1. Write down the entire polynomial-size formulation for Asymmetric Travelling Salesman Problem
2. Explain the constraints and the variables, and HOW the polynomially many constraints work in eliminating subtours.
Let a graph G = (V, E) be a given complete digraph, where V is a vertices V={1,…,n} is the vertex set and E={(i,j):i,j∈V} is the arc set, and let c ij be the cost associated with arc (i,j)∈E (with cii=+∞ for i∈V).
A Hamiltonian circuit (tour) of G is a circuit visiting each vertex of Vexactly once. The Asymmetric Traveling Salesman Problem (ATSP) is to find a Hamiltonian circuit G∗=(V,E∗) of G whose cost ∑(i,j)∈A∗cij is minimum..
The polynomial formulations can be directly solved by a general-purpose ILP solver.
The earliest polynomial formulation of the ATSP is owed to Miller (1960) (hereafter MTZ)
ui−uj+(n−1)xij ≤ n−2,i,j=2,…,n;
where ui,i=2,…,n,ui,i=2,…,n, is an arbitrary real number representing the order of vertex i in the optimal tour, and constraints (10) break subtours.
1. Write down the entire polynomial-size formulation for Asymmetric Travelling Salesman Problem 2. Explain the constraints...
Question 1.(20 points): For each LP problem below, write down the dual LP problem associated with it. Check if the dual problem is in standard or in canonical form (Explain why?). Explain how do you conduct the sign of the variables and the constraints in the dual problem? 2. max z= -x1 +2x3 st. x1 +x2 +13 = 2,
Question 1.(20 points): For each LP problem below, write down the dual LP problem associated with it. Check if the dual...
please write down detailed solution (do not copy
3. [Polynomial interpolation versus least squares fitting, 10pts] Recall how Q7 in HW3 required you to find the cubic best fit to six given data points. This led to a least squares optimization problem. We are given the same points as in HW3: i 01 | 2 | 3 | 4 | 5 X 0.0 0.5 1.0 1.5 2.0 2.5 Y 0.0 0.20 0.27 0.30 0.32 0.33 (a) Write down the least...
1 Write down and explain the advantages and disadvantages for shareholders in the control group. 2- Illustrate the different types of mergers in details, mentioning how- when - why it occurs?
3. Consider the initial value problem y'(t) = y2, y(0) = 1. a. Write down (i.e., write the formula which describes one step, Yn+1 = yn + ...) the second order Taylor method with step size h for this initial value problem. b. Write down the time stepping formula Yn+1 = Yn +... for the modified Euler method h Yn+1 := yn + hf(tn + h 2:9 » Yn + 5 f (tnYn)), for this initial value problem. c. What...
3. Consider the initial value problem y(t) = y, y(0) = 1. a. Write down (i.e., write the formula which describes one step, Yn+1 = yn + ...) the second order Taylor method with step size h for this initial value problem. b. Write down the time stepping formula Yn+1 = Yn +... for the modified Euler method 9n+1 := yn + hf(en +3.29 + s(tn, yn)), for this initial value problem. c. What is the difference between the two...
1. ?(?, ?) = ? ?^? ?^? a) Write down the utility maximization problem. b) Find the MRS if ? is 24, ? is 16, ? is 3, ? is 2 and ? is 1. How would you interpret this value?
Name.... * an10 a) Solve the following problem using graphical method (using the following graph): 2. Minimize f(x,y)=2x-y subject to the constraints x2+ y2s 20. y Sx (1) (2) (In the space provided below the graph, please write down your solution clearly) b) Suppose we wish to solve the above problem using Exterior Penalty Function approach. Define an augmented cost function and explain how to use it to find a solution to the above problem.
Name.... * an10 a) Solve...
Name.... * an10 a) Solve the following problem using graphical method (using the following graph): 2. Minimize f(x,y)=2x-y subject to the constraints x2+ y2s 20. y Sx (1) (2) (In the space provided below the graph, please write down your solution clearly) b) Suppose we wish to solve the above problem using Exterior Penalty Function approach. Define an augmented cost function and explain how to use it to find a solution to the above problem.
Name.... * an10 a) Solve...
Question 1 (a) Estimate the volume (in m3) and density (in kg m3) of a C nucleus. State your assumptions and comment on the value obtained Write down a relationship between a particle's energy, mass and momentum using relativistic (b) kinematics. The SuperKEKB b-factory will collide electrons with energy 7 GeV and positrons with energy 4 GeV in order to produce b-hadron pairs. Write down the four vectors of the electron and positron Hence work out the centre-of-mass energy of...
For this problem, we will consider the polynomial function f(1) = 414 - 1623 + 2422 - 32 +32 over the interval -3 <3 <3 (a) The degree of f(x)is Number (b) Which of the following choices describe the end behavior of f(x)? The graph of f(x) acts O like 22 (e. both ends up) like – 22 (ie, both ends down) O like 23 (e left end down, right end up) Olike - 23 (1e.left end up, right end...