Results can be found at the bottom of the page. Most of the students got it all right.
Preliminary deadline: Monday Feb 13th, 6pm
The students will program a primal-dual interior-point method and write 1-2 page report where you explain shortly how your code works and the solutions to the problem below. Submit both the report and the matlab codes in one zip file. Can be done in pairs (submit one package).
The task is to solve a convex problem of the form min f(x) s.t. g(x)<=0 with the interior point method. The algorithm is presented in Section 11.7 in Convex Optimization book https://stanford.edu/~boyd/cvxbook/ . (Note that there are no equality constraints Ax=b and the corresponding multipliers v.)
Solve the cantilever problem (an instance of geometric programming, Section 4.5.4) with N=8 and N=4. The objective is to minimize the volume of the beam depending on the widths w_i and heights h_i. The constraints are on aspect ratios h_i/w_i, maximum stress and deflection y_1.
(vertical deflection of the beam)
The method requires a feasible starting point . For N=8, you can use
You can use the matlab file given below to generate the functions, gradients and the hessians for the problem: f(x) objective function, df, hf are its gradient and hessian. g(x) is the vector of constraints, dg the jacobian andThe algorithm can be, for example, of the form: function [x,u]=pdip(x0,u0,f,df,Hf,g,dg,Hg,tol,maxiter), where tol=tolerance for optimality and maxiter=maximum number of iterations.
(Note that this is not a convex problem but can be converted into one using the transformation given in 4.5.3 by using variables y=log(x).You don't probably need to do this transformation but you can try.)
Here is the optimal shape with N=8: