Question 1: Consider the following problem, which was also a subject of the previous
min f(x) subject to x ≥ 0; (1)
where f : R3 ! R is defined as follows:
f(x) = max (x1)2 + (x2 −1)2 + (x3)2; (x1 −1)2 + (x2 −x1)2 + 2; −x1 + 3×2 + x3 −1:
Q1a. Formulate the master problem of the bundle method for problem (1) as a
quadratic optimization problem.
Q1b. Solve problem (1) by the bundle method starting from (2; 2; 2) and (1; 1; 1). To
simplify the implementation, do not remove cuts; that is, Step 5 only increases
the iteration counter. Compare with the cutting plane method.
Question 2: Consider the following problem:
max (x1 − 3)2 + (x2 − 5)2
subject to x2 1 + x2 2 ≤ 10
− 2×1 + x2 = 5:
Q2a. Formulate the quadratic penalty function. Solve problem (2) numerically, using
the quadratic penalty method; use the conjugate gradient method for solving
the unconstrained problem at each iteration.
Q2b. Solve the problem analytically using the necessary optimality conditions. Com
pare the theoretical and numerical solutions and comment on the result.
Question 3: Consider the following optimization problem:
max x1x2 + x2x3 + x1x3
subject to x1 + x2 + x3 ≤ 3
x ≥ 0:
Q3.a Is the objective function convex or concave?
Q3.b (Bonus question) Analyze problem (3) using the first and the second order
Q3.c Formulate the augmented Lagrangian function for problem (3) and determine
the parameter values for which the augmented Lagrangian function is convex.
Q3.d Solve the problem using the augmented Lagrangian method.
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