Problem 1 Consider the constrained minimization problem in RP for a given xo E R 1 (P) min (x xo)? (x xo) XER 2 subject to x ES where the set S is defined as S = {x = (x1, X2, X3)T E R ||X1| + |22| + |23| 0 what are the dimensions of the matrix A ? Hint: Express the definition of S by eliminating the use of absolute values. 2. 3 points Write down the Lagrangian and optimality conditions for (P). 3. 5 points Formulate the dual problem (D) to (P). Is there a duality gap ? 4. 5 points Given the point xo (-0.5, 0.4, -0.4)7 decide whether the point x* = (-0.4,0.3,-0.3)T is a minimizer of (P). Carefully justify your answer. = Problem 1 Consider the constrained minimization problem in RP for a given xo E R 1 (P) min (x xo)? (x xo) XER 2 subject to x ES where the set S is defined as S = {x = (x1, X2, X3)T E R ||X1| + |22| + |23| 0 what are the dimensions of the matrix A ? Hint: Express the definition of S by eliminating the use of absolute values. 2. 3 points Write down the Lagrangian and optimality conditions for (P). 3. 5 points Formulate the dual problem (D) to (P). Is there a duality gap ? 4. 5 points Given the point xo (-0.5, 0.4, -0.4)7 decide whether the point x* = (-0.4,0.3,-0.3)T is a minimizer of (P). Carefully justify your answer. =