Single Candidate Methods

Transcription

1 Single Candidate Methods In Heuristic Optimization Based on: [3] S. Luke, "Essentials of Metaheuristics," [Online]. Available: [Accessed 11 May 2015]. 1

2 Example Problem Many optimization problems don t have an explicit function to optimize, because the function is unknown. For example, the Travelling Salesman Problem: This function would input the distances of an arbitrary number of cities, in an arbitrary order, and compute the route distance. If we knew this function, and it was differentiable, we might be able to find its maxima by taking first and second order derivatives, and hence find the global optimal route. But, a closed form of this function cannot be simply written down. 2

3 Example Problem Another example, the Humanoid Robot simulator: You have different operations, and your candidate solutions are arbitrary-length strings of these operations. You can plug a string in the simulator and get a quality out (how far the robot moved forward before it fell over). How do you find a good solution? 3

4 Characteristic Problem In these types of problems, all you re given is a black box (in this case, the robot simulator) describing a problem that you d like to optimize. The box has a slot where you can submit a candidate solution to the problem (here, a string of timed robot operations). Then you press the big red button and out comes the assessed quality of that candidate solution. You have no idea what kind of surface the quality assessment function looks like when plotted. 4

5 Solution Procedure To optimize a candidate solution in this scenario, you need to be able to do four things: 1. Provide one or more initial candidate solutions. This is known as the initialization procedure. 2. Assess the Quality of a candidate solution. This is known as the assessment procedure. 3. Make a Copy of a candidate solution. 4. Tweak a candidate solution, which produces a randomly slightly different candidate solution. This, plus the Copy operation, are collectively known as the modification procedure. 5

6 Algorithm 4: Hill Climbing 1: S some initial candidate solution. //The Initialization Procedure 2: repeat 3: R Tweak(Copy(S)) //The Modification Procedure 4: if Quality(R) > Quality(S) then //The Assessment and Selection Procedures 5: S R 6: until S is the ideal solution or we have run out of time 7: return S This enables you to climb up the hill until you reach a local optimum. 6

7 Algorithm 1: Gradient Ascent 1: Random ini al vector. //The Initialization Procedure 2: repeat 3: + ( ) //The Modification Procedure 4: until is the ideal solution or we have run out of time 5: return S Notice the strong resemblance between Hill-Climbing and Gradient Ascent. The only real difference is that Hill-Climbing s more general Tweak operation must instead rely on a stochastic (partially random) approach to hunting around for better candidate solutions. Sometimes it finds worse ones nearby, sometimes it finds better ones. 7

8 Algorithm 5: Steepest Ascent Hill-Climbing 8

9 Algorithm 6: Steepest Ascent Hill-Climbing with Replacement 9

10 Comparing Algorithm 5 & 6: 10

11 Algorithm 7: Generate a Random Real-Valued Vector 11

12 Algorithm 8: Perturb a Random Real-Valued Vector 12

13 Tunable Parameter, the size of the bound on Tweak. If the size of is very small, then Hill-Climbing will march right up a local hill and be unable to make the jump to the next hill because the bound is too small for it to jump that far. Once it s on the top of a hill, everywhere it jumps will be worse than where it is presently, so it stays put. Further, the rate at which it climbs the hill will be bounded by its small size. On the other hand, if the size is large, then Hill-Climbing will bounce around a lot. This knob is one way of controlling the degree of Exploration versus Exploitation in our Hill-Climber. Optimization algorithms which make largely local improvements are exploiting the local gradient, and algorithms which mostly wander about randomly are thought to explore the space. 13

14 Algorithm 10: Hill-Climbing with Random Restarts 14

15 Algorithm 10: Hill-Climbing with Random Restarts If the randomly-chosen time intervals are generally extremely long, this algorithm is basically one big Hill-Climber. Likewise, if the intervals are very short, we re basically doing random search (by resetting to random new locations each time). 15

16 Analysis 1 Doing Hill-Climbing is close to optimal. Doing Random Search is a very bad pick. 16

17 Analysis 2 Hill-Climbing is quite bad. Random Search is expected to be as good you can do. 17

18 Analysis 3 Hill-Climbing is quite poor. For local search to be effective there must be an informative gradient which generally leads towards the best solutions. Random Search is better. 18

19 Analysis 4 Hill-Climbing: optimum not easily found. It is actively led away from the optimum. 19

20 Rules of Thumb 1 Adjust the Modification procedure Tweak occasionally makes large, random changes. If you run the algorithm long enough, this randomness will cause Tweak to eventually try every possible solution. The more large, random changes, the more exploration. 20

21 Rules of Thumb 2 Adjust the Selection procedure Change the algorithm so that you can go down hills at least some of the time. If you run the algorithm long enough, you ll go down enough hills that you ll eventually find the right hill to go up. The more often you go down hills, the more exploration. 21

22 Rules of Thumb 3 Jump to Something New Every once in a while start from a new location. If you try enough new locations, eventually you ll hit a hill which has the highest peak. The more frequently you restart, the more exploration. 22

23 Rules of Thumb 4 Use a Large Sample Try many candidate solutions in parallel. With enough parallel candidate solutions, one of them is bound to be on the highest peak. More parallel candidate solutions, more exploration. 23

24 Simulated Annealing 1. Choose a starting design. 2. Select,,, and calculate,, and. 3. Perform L cycles and do the following Z times 4. Randomly perturb the design to different discrete values close to the current design. 5. If the new design is better, accept it as the current design. 6. If the new design is worse, generate a random number between 0 and 1 using a uniform distribution. Compare this number to the Boltzmann probability. If the random number is lower than the Boltzmann probability, accept the worse design as the current design. 7. Continue perturbing the design randomly at the current temperature until steady state is reached, i.e., until Z=0. 8. Decrease temperature according to = 9. Go to step Repeat another cycle or until is reached. 24

25 Tabu Search 25

26 Iterated Local Search 26

27 References [1] J. D. Hedengren, "Optimization Techniques in Engineering," 5 April [Online]. Available: [Accessed 27 April 2015]. [2] A. R. Parkinson, R. J. Balling and J. D. Heden, "Optimization Methods for Engineering Design Applications and Theory," Brigham Young University, [3] S. Luke, "Essentials of Metaheuristics," [Online]. Available: [Accessed 11 May 2015]. [4] J. D. Hedengren, "Genetic Algorithms in Engineering Design," [Online]. Available: [Accessed 28 April 2015]. 27