Lec 3 | Optimization

Question 1

It is choosing the best option from a set of possible options

Answer 1

Optimization

Question 2

It is a search algorithm that maintains a single node and searches by moving to a neighboring node. It is interested in finding the best answer to a question.

Answer 2

Local Search

Question 3

Will local search give an optimal answer?

Answer 3

Local search will bring to an answer that is not optimal but “good enough”.

Although local search algorithms don’t always give the best possible solution, they can often give a good enough solution in situations where considering every possible state is computationally infeasible.

Question 4

It is a function that we use to maximize the value of the solution.

Answer 4

Objective Function

Question 5

It is a function that we use to minimize the cost of the solution

Answer 5

Cost Function

Question 6

It is the state that is currently being considered by the function.

Answer 6

Current State

Question 7

It is a state that the current state can transition to.

Answer 7

Neighbor State

Question 8

How do local search algorithms work?

Answer 8

Local search algorithm work is by considering one node in a current state, and then moving the node to one of the current state’s neighbors.

Question 9

It is one type of a local search algorithm. Where 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.

Answer 9

Hill Climbing

Question 10

What is the pseudocode for Hill Climbing?

Answer 10

function Hill-Climb(problem):
        current = initial state of problem
        repeat:
                  neighbor = best valued neighbor of current
                  if neighbor not better than current:
                           return current
                  current = neighbor

Question 11

It is short-sighted, often settling for solutions that are better than some others, but not necessarily the best of all possible solutions.

Answer 11

hill climbing algorithm

Question 12

It is a state that has a higher value than its neighboring states

Answer 12

Local Maximum/Maxima

Question 13

It is a state that has the highest value of all states in the state-space.

Answer 13

Global Maximum

Question 14

It is a state that has a lower value than its neighboring states.

Answer 14

Local Minimum/Minima

Question 15

It is a state that has the lowest value of all states in the state-space.

Answer 15

Global Minimum

Question 16

where multiple states of equal value are adjacent, forming a plateau whose neighbors have a worse value

Answer 16

Flat local maximum/minimum

Question 17

where multiple states of equal value are adjacent and the neighbors of the plateau can be both better and worse.

Answer 17

shoulder

Question 18

What is the problem when using the hill climbing algorithm?

Answer 18

The problem with hill climbing algorithms is that they may end up in local minima and maxima. What all variations of the algorithm have in common is that, no matter the strategy, each one still has the potential of ending up in local minima and maxima and no means to continue optimizing.

Question 19

What are the Hill Climbing Variants?

Answer 19

Steepest-ascent, Stochastic, First-choice, random-restart, and Local Beam Search

Question 20

Hill Climbing Variant

It chooses the highest-valued neighbor. It is the standard variation.

Answer 20

Steepest-ascent

Question 21

Hill Climbing Variant

It chooses randomly from higher-valued neighbors.

Answer 21

Stochastic

Question 22

Hill Climbing Variant

It chooses the first higher-valued neighbor.

Answer 22

First-choice

Question 23

Hill climbing Variant

It conducts hill climbing multiple times. Each time, start from a random state. Compare the maxima from every trial, and choose the highest amongst those.

Answer 23

Random-restart

Question 24

Hill climbing variant

It chooses the k highest-valued neighbors. It uses multiple nodes for the search, and not just one

Answer 24

Local Beam Search

Question 25

It allows the algorithm to “dislodge” itself if it gets stuck in a local maximum.

Answer 25

Simulated Annealing

Question 26

It is the process of heating metal and allowing it to cool slowly

Answer 26

Annealing

Question 27

What is the pseudocode for simulated annealing?

Answer 27

function Simulated-Annealing(problem, max):
        current = initial state of problem
              for t = 1 to max:
                        T = Temperature(t)
                        neighbor = random neighbor of current
                        ΔE = how much better neighbor is than current
                        if ΔE > 0:
                                 current = neighbor
                        with probability e^(ΔE/T) set current = neighbor
      return current

Question 28

The task is to connect all points while choosing the shortest possible distance. In this case, a neighbor state might be seen as a state where two arrows swap places. Calculating every possible combination makes this problem computationally demanding. By using the simulated annealing algorithm, a good solution can be found for a lower computational cost

Answer 28

Travelling Salesman Problem

Question 29

It is a family of problems that optimize a linear equation (an equation of the form y = ax₁ + bx₂ + …).

Answer 29

Linear Programming

Question 30

What bare the components of Linear Programming?

Answer 30

• A cost function that we want to minimize: c₁x₁ + c₂x₂ + … + cₙxₙ. Here, each x₋ is a variable and it is associated with some cost c₋.
• A constraint that’s represented as a sum of variables that is either less than or equal to a value (a₁x₁ + a₂x₂ + … + aₙxₙ ≤ b) or precisely equal to this value (a₁x₁ + a₂x₂ + … + aₙxₙ = b). In this case, x₋ is a variable, and a₋ is some resource associated with it, and b is how much resources we can dedicate to this problem.
• Individual bounds on variables (for example, that a variable can’t be negative) of the form lᵢ ≤ xᵢ ≤ uᵢ.

Question 31

What algorithms can we use in Linear Programming?

Answer 31

Simplex and Interior-Point.

Question 32

These are a class of problems where variables need to be assigned values while satisfying some conditions

Answer 32

Constraint Satisfaction problems

Question 33

What are the properties of Constraints satisfaction problems?

Answer 33

• Set of variables (x₁, x₂, …, xₙ)
• Set of domains for each variable {D₁, D₂, …, Dₙ}
• Set of constraints C

Question 34

terms worth knowing about constraint satisfaction problems:

It is a constraint that must be satisfied in a correct solution.

Answer 34

Hard Constraint

Question 35

A few more terms worth knowing about constraint satisfaction problems:

It is a constraint that expresses which solution is preferred over others

Answer 35

Soft Constraint

Question 36

A few more terms worth knowing about constraint satisfaction problems:

It is a constraint that involves only one variable. An example of this constraint would be saying that course A can’t have an exam on Monday {A ≠ Monday}.

Answer 36

Unary Constraint

Question 37

A few more terms worth knowing about constraint satisfaction problems:

It is a constraint that involves two variables. This is the type of constraint that we used in the example above, saying that some two courses can’t have the same value {A ≠ B}.

Answer 37

Binary Constraint

Question 38

It is when all the values in a variable’s domain satisfy the variable’s unary constraints.

Answer 38

Node Consistency

Question 39

It is when all the values in a variable’s domain satisfy the variable’s binary constraints (note that we are now using “arc” to refer to what we previously referred to as “edge”).

Answer 39

Arc Consistency

Question 40

What is the pseudocode that makes a variable arc-consistent with respect to some other variable?

Answer 40

function Revise(csp, X, Y):
       revised = false
       for x in X.domain:
             if no y in Y.domain satisfies constraint for (X,Y):
                     delete x from X.domain
                     revised = true
       return revised

Question 41

What is the pseudocode for the algorithm called AC-3, which uses Revise:

Answer 41

function AC-3(csp):
       queue = all arcs in csp
       while queue non-empty:
               (X, Y) = Dequeue(queue)
               if Revise(csp, X, Y):
                       if size of X.domain == 0:
                              return false
                       for each Z in X.neighbors - {Y}:
                              Enqueue(queue, (Z,X))
      return true

Question 42

A constraint satisfaction problem can be seen as a search problem:

Answer 42

• Initial state: empty assignment (all variables don’t have any values assigned to them).
• Actions: add a {variable = value} to assignment; that is, give some variable a value.
• Transition model: shows how adding the assignment changes the assignment. There is not much depth to this: the transition model returns the state that includes the assignment following the latest action.
• Goal test: check if all variables are assigned a value and all constraints are satisfied.
• Path cost function: all paths have the same cost. As we mentioned earlier, as opposed to typical search problems, optimization problems care about the solution and not the route to the solution.

Question 43

It is a type of a search algorithm that takes into account the structure of a constraint satisfaction search problem. It is a recursive function that attempts to continue assigning values as long as they satisfy the constraints.

Answer 43

Backtracking Search

Question 44

What is the pseudocode for backtracking search?

Answer 44

function Backtrack(assignment, csp):
       if assignment complete:
              return assignment
       var = Select-Unassigned-Var(assignment, csp)
       for value in Domain-Values(var, assignment, csp):
              if value consistent with assignment:
                      add {var = value} to assignment
                      result = Backtrack(assignment, csp)
                      if result ≠ failure:
                               return result
                      remove {var = value} from assignment
       return failure

Question 45

This algorithm will enforce arc-consistency after every new assignment of the backtracking search.

Answer 45

Maintaining Arc-Consistency algorithm.

Question 46

What is the revised pseudocode for backtracking search if it maintains arc-consistency?

Answer 46

function Backtrack(assignment, csp):
        if assignment complete:
                return assignment
        var = Select-Unassigned-Var(assignment, csp)
        for value in Domain-Values(var, assignment, csp):
                if value consistent with assignment:
                           add {var = value} to assignment
                           inferences = Inference(assignment, csp)
                           if inferences ≠ failure:
                                    add inferences to assignment
                          result = Backtrack(assignment, csp)
                          if result ≠ failure:
                                    return result
                          remove {var = value} and inferences from assignment
       return failure

Question 47

Heuristics

It is one such heuristic. The idea here is that if a variable’s domain was constricted by inference, and now it has only one value left (or even if it’s two values), then by making this assignment we will reduce the number of backtracks we might need to do later.

Answer 47

Minimum Remaining Values (MRV)

Question 48

Heuristics

It relies on the degrees of variables, where a __________ is how many arcs connect a variable to other variables.

Answer 48

Degree

Question 49

Heuristics

Where we select the value that will constrain the least other variables.

Answer 49

Least Constraining Values heuristic

Question 50

CS50 QUIZ

For which of the following will you always find the same solution, even if you re-run the algorithm multiple times?

Assume a problem where the goal is to minimize a cost function, and every state in the state space has a different cost.

• Steepest-ascent hill-climbing, each time starting from a different starting state
• Steepest-ascent hill-climbing, each time starting from the same starting state
• Stochastic hill-climbing, each time starting from a different starting state
• Stochastic hill-climbing, each time starting from the same starting state
• Both steepest-ascent and stochastic hill climbing, so long as you always start from the same starting state
• Both steepest-ascent and stochastic hill climbing, each time starting from a different starting state
• No version of hill-climbing will guarantee the same solution every time

Answer 50

Steepest-ascent hill-climbing, each time starting from the same starting state

Question 51

CS50 QUIZ

Consider this optimization problem:

A farmer is trying to plant two crops, Crop 1 and Crop 2, and wants to maximize his profits. The farmer will make $500 in profit from each acre of Crop 1 planted, and will make $400 in profit from each acre of Crop 2 planted.

However, the farmer needs to do all of his planting today, during the 12 hours between 7am and 7pm. Planting an acre of Crop 1 takes 3 hours, and planting an acre of Crop 2 takes 2 hours.

The farmer is also limited in terms of supplies: he has enough supplies to plant 10 acres of Crop 1 and enough supplies to plant 4 acres of Crop 2.

Assume the variable C1 represents the number of acres of Crop 1 to plant, and the variable C2 represents the number of acres of Crop 2 to plant.

What would be a valid objective function for this problem?

• 500 * C1 + 400 * C2
• 500 * 10 * C1 + 400 * 4 * C2
• 10 * C1 + 4 * C2
• -3 * C1 - 2 * C2
• C1 + C2

Answer 51

500 * C1 + 400 * C2

Question 52

CS50 QUIZ

Consider the same optimization problem as in Question 2. What are the constraints for this problem?

• 3 * C1 + 2 * C2 <= 12; C1 <= 10; C2 <= 4
• 3 * C1 + 2 * C2 <= 12; C1 + C2 <= 14
• 3 * C1 <= 10; 2 * C2 <= 4
• C1 + C2 <= 12; C1 + C2 <= 14

Answer 52

3 * C1 + 2 * C2 <= 12; C1 <= 10; C2 <= 4

Question 53

CS50 QUIZ

DI NAKO MABUTANG IRI

Answer 53

https://cs50.harvard.edu/ai/2024/quizzes/3/