1. Argue that, for any given value of k, an optimal solution of the penalized problem...

 

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1. Argue that, for any given value of k, an optimal solution of the penalized problem is given by x = 1/(1+ k) and x = 2/(1+ k). 2. What is the corresponding objective value g(k)? 3. To get a good approximation of the original problem, k needs to be selected ju- diciously. Indeed, if k = 0, then the penalized problems corresponds simply to ignoring the constraint. Alternatively, if k , then the optimal solution is simply x10 and 20. An approach to choose the value of k is to solve the optimization problem max g(k). k>0 Use binary search to solve the above optimization problem within 0.01 of optimal, with initial interval [0, 10]. You may assume that g'(k) = 1 + 5/(1 + k). 4. Is the solution x1 = 1/(1+k*) and x2 = 2/(1+k*), where k* is chosen according to point 3, feasible? If not, what is the violation of the constraint, i.e., the value x + x - 1? 1. Argue that, for any given value of k, an optimal solution of the penalized problem is given by x = 1/(1+ k) and x = 2/(1+ k). 2. What is the corresponding objective value g(k)? 3. To get a good approximation of the original problem, k needs to be selected ju- diciously. Indeed, if k = 0, then the penalized problems corresponds simply to ignoring the constraint. Alternatively, if k , then the optimal solution is simply x10 and 20. An approach to choose the value of k is to solve the optimization problem max g(k). k>0 Use binary search to solve the above optimization problem within 0.01 of optimal, with initial interval [0, 10]. You may assume that g'(k) = 1 + 5/(1 + k). 4. Is the solution x1 = 1/(1+k*) and x2 = 2/(1+k*), where k* is chosen according to point 3, feasible? If not, what is the violation of the constraint, i.e., the value x + x - 1?