Mechanical Engineering, Indian Institute of
Science, Bangalore 560 012, India Optimization hinders evolution! ME256
Variational Methods and Structural Optimization
Jan.-May., 2017
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
Homework #2
Assigned: Jan. 12th, 2017
Due: Jan. 19th, 2017
Points: 60 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".
35 points
Consider f = a*x12 + b*x22 + c*x1;
h = d*x2 - e*x12 + k*x1; and
g = m*x1 + n*x2 + p
(a) Find all solutions if f is minimized subject to h = 0 and g <= 0. Write the sufficiency conditions for the constrained minimum. Use symbolic manipulation software (e.g., in Matlab or Mathematica) if you wish.
(b) Use a = 1; b = 1; c = -2; d = -3; e = -1; k = 6; m = 1; n = -2; p = 5 and draw the contours of f and curves of h and g and show your solution(s) graphically. Also, check KKT conditions and sufficiency conditions through numerical substituitions.
(c) In problem 1, change the values of m, n, and p (and others if necessary) such that the inequality constraint g is active. Solve the problem and obtain the Lagrange multipliers. Now, if you perturb your value of p by 1%, what is the % change in optimum f?
25 points
In problem 1, change the values of constants to make the gradients of the two constraints linearly dependent (i.e., constraint qualification is not satisfied) at the minimum point. Chech to see what happens to the KKT conditions now. Show your result graphically in addition to computation.
