Apr 20, 2023 · Practice. Uniform-Cost Search is a variant of Dijikstra’s algorithm. Here, instead of inserting all vertices into a priority queue, we insert only the source, then one by one insert when needed. In every step, we check if the item is already in the priority queue (using the visited array). If yes, we perform the decrease key, else we insert it. hill climbing search algorithm1 hill climbing algorithm evaluate initial state, if its goal state quit, otherwise make current state as initial state2 select...Beam Search : A heuristic search algorithm that examines a graph by extending the most promising node in a limited set is known as beam search. Beam search is a heuristic search technique that always expands the W number of the best nodes at each level. It progresses level by level and moves downwards only from the best W nodes at each level.HILL CLIMBING: AN INTRODUCTION • Hill Climbing is a heuristic search used for mathematical optimization problems in the field of Artificial Intelligence. • Given a large set of inputs and a good heuristic function, it tries to find a sufficiently good solution to the problem.Say the hidden function is: f (x,y) = 2 if x> 9 & y>9. f (x,y) = 1 if x>9 or y>9 f (x,y) = 0 otherwise. GA Works Well here. Individual = point = (x,y) Mating: something from each so: mate ( {x,y}, {x’,y’}) is {x,y’} and {x’,y}. No mutation Hill-climbing does poorly, GA does well.Hill-climbing The “biggest” hill in the solution landscape is known as the global maximum. The top of any other hill is known as a local maximum (it’s the highest point in the local area). Standard hill-climbing will tend to get stuck at the top of a local maximum, so we can modify our algorithm to restart the hill-climb if need be.Local search algorithms • Hill-climbing search – Gradient descent in continuous state spaces – Can use, e.g., Newton’s method to find roots • Simulated annealing search • Local beam search • Genetic algorithms • Linear Programming (for specialized problems) First, let's talk about the Hill climbing in Artificial intelligence. Hill Climbing Algorithm. It is a technique for optimizing the mathematical problems. Hill Climbing is widely used when a good heuristic is available. It is a local search algorithm that continuously moves in the direction of increasing elevation/value to find the mountain's ...Local search algorithms • Hill-climbing search – Gradient descent in continuous state spaces – Can use, e.g., Newton’s method to find roots • Simulated annealing search • Local beam search • Genetic algorithms • Linear Programming (for specialized problems)Nov 25, 2020 · The algorithm is as follows : Step1: Generate possible solutions. Step2: Evaluate to see if this is the expected solution. Step3: If the solution has been found quit else go back to step 1. Hill climbing takes the feedback from the test procedure and the generator uses it in deciding the next move in the search space. A node of hill climbing algorithm has two components which are state and value. Hill climbing algorithm is a technique which is used for optimizing the mathematical problems. One of the widely discussed examples of Hill climbing algorithm is Traveling-salesman Problem in which we need to minimize the distance traveled by the salesman.Let's take an example of two-player search tree to understand the working of Alpha-beta pruning. Step 1: At the first step the, Max player will start first move from node A where α= -∞ and β= +∞, these value of alpha and beta passed down to node B where again α= -∞ and β= +∞, and Node B passes the same value to its child D. Artificial Intelligence is the study of building agents that act rationally. Most of the time, these agents perform some kind of search algorithm in the background in order to achieve their tasks. A search problem consists of: A State Space. Set of all possible states where you can be. A Start State.Artificial Intelligence Methods Graham Kendall Hill Climbing Hill Climbing Hill Climbing - Algorithm 1. Pick a random point in the search space 2. Consider all the neighbours of the current state 3. Choose the neighbour with the best quality and move to that state 4. Repeat 2 thru 4 until all the neighbouring states are of lower quality 5. Mar 22, 2023 · Artificial Intelligence is the study of building agents that act rationally. Most of the time, these agents perform some kind of search algorithm in the background in order to achieve their tasks. A search problem consists of: A State Space. Set of all possible states where you can be. A Start State. Hill-climbing (or gradient ascent/descent) function Hill-Climbing (problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(problem.Initial-State) loop do neighbor a highest-valued successor of current if neighbor.Value current.Value then return current.State Jul 21, 2019 · Hill Climbing Algorithm: Hill climbing search is a local search problem. The purpose of the hill climbing search is to climb a hill and reach the topmost peak/ point of that hill. It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak. Hill Climbing Algorithm: Hill climbing search is a local search problem. The purpose of the hill climbing search is to climb a hill and reach the topmost peak/ point of that hill. It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak.Ex:- Some games like chess, hill climbing, certain design and scheduling problems. Figure 5: AI Search Algorithms Classification (Image designed by Author ) Search algorithm evaluating criteria:A node of hill climbing algorithm has two components which are state and value. Hill climbing algorithm is a technique which is used for optimizing the mathematical problems. One of the widely discussed examples of Hill climbing algorithm is Traveling-salesman Problem in which we need to minimize the distance traveled by the salesman.Hill Climbing. Hill climbing is one type of a local search algorithm. In this algorithm, the neighbor states are compared to the current state, and if any of them is better, we change the current node from the current state to that neighbor state.Mohammad Faizan Follow Recommended Heuristc Search Techniques Jismy .K.Jose 9.6K views•49 slides Hill climbing algorithm in artificial intelligence sandeep54552 4.7K views•7 slides Control Strategies in AI Amey Kerkar 28.6K views•76 slides Hill climbing algorithm Dr. C.V. Suresh Babu 2.4K views•14 slidesAlgorithm for Simple Hill Climbing: Step 1: Evaluate the initial state, if it is goal state then return success and Stop. Step 2: Loop Until a solution is found or there is no new operator left to ...• Steepest ascent, hill-climbing with limited sideways moves, stochastic hill-climbing, first-choice hill-climbing are all incomplete. • Complete: A local search algorithm is complete if it always finds a goal if one exists. • Optimal: A local search algorithm is complete if it always finds the global maximum/minimum.Here’s the pseudocode for the best first search algorithm: 4. Comparison of Hill Climbing and Best First Search. The two algorithms have a lot in common, so their advantages and disadvantages are somewhat similar. For instance, neither is guaranteed to find the optimal solution. For hill climbing, this happens by getting stuck in the local ...Mar 25, 2018 · In the depth-first search, the test function will merely accept or reject a solution. But in hill climbing the test function is provided with a heuristic function which provides an estimate of how close a given state is to goal state. The hill climbing test procedure is as follows : 1. HILL CLIMBING: AN INTRODUCTION • Hill Climbing is a heuristic search used for mathematical optimization problems in the field of Artificial Intelligence. • Given a large set of inputs and a good heuristic function, it tries to find a sufficiently good solution to the problem.In-and-Out of A* Algorithm • This formula is the heart and soul of this algorithm • These help in optimizing and finding the efficient path www.edureka.co In-and-Out of A* Algorithm • This parameter is used to find the least cost from one node to the other F = G + H • Responsible to find the optimal path between source and destination ...Title: Hill-climbing Search 1 Hill-climbing Search. Goal Optimizing an objective function. Can be applied to goal predicate type of problems. BSAT with objective function number of clauses satisfied. Intuition Always move to a better state ; 2 Some Hill-Climbing Algos. Start State empty state or random state or special state ; Until (no ...Greedy search example Arad (366) 6 februari Pag. 2008 7 AI 1 Assume that we want to use greedy search to solve the problem of travelling from Arad to Bucharest. The initial state=Arad Greedy search example Arad Sibiu(253) Zerind(374) Pag. 2008 8 AI 1 The first expansion step produces: – Sibiu, Timisoara and Zerind Greedy best-first will ...Beam Search is a greedy search algorithm similar to Breadth-First Search (BFS) and Best First Search (BeFS). In fact, we’ll see that the two algorithms are special cases of the beam search. Let’s assume that we have a Graph that we want to traverse to reach a specific node. We start with the root node.Implementation of Best First Search: We use a priority queue or heap to store the costs of nodes that have the lowest evaluation function value. So the implementation is a variation of BFS, we just need to change Queue to PriorityQueue. // Pseudocode for Best First Search Best-First-Search (Graph g, Node start) 1) Create an empty PriorityQueue ...Note that the way local search algorithms work is by considering one node in a current state, and then moving the node to one of the current state’s neighbors. This is unlike the minimax algorithm, for example, where every single state in the state space was considered recursively. Hill Climbing. Hill climbing is one type of a local search ... Random-restart hill climbing is a series of hill-climbing searches with a randomly selected start node whenever the current search gets stuck. See also simulated annealing -- in a moment. A hill climbing example A hill climbing example (2) A local heuristic function Count +1 for every block that sits on the correct thing.Using Computational Intelligence • Heuristic algorithms, ... Illustrative Example Hill-Climbing (assuming maximisation) 1. current_solution = generate initialMar 4, 2021 · Introduction. Hill Climbing In Artificial Intelligence is used for optimizing the mathematical view of the given problems. Thus, in the sizable set of imposed inputs and heuristic functions, an algorithm tries to get the possible solution for the given problem in a reasonable allotted time. Hill climbing suits best when there is insufficient ... ICS 171 Fall 2006 Summary Heuristics and Optimal search strategies heuristics hill-climbing algorithms Best-First search A*: optimal search using heuristics Properties of A* admissibility, monotonicity, accuracy and dominance efficiency of A* Branch and Bound Iterative deepening A* Automatic generation of heuristics Problem: finding a Minimum Cost Path Previously we wanted an arbitrary path to ... Best first search algorithm: Step 1: Place the starting node into the OPEN list. Step 2: If the OPEN list is empty, Stop and return failure. Step 3: Remove the node n, from the OPEN list which has the lowest value of h (n), and places it in the CLOSED list. Step 4: Expand the node n, and generate the successors of node n.May 15, 2023 · Here’s the pseudocode for the best first search algorithm: 4. Comparison of Hill Climbing and Best First Search. The two algorithms have a lot in common, so their advantages and disadvantages are somewhat similar. For instance, neither is guaranteed to find the optimal solution. For hill climbing, this happens by getting stuck in the local ... Hill-climbing The “biggest” hill in the solution landscape is known as the global maximum. The top of any other hill is known as a local maximum (it’s the highest point in the local area). Standard hill-climbing will tend to get stuck at the top of a local maximum, so we can modify our algorithm to restart the hill-climb if need be.Simple Hill Climbing: Simple hill climbing is the simplest way to implement a hill climbing algorithm. It only evaluates the neighbor node state at a time and selects the first one which optimizes current cost and set it as a current state.In this video we will talk about local search method and discuss one search algorithm hill climbing which belongs to local search method. We will also discus...Hill-climbing (or gradient ascent/descent) \Like climbing Everest in thick fog with amnesia" function Hill-Climbing(problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(Initial-State[problem]) loop do neighbor a highest-valued successor of currentHill-climbing (or gradient ascent/descent) function Hill-Climbing (problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(problem.Initial-State) loop do neighbor a highest-valued successor of current if neighbor.Value current.Value then return current.State CSCI 5582 Artificial Intelligence. CS 2710, ISSP 2610 R&N Chapter 4.1 Local Search and Optimization * Example Local Search Problem Formulation Group travel: people traveling from different places: See chapter4example.txt on the course schedule. From Segaran, T. Programming Collective Intelligence, O’Reilly, 2007.Jul 21, 2019 · Hill Climbing Algorithm: Hill climbing search is a local search problem. The purpose of the hill climbing search is to climb a hill and reach the topmost peak/ point of that hill. It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak. Jan 28, 2022 · Hill Climbing Search Solved Example using Local and Global Heuristic Function by Dr. Mahesh HuddarThe following concepts are discussed:_____... Best first search algorithm: Step 1: Place the starting node into the OPEN list. Step 2: If the OPEN list is empty, Stop and return failure. Step 3: Remove the node n, from the OPEN list which has the lowest value of h (n), and places it in the CLOSED list. Step 4: Expand the node n, and generate the successors of node n. This information can be in the form of heuristics, estimates of cost, or other relevant data to prioritize which states to expand and explore. Examples of informed search algorithms include A* search, Best-First search, and Greedy search. Example: Greedy Search and Graph Search. Here are some key features of informed search algorithms in AI:Abstract: The paper proposes artificial intelligence technique called hill climbing to find numerical solutions of Diophantine Equations. Such equations are important as they have many applications in fields like public key cryptography, integer factorization, algebraic curves, projective curves and data dependency in super computers. Future of Artificial Intelligence. Undoubtedly, Artificial Intelligence (AI) is a revolutionary field of computer science, which is ready to become the main component of various emerging technologies like big data, robotics, and IoT. It will continue to act as a technological innovator in the coming years. In just a few years, AI has become a ... Hill Climbing. Hill climbing is one type of a local search algorithm. In this algorithm, the neighbor states are compared to the current state, and if any of them is better, we change the current node from the current state to that neighbor state.Local search algorithms • Hill-climbing search – Gradient descent in continuous state spaces – Can use, e.g., Newton’s method to find roots • Simulated annealing search • Local beam search • Genetic algorithms • Linear Programming (for specialized problems) Feb 8, 2022 · Ex:- Some games like chess, hill climbing, certain design and scheduling problems. Figure 5: AI Search Algorithms Classification (Image designed by Author ) Search algorithm evaluating criteria: A sufficiently good solution to the desired function, given sufficient training data goal from the state!: when reaching a plateau, jump somewhere hill climbing algorithm in artificial intelligence with example ppt and restart the algorithm, the algorithm with. Is a heuristic search Puzzle problem in AI ( Artificial Intelligence...Dec 31, 2017 · A* search. Renas R. Rekany Artificial Intelligence Nawroz University Keep Reading as long as you breathComSci: Renas R. Rekany Oct2016 5 Hill Climbing • Hill climbing search algorithm (also known as greedy local search) uses a loop that continually moves in the direction of increasing values (that is uphill). Jul 21, 2019 · Hill Climbing Algorithm: Hill climbing search is a local search problem. The purpose of the hill climbing search is to climb a hill and reach the topmost peak/ point of that hill. It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak. Stochastic Hill climbing is an optimization algorithm. It makes use of randomness as part of the search process. This makes the algorithm appropriate for nonlinear objective functions where other local search algorithms do not operate well. It is also a local search algorithm, meaning that it modifies a single solution and searches the ...Hill climbing algorithm is a local search algorithm that continuously moves in the direction of increasing elevation/value to find the peak of the mountain o... Hill-climbing (or gradient ascent/descent) function Hill-Climbing (problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(problem.Initial-State) loop do neighbor a highest-valued successor of current if neighbor.Value current.Value then return current.State Example 1 Apply the hill climbing algorithm to solve the blocks world problem shown in Figure. Solution To use the hill climbing algorithm we need an evaluation function or a heuristic function.Local search algorithms • Hill-climbing search – Gradient descent in continuous state spaces – Can use, e.g., Newton’s method to find roots • Simulated annealing search • Local beam search • Genetic algorithms • Linear Programming (for specialized problems) Hill Climbing Search Solved Example using Local and Global Heuristic Function by Dr. Mahesh HuddarThe following concepts are discussed:_____...Hill-climbing Search The successor function is where the intelligence lies in hill-climbing search It has to be conservative enough to preserve significant “good” portions of the current solution And liberal enough to allow the state space to be preserved without degenerating into a random walk Hill-climbing search Problem: depending on ...Introduction HillHill climbingclimbing. Artificial Intelligence search algorithms Search techniques are general problem-solving methods. When there is a formulated search problem, a set of states, a set of operators, an initial state, and a goal criterion we can use search techniques to solve the problem (Pearl & Korf, 1987)Disadvantages: The question that remains on hill climbing search is whether this hill is the highest hill possible. Unfortunately without further extensive exploration, this question cannot be answered. This technique works but as it uses local information that’s why it can be fooled. The algorithm doesn’t maintain a search tree, so the ...Mar 27, 2022 · INTRODUCTION Hill Climbing is a heuristic search that tries to find a sufficiently good solution to the problem, according to its current position. Types of Hill climbing: • Simple Hill climbing: select first node that is closer to the solution state than current node. • Steepest-Ascent Hill climbing: examines all nodes then selects closest ... Hill-Climbing Search The hill-climbing search algorithm (or steepest-ascent) moves from the current state towards the neighbor-ing state that increases the objective value the most. The algorithm does not maintain a search tree but only the states and the corresponding values of the objective. The “greediness" of hill-climbing makes it vulnera-Hill climbing is basically a search technique or informed search technique having different weights based on real numbers assigned to different nodes, branches, and goals in a path. In AI, machine learning, deep learning, and machine vision, the algorithm is the most important subset. With the help of these algorithms, ( What Are Artificial ...N-Queens Problem. N - Queens problem is to place n - queens in such a manner on an n x n chessboard that no queens attack each other by being in the same row, column or diagonal. It can be seen that for n =1, the problem has a trivial solution, and no solution exists for n =2 and n =3. So first we will consider the 4 queens problem and then ...Example 1 Apply the hill climbing algorithm to solve the blocks world problem shown in Figure. Solution To use the hill climbing algorithm we need an evaluation function or a heuristic function.The less optimal solution and the solution is not guaranteed. Algorithm for Simple Hill Climbing: Step 1: Evaluate the initial state, if it is a goal state then return success and Stop. Step 2 ...See also Steps to Solve Problems in Artificial Intelligence. 1. Current state = (0, 0) 2. Loop until the goal state (2, 0) reached. – Apply a rule whose left side matches the current state. – Set the new current state to be the resulting state. (0, 0) – Start State. (0, 3) – Rule 2, Fill the 3-liter jug. There are mainly four ways of knowledge representation which are given as follows: Logical Representation. Semantic Network Representation. Frame Representation. Production Rules. 1. Logical Representation. Logical representation is a language with some concrete rules which deals with propositions and has no ambiguity in representation. Hill Climbing Search Solved Example using Local and Global Heuristic Function by Dr. Mahesh HuddarThe following concepts are discussed:_____...move. For example, we could try 3-opt, rather than a 2-opt move when implementing the TSP. Unfortunately, neither of these have proved satisfactory in practice when using a simple hill climbing algorithm. Simulated annealing solves this problem by allowing worse moves (lesser quality) to be taken some of the time.Algorithm for Simple Hill Climbing: Step 1: Evaluate the initial state, if it is goal state then return success and Stop. Step 2: Loop Until a solution is found or there is no new operator left to apply. Step 3: Select and apply an operator to the current state. If it is goal state, then return success and quit.More on hill-climbing • Hill-climbing also called greedy local search • Greedy because it takes the best immediate move • Greedy algorithms often perform quite well 16 Problems with Hill-climbing n State Space Gets stuck in local maxima ie. Eval(X) > Eval(Y) for all Y where Y is a neighbor of X Flat local maximum: Our algorithm terminates ...Note that the way local search algorithms work is by considering one node in a current state, and then moving the node to one of the current state’s neighbors. This is unlike the minimax algorithm, for example, where every single state in the state space was considered recursively. Hill Climbing. Hill climbing is one type of a local search ...INTRODUCTION Hill Climbing is a heuristic search that tries to find a sufficiently good solution to the problem, according to its current position. Types of Hill climbing: • Simple Hill climbing: select first node that is closer to the solution state than current node. • Steepest-Ascent Hill climbing: examines all nodes then selects closest ...There are several variations of Hill Climbing, including steepest ascent Hill Climbing, first-choice Hill Climbing, and simulated annealing. In steepest ascent Hill Climbing, the algorithm evaluates all the possible moves from the current solution and selects the one that leads to the best improvement.In Artificial Intelligence, Search techniques are universal problem-solving methods. Rational agents or Problem-solving agents in AI mostly used these search strategies or algorithms to solve a specific problem and provide the best result. Problem-solving agents are the goal-based agents and use atomic representation.Jan 28, 2022 · Hill Climbing Search Solved Example using Local and Global Heuristic Function by Dr. Mahesh HuddarThe following concepts are discussed:_____... Step1: Generate possible solutions. Step2: Evaluate to see if this is the expected solution. Step3: If the solution has been found quit else go back to step 1. Hill climbing takes the feedback from the test procedure and the generator uses it in deciding the next move in the search space.Aug 16, 2021 · Hill climbing algorithm. HILL CLIMBING ALGORITHM Dr. C.V. Suresh Babu (CentreforKnowledgeTransfer) institute HILL CLIMBING: AN INTRODUCTION • Hill Climbing is a heuristic search used for mathematical optimization problems in the field of Artificial Intelligence. • Given a large set of inputs and a good heuristic function, it tries to find a ...
For example, the travelling salesman problem, the eight-queens problem, circuit design, and a variety of other real-world problems. Hill Climbing has been used in inductive learning models. One such example is PALO, a probabilistic hill climbing system which models inductive and speed-up learning.. Coca cola auburndale shipping
Hill Climbing Algorithm: Hill climbing search is a local search problem. The purpose of the hill climbing search is to climb a hill and reach the topmost peak/ point of that hill. It is based on the heuristic search technique where the person who is climbing up on the hill estimates the direction which will lead him to the highest peak.Beam Search : A heuristic search algorithm that examines a graph by extending the most promising node in a limited set is known as beam search. Beam search is a heuristic search technique that always expands the W number of the best nodes at each level. It progresses level by level and moves downwards only from the best W nodes at each level.A class of general purpose algorithms that operates in a brute force way The search space is explored without leveraging on any information on the problem Also called blind search, or naïve search Since the methods are generic they are intrinsically inefficient E.g. Random Search Feb 14, 2020 · In-and-Out of A* Algorithm • This formula is the heart and soul of this algorithm • These help in optimizing and finding the efficient path www.edureka.co In-and-Out of A* Algorithm • This parameter is used to find the least cost from one node to the other F = G + H • Responsible to find the optimal path between source and destination ... Techniques of knowledge representation. There are mainly four ways of knowledge representation which are given as follows: Logical Representation. Semantic Network Representation. Frame Representation. Production Rules. 1. Logical Representation. Logical representation is a language with some concrete rules which deals with propositions and has ...Hill-climbing Search The successor function is where the intelligence lies in hill-climbing search It has to be conservative enough to preserve significant “good” portions of the current solution And liberal enough to allow the state space to be preserved without degenerating into a random walk Hill-climbing search Problem: depending on ... Jul 21, 2022 · Hill climbing is basically a search technique or informed search technique having different weights based on real numbers assigned to different nodes, branches, and goals in a path. In AI, machine learning, deep learning, and machine vision, the algorithm is the most important subset. With the help of these algorithms, ( What Are Artificial ... Dec 27, 2019 · 👉Subscribe to our new channel:https://www.youtube.com/@varunainashots 🔗Link for AI notes: https://rb.gy/9kj1z👩🎓Contributed by: Nisha GuptaHill Climbing ... N-Queens Problem. N - Queens problem is to place n - queens in such a manner on an n x n chessboard that no queens attack each other by being in the same row, column or diagonal. It can be seen that for n =1, the problem has a trivial solution, and no solution exists for n =2 and n =3. So first we will consider the 4 queens problem and then ...Hill-climbing (or gradient ascent/descent) function Hill-Climbing (problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(problem.Initial-State) loop do neighbor a highest-valued successor of current if neighbor.Value current.Value then return current.StateFor example, the travelling salesman problem, the eight-queens problem, circuit design, and a variety of other real-world problems. Hill Climbing has been used in inductive learning models. One such example is PALO, a probabilistic hill climbing system which models inductive and speed-up learning.Hill Climbing Algorithm In Artificial Intelligence | Artificial Intelligence Tutorial | Simplilearn. This presentation on the Hill Climbing Algorithm will help you understand what Hill Climbing Algorithm is and its features. You will get an idea about the state and space diagrams and learn the Hill Climbing Algorithms types.Hill-climbing (or gradient ascent/descent) \Like climbing Everest in thick fog with amnesia" function Hill-Climbing(problem) returns a state that is a local maximum inputs: problem, a problem local variables: current, a node neighbor, a node current Make-Node(Initial-State[problem]) loop do neighbor a highest-valued successor of current More on hill-climbing • Hill-climbing also called greedy local search • Greedy because it takes the best immediate move • Greedy algorithms often perform quite well 16 Problems with Hill-climbing n State Space Gets stuck in local maxima ie. Eval(X) > Eval(Y) for all Y where Y is a neighbor of X Flat local maximum: Our algorithm terminates ....