Search Algorithms for Discrete Optimization Problems

Discrete optimization arises in various computational incidences.  The advancement of technology is accompanied by several data problems thatoccur as a result of increased demand for faster processors. In the modern world, discrete optimization problems are witnessed in various areas such as gaming, robotics, and other places where a high level of intelligence is required. Due to this reason, the machine develops some complexity in data interpretation. Discrete optimization problems are formulated by trying to find the low-costpath that the information can adopt to reach the target node. The research conducted by Afsar et al. (2017) affirms that the more the data, the more the problems becomechallenging to solve. In the realm of computation, several algorithms are employed to arrive at the final answer toa particular set of data. Commentaries made by Lampoudi et al.(2015)affirm that the only solution to deal with multiple data would involve the use of universally agreed procedures. Basing the argument on the topic under study, DOP can be solved by methods such as backs trapping, dynamic programming and heuristic search. They offer the most accurate output, which can be relied upon for further processes.  Discrete optimization forms a class of expensive problems, but of practical and theoretical importance. The only way to solve such challenges would require more inventive methods, which would guarantee the end user about the efficiency.

In the same vein, search algorithms locate possible space for the solutions, subjects of the constraints.  When the problem is fed into the processor, the algorithmtypically calculate the answer, by either sharing the difficulties among the processors available or executing the task instantly. In the case of the discrete optimization problem, the search algorithm first identifies the space to store input and the final output. This is because complex data consume huge space that can lower the performance (Astrachan et al., 2015). In most cases, such problems are solved through parallel arrangements of the processors.  The primary task is shared among the multiple processors, and the final solution is combined to one standard answer. The following paragraphs will discuss the various subsections of the search algorithms for discrete optimization problems.

Definitions and Examples

Discrete optimization problem can be expressed in a function form referred to as tuple.  It can berepresented as (s, f), where each term has a computational implication. From the representation that has been detailed, S stands for the all sets of solutions that are arrived at after computing the complex challenge. As it was  previously indicated, sophisticated data is usually, solved in pieces (Afsar et al., 2017). Every processor is assigned a specific set, and the final output comprises the summation of all results.  Furthermore, f is the cost of mapping each element in set S to real number R. Some of the costs that are associated with mapping are; energy cost, memory space and also expenses related to the computer hardware. Of all the charges, memory is the most crucial in discrete optimizationproblems (Lampoudi et al., 2015).The objective of DOP is to reduce the cost of mapping all the elements that are located in the set S. The process can be represented mathematically since the computer memory works based on numbers;f(xopt) ≤ f(x) .

From the above equation, f (xopt) represents the maximum cost of mapping all the elements, into a new location. Moreover, f(x) is the cost of each component that is being translated to the new position. Based on the above representation, theoptimum cost of moving all elements should be lower or equal to the summation of all values. Thisis the principle that governs the discrete optimization problems. The search algorithms search for shorter routes that the data can take, to increase the rate at which it’s processed. Longer pathways are linked to latency in communication and delayed rate of output.Some problems such as robots motion planning and test pattern generation can be classified as DoP(Astrachan et al.,2015). This is because they use high levels of data, which need preciseness and accuracy. A slight error could result in halting the entire functioning of particular systems. Splitting the major probleminto smaller sections reduces data redundancy.

The 8-puzzle problem

The puzzle consists of a 3×3 grid which contains a total of eight tiles. One of the spaces is blank, which allows the adjacent tiles to move into the area. When a tileis transferred to the open space, the original position is left empty. Another tile can occupy the new site. This allows rearrangements of the tiles, utilizing the minimum energy possible.  Also,the distance traveled to reach the destination is significantly reduced (Astrachan et al., 2015). Assuming each position in the tile is represented in form of pairs, the distance between the pair [i.j] and [k,l) is illustrated by s | I −k| + |j − l|. Itis referred to as Manhattan distance. The data assumes the shortest distance, hence increasing the productivity. This is the idea behind the 8-puzzle problem thatensures enormous information is solved in high speed. However, it is associated with the challenges of inaccuracies. In the process of relocating the tiles, in this case, the data, a mix up might occur. It is difficult to control the speed of migration from on point to another. Some of the points might lag behind hence confusing. It is always possible to estimate the cost to reach the final destination (Lampoudi et al., 2015). Thishelps in determining the most appropriate algorithm that can be adopted to execute the task. Despite the simplicity offered by the technique, it is equally essential to understand the environment under which the machine is functioning. Gaming programming might be different from the robotic, which requires high technology artificial intelligence. A device that would require logical thinking to perform a particular task would need more modifications to conserve space. Moreover, understanding the nature of the problem helps in establishing the right algorithm to use.

The 0/1 integer-linear-programming problem revisited

In this kind of programming, three pairs of elements are provided.  They are indicated as mxn matrix, vector mx1,and vector nx1. The principal objective is to find nx1 vector factorial that can only take the value0 or 1(Afsar et al., 2017). Based on the rules of the computation, computers accept the data that is fedin forms of zeros and ones.  Whatever is displayed on the screen is different from what happens in the background. The challenge of this problem is to determine how to multiply the matrixes and reorganize the vectors, to fit the programming language of 0/1.  Once the vector that can provide the digits is established, solving the original problem becomes easy. Therefore, the major task is determining the correct path, through which the data can be relayed to the destination node, which is the primary goal in this case (Lampoudi et al.,2015). Conclusively, discrete optimization problem utilizes search algorithms to find a viable route through which information can be transmitted, with minimal abstractions. It enables the processors to decongest the system, by splitting the large data into subsets that can be handled easily. This programming technique is usually helpful in areas where recognition attributes are required. Reduction of the problem discussed improves the performance of the machine.

References

Afsar, B., Aydin, D., Ugur, A., &Korukoglu, S. (2017). Self-Adaptive and Adaptive Parameter Control in Improved Artificial Bee Colony Algorithm. Informatica28(3), 415-438.

Astrachan, O., Morelli, R., Chapman, G., & Gray, J. (2015). Scaling High School Computer Science. Proceedings of the 46th ACM Technical Symposium on Computer Science Education – SIGCSE ’15. Lampoudi, S., Saunders, E., & Eastman, J. (2015). An Integer Linear Programming Solution to the Telescope Network Scheduling Problem. Proceedings of the International Conference on Operations Research and Enterprise Systems.

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