Ee364a Homework 6 Solutions To Equations

Unformatted text preview: EE364a, Winter 2014-15 Prof. S. Boyd EE364a Homework 6 solutions 8.16 Maximum volume rectangle inside a polyhedron. Formulate the following problem as a convex optimization problem. Find the rectangle R = { x ∈ R n | l ± x ± u } of maximum volume, enclosed in a polyhedron P = { x | Ax ± b } . The variables are l,u ∈ R n . Your formulation should not involve an exponential number of constraints. Solution. A straightforward, but very inefficient, way to express the constraint R ⊆ P is to use the set of m 2 n inequalities Av i ± b , where v i are the (2 n ) corners of R . (If the corners of a box lie inside a polyhedron, then the box does.) Fortunately it is possible to express the constraint in a far more efficient way. Define a + ij = max { a ij , } , a-ij = max {-a ij , } . Then we have R ⊆ P if and only if n X j =1 ( a + ij u j-a-ij l j ) ≤ b i , i = 1 ,...,m, The maximum volume rectangle is the solution of maximize ( Q n i =1 ( u i-l i )) 1 /n subject to ∑ n j =1 ( a + ij u j-a-ij l j ) ≤ b i , i = 1 ,...,m, with implicit constraint u ² l . Another formulation can be found by taking the log of the objective, which yields maximize ∑ n i =1 log( u i-l i ) subject to ∑ n j =1 ( a + ij u j-a-ij l j ) ≤ b i , i = 1 ,...,m. A3.28 Probability bounds. Consider random variables X 1 ,X 2 ,X 3 ,X 4 that take values in { , 1 } . We are given the following marginal and conditional probabilities: prob ( X 1 = 1) = 0 . 9 , prob ( X 2 = 1) = 0 . 9 , prob ( X 3 = 1) = 0 . 1 , prob ( X 1 = 1 ,X 4 = 0 | X 3 = 1) = 0 . 7 , prob ( X 4 = 1 | X 2 = 1 ,X 3 = 0) = 0 . 6 . 1 Explain how to find the minimum and maximum possible values of prob ( X 4 = 1), over all (joint) probability distributions consistent with the given data. Find these values and report them. Hints. (You should feel free to ignore these hints.) • Matlab: – CVX supports multidimensional arrays; for example, variable p(2,2,2,2) declares a 4-dimensional array of variables, with each of the four indices taking the values 1 or 2. – The function sum(p,i) sums a multidimensional array p along the i th index. – The expression sum(a(:)) gives the sum of all entries of a multidimensional array a . You might want to use the function definition sum_all = @(A) sum( A(:)); , so sum_all(a) gives the sum of all entries in the multidimensional array a . • Python: – Create a 1-d Variable and manually index the entries. You should come up with a reasonable scheme to avoid confusion. • Julia: – You can create a multidimensional array of variables in Convex.jl. For exam-ple, the following creates a 4-dimensional array of variables, with each of the four indices taking the values 1 or 2....
View Full Document

Консьерж бросил внимательный взгляд в его спину, взял конверт со стойки и повернулся к полке с номерными ячейками. Когда он клал конверт в одну из ячеек, Беккер повернулся, чтобы задать последний вопрос: - Как мне вызвать такси. Консьерж повернул голову и. Но Беккер не слушал, что тот. Он рассчитал все .

0 comments

Leave a Reply

Your email address will not be published. Required fields are marked *