- 18.104.22.168: What is the difference between constrained and unconstrained optimi...
- 22.214.171.124: State the idea and the basic formulas of the method of steepest des...
- 126.96.36.199: Write down an algorithm for the method of steepest descent.
- 188.8.131.52: Design a "method of steepest ascent" for determining maxima.
- 184.108.40.206: What i~ linear programming? lt~ ba~ic idea? An objective function?
- 220.127.116.11: Why can we not use methods of calculus for extrema in linear progra...
- 18.104.22.168: Whar are slack variables? Artificial variables? Why did we use them'?
- 22.214.171.124: Apply the method of steepest descent to f(x) = X12 + 1.5X22. starti...
- 126.96.36.199: What does the method of steepe~t de~cent amount to in the case of a...
- 188.8.131.52: In Prob. 8 start from Xo = [1.5 I ] T. Show that the next even-numb...
- 184.108.40.206: What happens in Example I of Sec. 22.1 if you replace the function ...
- 220.127.116.11: Apply the method of steepest descent to f(x) = 9X12 + X22 + 18.\'1 ...
- 18.104.22.168: In Prob. 12, could you start from [0 O]T and do 5 steps?
- 22.214.171.124: Show that the gradients in Prob. 13 are orthogonal. Give a reason.
- 126.96.36.199: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 188.8.131.52: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 184.108.40.206: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 220.127.116.11: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 18.104.22.168: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 22.214.171.124: Graph or sketch the region in the first quadrant of the X1x2-plane ...
- 126.96.36.199: Maximize f = lOx. + 20X2 subject to Xl ~ 5.Xl + X2 ~ 6. X2 ~ 4.
- 188.8.131.52: Maximize f = Xl + X2 subject to Xl + 2X2 ~ 10. 2X1 + X2 ~ LO. X2 ~ 4.
- 184.108.40.206: Minimize f = 2X1 - IOx2 subject to Xl - X2 ~ 4. 2xI + x2 ~ 14. Xl +...
- 220.127.116.11: A factory produces two kinds of gaskets, GI G2 with net profit of $...
- 18.104.22.168: Maximize the daily output in producing XI chairs by a process PI an...
Solutions for Chapter 22: Unconstrained Optimization. Linear Programming
Full solutions for Advanced Engineering Mathematics | 9th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.