CS 205A class_13 notes

Scientific Computing

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CS205 – Class 13 Covered in class: 1, 3 Readings: 6.7, 7.2 to 7.3.3, 7.4 1. Constrained Optimization a. Minimize () f x G subject to constraints () 0 gx = G G i. Here n x R G and = GG is as system of mn equations ii. One can show that a solution x G must satisfy T g f xJ x λ −∇ = G G G 1. g Jx G is the Jacobian matrix of g 2. G is an m-vector of Lagrange multipliers 3. This condition says that we cannot reduce the objective function without violating the constraints iii. Define (, ) T Lx f x λλ =+ GGG 1. The critical points are found by setting 0 T g fx J x ⎡⎤ ∇+ == ⎢⎥ ⎣⎦ G G G G G 2. Suppose for simplicity that g is a linear function. Then the Hessian is 0 T fg g Hx Jx Hx = G G G where the x partial derivatives of T g G G vanish because g is linear. a. Note that H is not positive definite b. It turns out that positive definiteness is only needed on the tangent space to the constraint surface, i.e. on the null space of g J . iv. Consider 22 12 () . 5 2 . 5 f xx x with 1 0 x x = −−= 1. ( ) 1 2 (, ) . 5 2 . 5 1 x x + G G 2. 1 2 5 0 1 x x + ∇= = −− G G G 3. so we solve 1 2 10 1 0 05 1 0 11 0 1 x x −= to obtain 1 2 .833 .167 .833 x x ⎤⎡ ⎥⎢ =− ⎦⎣
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The gradient of the function is perpendicular to the constraint surface at the constrained minimum. 2. Linear Programming a. Minimize cx GG subject to constraints A xb = G G and 0 x G G b. The feasible region is a convex polyhedron in n-dimensional space c. The minimum must occur at one of the vertices of the polyhedron d. Simplex method - systematically examine a sequence of vertices to find the one yielding the minimum 3. Interpolation a.
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CS 205A class_13 notes - CS205 Class 13 Covered in class 1...

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