Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May., 2014
Instructor: G. K. Ananthasuresh , Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 10:00 AM; Venue: ME MMCR (Multimedia Classroom)

Homework #3
Assigned: Jan. 28th, 2014
Due: Feb. 4, 2014
Points: 40
Additional points for work that is beyond instructor's expectation!
Look up Homework #1 of 2010 offering of this course and the solution posted there to know what is meant by "beyond instructor's expectation".

  1. 30 points
    Min f = x1 + x22 + x2x3 + 2x32 with respect to x1, x2, and x3 subject to h = x12 + x22 + x32 - 1 = 0.
    Verify the KKT necessary conditions for this problem at (1,0,0) and find the Lagrange multiplier.
    In Slide 20 of Lecture 6, it is argued that (1,0,0) is a minimum using the concept of Bordered Hessian. Now, verify graphically that (1,0,0) is indeed a minimum. For this, draw contours of f in x1-x2, x2-x3, and x3-x1 planes while keeping the third variable at its minimizing value. Show also h in all cases.
    In all three cases, graphically solve the problem starting from (2,2) (i.e., x1 = x2 = 2 in the x1-x2 plane; x2 = x3 = 2 in the x2-x3 plane, and x3 = x1 = 2 in the x3-x1 plane) by using steepest descent method.
  2. 10 points
    Use "fmincon" function in Matlab to find the solution of Problem 1. Use (2,2,2) as the initial guess. Show the path taken by "fimincon" in the three contour plots drawn in x1-x2, x2-x3, and x3-x1 planes.