Question

Consider the following optimization problem: min: ​2x₁​+3x₂​+4x₃​+5x₄+6x₅+7x₆ s.t.:​ x₁​−3x₂​+3x₃​=5 x₄​−x₅​+x₆​≥5 0≤xᵢ​≤9∀ᵢ=1,…,6​ where the variables of the multivariable optimization problem are integers. Use Extension-2 type approach, (i.e., Treat the multiple variables together as one structure). Answer the following: (a) Let [4,5,3,1,1,1] ᵀ be the current solution. Ignore Constraints (2) & (3), and write all the possible feasible immediate neighbors of the current solution. (b) Let [4,5,3,1,1,1] ᵀ be the current solution. Ignore Constraints (2)& (3), and write 3 possible feasible extended neighbors of the current solution. (c) Execute one full iteration of the greedy search with the immediate/star neighborhood. Use starting solution as [4,5,3,1,1,1]ᵀ Handle Constraint(2) & (3) by creating a penalized objective function. Assume all the penalty coefficients are equal to 1000 . (d) Execute 3 full iterations of the random-walk search with the extended/expanded neighborhood. Use starting solution as [4,5,3,1,1,1]ᵀ . Handle Constraint(2) & (3) by creating a penalized objective function. Assume the penalty coefficients are equal to 1000 .

Answer 1

The given optimization problem involves finding the optimal solution while satisfying constraints. Different strategies such as finding immediate and extended neighbors, greedy search, and random-walk search can be used to improve the solution.

Step-by-step explanation:

The given optimization problem aims to minimize an objective function while satisfying certain constraints. To solve this problem, we are provided with a current solution [4, 5, 3, 1, 1, 1] which serves as the starting point for further analysis.

(a) To find feasible immediate neighbors, we need to consider all possible integer values for each variable in the solution except x₁ and x₄. Since x₁ and x₄ have constraints on them, we cannot vary their values freely. By varying the other variables, we can find all possible immediate neighbors of the current solution.

(b) For feasible extended neighbors, we can consider changing multiple variables simultaneously. By varying the values of x₂, x₃, x₅, and x₆ while keeping the values of x₁ and x₄ within their constraints, we can create extended neighbors for the current solution.

(c) Greedy search involves choosing the best possible feasible solution at each iteration based on a penalized objective function. By using the starting solution and penalizing violation of constraints (2) and (3), we can iteratively improve the solution.

(d) Random-walk search involves randomly exploring the solution space. By considering the extended neighborhood and penalizing constraint violations, we can perform multiple iterations to search for a better solution.