-circuit design, factory-floor layout, job-shop scheduling, automatic programming, telecommunications network optimization, vehicle routing, and portfolio management.

-Local search algorithms operate using a single current state (rather than multiple paths)

-Paths followed are typically not retained-Local search algorithms have two key advantages:

1. They use very little memory2. They can often find reasonable solutions in

large or infinite state spaces for which systematic algorithms are not suitable

-Local search algorithms are also useful for solving pure optimization problems, which aim to find the best state according to an objective function

-The state space landscape is useful for understanding local search; a landscape has both “location” (defined by the state) and “elevation” (defined by the value of the heuristic cost function or objective function)

State Space Landscape (continued)

-If elevation corresponds to cost, then the aim is to find the lowest valley, called a global minimum

-If elevation corresponds to an objective function, then the aim is to find the highest peak, called a global maximum

State Space Landscape (continued)

In a state space landscape:

-A complete local search algorithm always finds a goal if one exists

-An optimal algorithm always finds a global maximum/minimum

Hill-Climbing Search

-The hill-climbing search is a loop that continually moves uphill in the direction of increasing value

-It terminates when it reaches a “peak” where no neighbour has a higher value

-It does not maintain a search tree-This algorithm does not look beyond the immediate

neighbours of the current state-Hill climbing is sometimes called “greedy local

search” because it grabs a good neighbour state without thinking ahead about where to go next

-While hill-climbing searches often perform quite well, they also often get stuck due to:

1. Local maxima – a peak that is higher than each of its neighbouring states, but lower than the global maximum. Hill-climbing algorithms that reach the vicinity of a local maximum will be drawn upwards towards the peak, but then be stuck with now where else to go.

2. Ridges – a sequence of local maxima3. Plateaux – an area of the state space

landscape where the evaluation function is flat

Hill-climbing variations-Stochastic hill climbing chooses at random from

among the uphill moves; the probability of selection can vary with the steepness of the uphill move

-First-choice hill climbing implements stochastic hill climbing by generating successors randomly until one is generated that is better than the current state. This is good when a state has many (e.g., thousands) of successors

-Random-restart hill climbing conducts a series of hill-climbing searches from randomly generated initial states, stopping when a goal is found

-Simulated annealing combines hill climbing with a random walk; this yields both efficiency and completeness

Local Beam Search

-Local beam search - this algorithm keeps track of multiple states as opposed to just one. It begins with k randomly generated states. At each step, all the successors of all k states are generated, and if any is a goal the algorithm halts; otherwise it selects the k best successors from the complete list and repeats. This is different from a random-restart search in that useful information is passed among the k parallel search threads. Therefore, unfruitful searches are quickly abandoned and resources are moved to where the most progress is being made

Stochastic beam search

-Stochastic beam search – this variant of the local beam search chooses k successors at random, as opposed to choosing k from a pool of candidate successors, with the probability of choosing a given successor being an increasing function of its value.

-This helps alleviate the problem local beam search algorithms can have when they become too concentrated in a small region of the state space

Genetic Algorithms

-A genetic algorithm is a variant of stochastic beam search in which successor states are generated by combining two parent states, rather than by just modifying a single state.

Genetic Algorithms (continued)

-Genetic algorithm's begin with a set of k randomly generated states, called the population-Each state, or individual, is represented as a string over a finite alphabet (most commonly a string of 0s and 1s)-Each state is evaluated by the fitness function, which rates better states more highly

Local Search in Continuous Spaces

Local Search in Continuous Spaces

None of the previously described algorithms can handle continuous state spaces. Introduced here are some local search techniques for finding optimal solutions in continuous spaces. Basically, anything that deals with the real world is in such a space.

Problems from local maxima, ridges, and plateaux are just as prevalent in continuous state spaces as they are in local search methods.

-One way to avoid continuous problems is to simply discretize the neighbourhood of each state. This means changing continuous models into discrete models.

-The gradient of the landscape can be used to find a maximum

-When the objective function is not available in a differential form at all, an empirical gradient can be determined by evaluating the response to small increments and decrements in each coordinate. (Empirical gradient search is the same as steepest-ascent hill climbing in a discretized version of the state space.)

Newton-Raphson method

-Newton-Raphson method – for may problems this algorithm is the most effective; this is a general technique for finding roots of functions, a.k.a. Solving equations of the form g(x)=0

-The Newton-Raphson method works by computing a new estimate for the root x according to Newton's formula

x <- x – g(x)/g1(x)

-x must be found so that the gradient is zero. Thus g(x) in Newton's formula becomes ▼f(x) and the updated equation can be written as

x <- x – Hf

-1(x)▼f(x)

where Hf (x) is the Hessian matrix of second

derivatives

Constrained optimization

-An optimization problem is constrained if solutions must satisfy some hard constraints on the values of each variable.

-The difficulty of constrained optimization problems depends upon the nature of the constraints and the objective function

Sources

-Textbook-Wikipedia