Matlab代写|Assignment 7 of MATP4820/6610



Requirement: you must write the solutions by yourself. As difffferent problems may be assigned to MATP4820 and 6610 students, you must explicitly write whether you are taking MATP4820 or 6610.

Code submission: you are required submit your source code for any coding question in LMS for verifification. In addition, include your code, if any, together with your output in one single PDF fifile.

Bonus: In addition to the bonus problems (that will be assigned occasionally), you can earn up to 5% bonus credit if you write your homework in LaTex.

Problem 1

Consider the problem

  1. Argue that (1, 0) is the unique minimizer.
  2. The augmented Lagrangian function with penalty parameter β > 0 is

Argue that for any β (0, 1], the augmented Lagrangian method with initial multiplier

v (0) = 0 does not work for this instance, i.e., the generated sequence (x(k) , y(k) ) does not converge to a KKT point.

  1. For β = 2, do by hand two iterations of the augmented Lagrangian method with initial multiplier v (0) = 0.
  1. (bonus question) For any β > 2, does the generated sequence (x(k) , y(k) ) converge to the unique minimizer (1, 0), starting from any v (0)? If yes, prove it; if not, give a value of β > 2 and v (0) such that the sequence does not converge to (1, 0).

Problem 2

Consider the nonnegative quadratic program:

where Q is a symmetric and positive semidefifinite matrix, and X = {x Rn : xi 0, i =1, . . . , n}.Use the instructor’s provided fifile alm_qp.m to write a Matlab function alm_qp with input Q, c, A, b, initial vector x0, stopping tolerance tol, and penalty parameter β > 0. Also test your function by running the provided test fifile test_alm_qp.m and compare to the instructor’s function. Print your code and the results you get.

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