(1) Consider the optimization problem: minimize |Ar bll where A E Rmxn, m 2 n and bE Rm. Show tha...
Given three numbers n, m, r and a constant matrix Z E R"Xm, consider the optimization problem minimize Z- XY subject toX20, Y20 (note that the sign"2" means that all elements of the corresponding matrix are nonnegative, and thatIF denotes the Frobenius norm). (10 points) Write the first-order optimality conditions for (1). (10 points) Describe how to solve (1) using the gradient projection method with the step size along the feasible direction chosen to be and the step size along...
Exercise 7.3. Consider the nonlinearly constrained problem minimize xER2 to (7.1) a x2 1 = 0. subject 1)T is a feasible path for the nonlinear constraint (a) Show that x(a) x x - 1 = 0 of problem (7.1). Compute the tangent to the feasible path at E = (0, 0)7 (sin a, cos a - + X (b) Find another feasible path for the constraint x? + (x2 + 1)2 - 1 = 0. Compute the tangent to the...
Consider the following optimization problem: minimize 71 subject tox,- r, where r > 0 is a given scalar 1. Write down the FONC and SONC for this problem. (5 points) 2. Shw ihai whken f is vx, nxxssary conditions a: also sufiint. (10 poimis)
Consider the following optimization problem: minimize 71 subject tox,- r, where r > 0 is a given scalar 1. Write down the FONC and SONC for this problem. (5 points) 2. Shw ihai whken f is...
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]
Problem 8. Let n 2 2. Consider the following optimization problem min n(n-1) 1 i=1 Show that (0,1/(n 11/(n 1) is an optimal solution SolutionType your solution here.]
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
Consider the optimization problem minimize f(x) subject to αεΩ where f(x) = x122, where x = [11, [2], and N = {x € R2 : x1 = 22, Xı >0}. (a) Find all points satisfying the KKT condition. (b) Do each of the points found in part (a) satisfy the second-order necessary condition? (c) Do each of the points found in part (a) satisfy the second-order sufficient condition?
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pseudo-inverse of a vector u R" regarded as an m x 1 matrix? 3. (5 pts) Let B AT where A is the matrix in Problem 1. Use Matlab to find the singular value decomposition and the Moore-Penrose pseudo-inverse of B. Then solve minimum-norm least squares problem minl-ll : FE R minimizes IBr-ey where c- [1,2. Compare...
2. Steepest descent for unconstrained quadratic function minimization The steepest descent method for minimize f(x) is the gradient descent method using exact line search, that is, the step size of the kth iteration is chosen as Ok = argmin f(x“ – av f(x)). a20 (a) (3 points) Consider the objective function f(x):= *xAx- Ax - c^x + d. where A e RrXnCER”, d E R are given. Assume that A is symmetric positive definite and, at xk, f(x) = 0....