Hill Climbing: an Exploratory Local Search
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AI-KI-B Heuristic Search
AI-KI-B Heuristic Search Ute Schmid & Diedrich Wolter Practice: Johannes Rabold Cognitive Systems and Smart Environments Applied Computer Science, University of Bamberg last change: 3. Juni 2019, 10:05 Outline Introduction of heuristic search algorithms, based on foundations of problem spaces and search • Uniformed Systematic Search • Depth-First Search (DFS) • Breadth-First Search (BFS) • Complexity of Blocks-World • Cost-based Optimal Search • Uniform Cost Search • Cost Estimation (Heuristic Function) • Heuristic Search Algorithms • Hill Climbing (Depth-First, Greedy) • Branch and Bound Algorithms (BFS-based) • Best First Search • A* • Designing Heuristic Functions • Problem Types Cost and Cost Estimation • \Real" cost is known for each operator. • Accumulated cost g(n) for a leaf node n on a partially expanded path can be calculated. • For problems where each operator has the same cost or where no information about costs is available, all operator applications have equal cost values. For cost values of 1, accumulated costs g(n) are equal to path-length d. • Sometimes available: Heuristics for estimating the remaining costs to reach the final state. • h^(n): estimated costs to reach a goal state from node n • \bad" heuristics can misguide search! Cost and Cost Estimation cont. Evaluation Function: f^(n) = g(n) + h^(n) Cost and Cost Estimation cont. • \True costs" of an optimal path from an initial state s to a final state: f (s). • For a node n on this path, f can be decomposed in the already performed steps with cost g(n) and the yet to perform steps with true cost h(n). • h^(n) can be an estimation which is greater or smaller than the true costs. -
CAP 4630 Artificial Intelligence
CAP 4630 Artificial Intelligence Instructor: Sam Ganzfried [email protected] 1 • http://www.ultimateaiclass.com/ • https://moodle.cis.fiu.edu/ • HW1 out 9/5 today, due 9/28 (maybe 10/3) – Remember that you have up to 4 late days to use throughout the semester. – https://www.cs.cmu.edu/~sganzfri/HW1_AI.pdf – http://ai.berkeley.edu/search.html • Office hours 2 • https://www.cs.cmu.edu/~sganzfri/Calendar_AI.docx • Midterm exam: on 10/19 • Final exam pushed back (likely on 12/12 instead of 12/5) • Extra lecture on NLP on 11/16 • Will likely only cover 1-1.5 lectures on logic and on planning • Only 4 homework assignments instead of 5 3 Heuristic functions 4 8-puzzle • The object of the puzzle is to slide the tiles horizontally or vertically into the empty space until the configuration matches the goal configuration. • The average solution cost for a randomly generated 8-puzzle instance is about 22 steps. The branching factor is about 3 (when the empty tile is in the middle, four moves are possible; when it is in a corner, two; and when it is along an edge, three). This means that an exhaustive tree search to depth 22 would look at about 3^22 = 3.1*10^10 states. A graph search would cut this down by a factor of about 170,000 because only 181,440 distinct states are reachable (homework exercise). This is a manageable number, but for a 15-puzzle, it would be 10^13, so we will need a good heuristic function. -
Guided Local Search
GUIDED LOCAL SEARCH Christos Voudouris, Edward P. K. Tsang and Abdullah Alsheddy Abstract Combinatorial explosion problem is a well known phenomenon that pre- vents complete algorithms from solving many real-life combinatorial optimization problems. In many situations, heuristic search methods are needed. This chapter de- scribes the principles of Guided Local Search (GLS) and Fast Local Search (FLS) and surveys their applications. GLS is a penalty-based meta-heuristic algorithm that sits on top of other local search algorithms, with the aim to improve their efficiency and robustness. FLS is a way of reducing the size of the neighbourhood to improve the efficiency of local search. The chapter also provides guidance for implement- ing and using GLS and FLS. Four problems, representative of general application categories, are examined with detailed information provided on how to build a GLS- based method in each case. Key words: Heuristic Search, Meta-Heuristics, Penalty-based Methods, Guided Local Search, Tabu Search, Constraint Satisfaction. 1 INTRODUCTION Many practical problems are NP-hard in nature, which means complete, construc- tive search is unlikely to satisfy our computational demand. For example, suppose Christos Voudouris Group Chief Technology Office, BT plc, Orion Building, mlb1/pp12, Marthlesham Heath, Ipswich, IP5 3RE, United Kingdom, e-mail: [email protected] Edward P. K. Tsang Department of Computer Science, Univeristy of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom, e-mail: [email protected] Abdullah Alsheddy Department of Computer Science, Univeristy of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom, e-mail: [email protected] 1 2 Christos Voudouris, Edward P. -
(Stochastic) Local Search Algorithms
DM841 DISCRETE OPTIMIZATION Part 2 – Heuristics (Stochastic) Local Search Algorithms Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Local Search Algorithms Basic Algorithms Outline Local Search Revisited 1. Local Search Algorithms 2. Basic Algorithms 3. Local Search Revisited Components 2 Local Search Algorithms Basic Algorithms Outline Local Search Revisited 1. Local Search Algorithms 2. Basic Algorithms 3. Local Search Revisited Components 3 Local Search Algorithms Basic Algorithms Local Search Algorithms Local Search Revisited Given a (combinatorial) optimization problem Π and one of its instances π: 1. search space S(π) I specified by the definition of (finite domain, integer) variables and their values handling implicit constraints I all together they determine the representation of candidate solutions I common solution representations are discrete structures such as: sequences, permutations, partitions, graphs (e.g., for SAT: array, sequence of truth assignments to propositional variables) Note: solution set S 0(π) ⊆ S(π) (e.g., for SAT: models of given formula) 4 Local Search Algorithms Basic Algorithms Local Search Algorithms (cntd) Local Search Revisited 2. evaluation function fπ : S(π) ! R I it handles the soft constraints and the objective function (e.g., for SAT: number of false clauses) S(π) 3. neighborhood function, Nπ : S ! 2 I defines for each solution s 2 S(π) a set of solutions N(s) ⊆ S(π) that are in some sense close to s. (e.g., for SAT: neighboring variable assignments differ in the truth value of exactly one variable) 5 Local Search Algorithms Basic Algorithms Local Search Algorithms (cntd) Local Search Revisited Further components [according to [HS]] 4. -
Fast and Efficient Black Box Optimization Using the Parameter-Less Population Pyramid
Fast and Efficient Black Box Optimization Using the Parameter-less Population Pyramid B. W. Goldman [email protected] Department of Computer Science and Engineering, Michigan State University, East Lansing, 48823, USA W. F. Punch [email protected] Department of Computer Science and Engineering, Michigan State University, East Lansing, 48823, USA doi:10.1162/EVCO_a_00148 Abstract The parameter-less population pyramid (P3) is a recently introduced method for per- forming evolutionary optimization without requiring any user-specified parameters. P3’s primary innovation is to replace the generational model with a pyramid of mul- tiple populations that are iteratively created and expanded. In combination with local search and advanced crossover, P3 scales to problem difficulty, exploiting previously learned information before adding more diversity. Across seven problems, each tested using on average 18 problem sizes, P3 outperformed all five advanced comparison algorithms. This improvement includes requiring fewer evaluations to find the global optimum and better fitness when using the same number of evaluations. Using both algorithm analysis and comparison, we find P3’s effectiveness is due to its ability to properly maintain, add, and exploit diversity. Unlike the best comparison algorithms, P3 was able to achieve this quality without any problem-specific tuning. Thus, unlike previous parameter-less methods, P3 does not sacrifice quality for applicability. There- fore we conclude that P3 is an efficient, general, parameter-less approach to black box optimization which is more effective than existing state-of-the-art techniques. Keywords Genetic algorithms, linkage learning, local search, parameter-less. 1 Introduction A primary purpose of evolutionary optimization is to efficiently find good solutions to challenging real-world problems with minimal prior knowledge about the problem itself. -
Artificial Intelligence 1: Informed Search
Informed Search II 1. When A* fails – Hill climbing, simulated annealing 2. Genetic algorithms When A* doesn’t work AIMA 4.1 A few slides adapted from CS 471, UBMC and Eric Eaton (in turn, adapted from slides by Charles R. Dyer, University of Wisconsin-Madison)...... Outline Local Search: Hill Climbing Escaping Local Maxima: Simulated Annealing Genetic Algorithms CIS 391 - Intro to AI 3 Local search and optimization Local search: • Use single current state and move to neighboring states. Idea: start with an initial guess at a solution and incrementally improve it until it is one Advantages: • Use very little memory • Find often reasonable solutions in large or infinite state spaces. Useful for pure optimization problems. • Find or approximate best state according to some objective function • Optimal if the space to be searched is convex CIS 391 - Intro to AI 4 Hill Climbing Hill climbing on a surface of states h(s): Estimate of distance from a peak (smaller is better) OR: Height Defined by Evaluation Function (greater is better) CIS 391 - Intro to AI 6 Hill-climbing search I. While ( uphill points): • Move in the direction of increasing evaluation function f II. Let snext = arg maxfs ( ) , s a successor state to the current state n s • If f(n) < f(s) then move to s • Otherwise halt at n Properties: • Terminates when a peak is reached. • Does not look ahead of the immediate neighbors of the current state. • Chooses randomly among the set of best successors, if there is more than one. • Doesn’t backtrack, since it doesn’t remember where it’s been a.k.a. -
A. Local Beam Search with K=1. A. Local Beam Search with K = 1 Is Hill-Climbing Search
4. Give the name that results from each of the following special cases: a. Local beam search with k=1. a. Local beam search with k = 1 is hill-climbing search. b. Local beam search with one initial state and no limit on the number of states retained. b. Local beam search with k = ∞: strictly speaking, this doesn’t make sense. The idea is that if every successor is retained (because k is unbounded), then the search resembles breadth-first search in that it adds one complete layer of nodes before adding the next layer. Starting from one state, the algorithm would be essentially identical to breadth-first search except that each layer is generated all at once. c. Simulated annealing with T=0 at all times (and omitting the termination test). c. Simulated annealing with T = 0 at all times: ignoring the fact that the termination step would be triggered immediately, the search would be identical to first-choice hill climbing because every downward successor would be rejected with probability 1. d. Simulated annealing with T=infinity at all times. d. Simulated annealing with T = infinity at all times: ignoring the fact that the termination step would never be triggered, the search would be identical to a random walk because every successor would be accepted with probability 1. Note that, in this case, a random walk is approximately equivalent to depth-first search. e. Genetic algorithm with population size N=1. e. Genetic algorithm with population size N = 1: if the population size is 1, then the two selected parents will be the same individual; crossover yields an exact copy of the individual; then there is a small chance of mutation. -
Exploring High-Order Neighborhoods by Pattern Mining and Injection
PILS: Exploring high-order neighborhoods by pattern mining and injection Florian Arnolda, ´Italo Santanab, Kenneth S¨orensena, Thibaut Vidalb∗ a University of Antwerp, Department of Engineering Management, Belgium fflorian.arnold,[email protected] bDepartamento de Inform´atica,Pontif´ıciaUniversidade Cat´olicado Rio de Janeiro (PUC-Rio) fisantana,[email protected] Abstract. We introduce pattern injection local search (PILS), an optimization strategy that uses pattern mining to explore high-order local-search neighborhoods, and illustrate its application on the vehicle routing problem. PILS operates by storing a limited number of frequent patterns from elite solutions. During the local search, each pattern is used to define one move in which 1) incompatible edges are disconnected, 2) the edges defined by the pattern are reconnected, and 3) the remaining solution fragments are optimally reconnected. Each such move is accepted only in case of solution improvement. As visible in our experiments, this strategy results in a new paradigm of local search, which complements and enhances classical search approaches in a controllable amount of computational time. We demonstrate that PILS identifies useful high-order moves (e.g., 9-opt and 10-opt) which would otherwise not be found by enumeration, and that it significantly improves the performance of state-of-the-art population-based and neighborhood-centered metaheuristics. Keywords. Local search, Pattern mining, Combinatorial optimization, Vehicle routing problem ∗ Corresponding author 1. Introduction Since the \no free lunch" theorem [46], it is known that no method can | on average | perform arXiv:1912.11462v1 [cs.AI] 24 Dec 2019 better than any other method on all possible problems. -
On the Design of (Meta)Heuristics
On the design of (meta)heuristics Tony Wauters CODeS Research group, KU Leuven, Belgium There are two types of people • Those who use heuristics • And those who don’t ☺ 2 Tony Wauters, Department of Computer Science, CODeS research group Heuristics Origin: Ancient Greek: εὑρίσκω, "find" or "discover" Oxford Dictionary: Proceeding to a solution by trial and error or by rules that are only loosely defined. Tony Wauters, Department of Computer Science, CODeS research group 3 Properties of heuristics + Fast* + Scalable* + High quality solutions* + Solve any type of problem (complex constraints, non-linear, …) - Problem specific (but usually transferable to other problems) - Cannot guarantee optimallity * if well designed 4 Tony Wauters, Department of Computer Science, CODeS research group Heuristic types • Constructive heuristics • Metaheuristics • Hybrid heuristics • Matheuristics: hybrid of Metaheuristics and Mathematical Programming • Other hybrids: Machine Learning, Constraint Programming,… • Hyperheuristics 5 Tony Wauters, Department of Computer Science, CODeS research group Constructive heuristics • Incrementally construct a solution from scratch. • Easy to understand and implement • Usually very fast • Reasonably good solutions 6 Tony Wauters, Department of Computer Science, CODeS research group Metaheuristics A Metaheuristic is a high-level problem independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms. Instead of reinventing the wheel when developing a heuristic -
A Novel LP-Based Local Search Technique – Fast and Quite Good –
A Novel LP-based Local Search Technique – Fast and Quite Good – Vom Fachbereich Informatik der Technische Universit¨at Darmstadt zur Erlangung des akademischen Grades eines Dr. rer. nat. genehmigte Dissertation von Herrn Alaubek Avdil (M.Sc.) aus der Ulaanbaatar, Mongolei Referent: Professor Dr. Karsten Weihe Korreferent: Professor Dr. Matthias M¨uller-Hannemann Tag der Einreichung: 04. Juni 2009 Tag der m¨undlichen Pr¨ufung: 17. Juli 2009 Darmstadt 2009 D 17 ii Acknowledgments I want to express my deep and sincere gratitude to my supervisor Prof. Karsten Weihe for enabling my doctoral study, inspiring with scientific work and motivating and providing me pleasant research and work atmosphere. Without his trust and encouragement my research would not be done and this theses would not exist. I am extremely grateful to Prof. Matthias M¨uller-Hannemann for his invaluable support and advice on my research, excellent and critical review to improve the presentation of my work, and for his friendship, and making it possible for my young family be close with me during my doctoral study in Germany. I would like to express my warm and sincere thanks to the colleagues and members of Algorithmics Group at the Department of Computer Science of Technische Universit¨at Darmstadt for their support and valuable comments on my research. My special ackno- wledgment goes to Dr. Roland Martin for his unforgettable support and encouragement during my doctoral study, and his detailed review and constructive criticism during the writing process of my dissertation. I wish to extend my warmest thanks to all people, who have helped me with my work at the Department of Computer Science at Technische Universit¨at Darmstadt. -
Job Shop Scheduling with Metaheuristics for Car Workshops
Job Shop Scheduling with Metaheuristics for Car Workshops Hein de Haan s2068125 Master Thesis Artificial Intelligence University of Groningen July 2016 First Supervisor Dr. Marco Wiering (Artificial Intelligence, University of Groningen) Second Supervisor Prof. Dr. Lambert Schomaker (Artificial Intelligence, University of Groningen) Abstract For this thesis, different algorithms were created to solve the problem of Jop Shop Scheduling with a number of constraints. More specifically, the setting of a (sim- plified) car workshop was used: the algorithms had to assign all tasks of one day to the mechanics, taking into account minimum and maximum finish times of tasks and mechanics, the use of bridges, qualifications and the delivery and usage of car parts. The algorithms used in this research are Hill Climbing, a Genetic Algorithm with and without a Hill Climbing component, two different Particle Swarm Opti- mizers with and without Hill Climbing, Max-Min Ant System with and without Hill Climbing and Tabu Search. Two experiments were performed: one focussed on the speed of the algorithms, the other on their performance. The experiments point towards the Genetic Algorithm with Hill Climbing as the best algorithm for this problem. 2 Contents 1 Introduction 5 1.1 Scheduling . 5 1.2 Research Question . 7 2 Theoretical Background of Job Shop Scheduling 9 2.1 Formal Description . 9 2.2 Early Work . 11 2.3 Current Algorithms . 13 2.3.1 Simulated Annealing . 13 2.3.2 Neural Networks . 14 3 Methods 19 3.1 Introduction . 19 3.2 Hill Climbing . 20 3.2.1 Introduction . 20 3.2.2 Hill Climbing for Job Shop Scheduling . -
Coupling Ant Colony System with Local Search
UNIVERSITE´ LIBRE DE BRUXELLES Ecole´ Polytechnique de Bruxelles IRIDIA - Institut de Recherches Interdisciplinaires et de D´eveloppements en Intelligence Artificielle Coupling Ant Colony System with Local Search Luca Maria Gambardella Promoteur de Th`ese: Prof. Marco DORIGO Th`esepr´esent´eeen vue de l'obtention du titre de Docteur en Sciences de l'Ing´enieur Ann´eeacad´emique2014-2015 2 Summary In the last decades there has been a lot of interest in computational models and algorithms inspired by the observation of natural systems. Nature seems to be very good at inventing processes that have many of the qualities we would like to be able to put into artificial systems. Consider for example genetic algorithms and evolutionary computation (Holland 1975): by reproducing in a simplified way the approach taken by Nature to develop living beings, researchers were able to develop both control systems for learning autonomous agents, and robust optimiza- tion algorithms for function optimization or combinatorial optimization. We can also think of work in reinforcement learning (Kaelbling et al. 1996): again, the observation of how animals learn to accomplish some simple tasks has suggested a useful and productive set of reinforcement- based learning algorithms. There are many such examples, some already very widespread, like simulated annealing (Kirkpatrick et al. 1983), ge- netic algorithms (Holland 1975), neural networks (Rumelhart et al. 1986), and others that still exist mostly in research labs, like immune networks (Bersini & F.Varela 1993) and classifier systems (Holland & Reitman 1978). In this thesis we present another ethological approach to combinatorial optimization, which has been inspired by the observation of ant colonies: the Ant Colony System (ACS, Gambardella & Dorigo 1996; Dorigo & Gambardella 1997).