3 10 pts The following shows the sequence of points visited

3 (10 pts) The following shows the sequence of points visited by an improving search. Compute the corresponding sequence of move directions assuming that all step sizes 1. w(0) (0,1,1), w C1) (4,-1,7), w (2) (4,-3,19), w 3) (3,-3,22), 4 (20 pts) Consider a mathematical program with constraints 1 2x2 3x3 25 x1,x2, x3 0 Determine the maximum step (possibly +oo) that preserves feasibility in the direction indicated from the point specified. Also indicate whether that step indicates that the model is bounded, assuming that directions improve everywhere a) Ax (-3, -3,9) from x (9,4,60 b) Ax (-4,0,3 from x (16,2,1) 5 (15 pts) Construct an improving direction from the gradient of each objective function at the point indicated a) min 4w1 w3 5w4 at w (1,1,7,-1) b) max (wy) 4wy 6wo at w (2,5)

Solution

3) Let us consider it into x,y,z directions then

from 0 to1 the movement will be 4 steps forward in positive x direction , 2 steps in negative y direction and 6 steps in positive z direction

from 1 to 2 the movement will be no movment in x direction again 2 steps in negative direction and 12 steps forward in z direction

from 2 to 3 the movement will be 1 step in negative direction and no movement in y direction and 3 steps in positive y direction

4) a) the first movement will lead us to a position of (6,1,15) which does not satisfy the first condition and thus the movement isnot valid

b) the second movement will lead us to a position of (12,2,4) which is validated by both the equations and thus itis valid movement

5) a) min will be 16

b) mx will be 10