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Complexity and Ellipsoid Method

# Complexity and Ellipsoid Method - IE411 Linear Optimization...

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Unformatted text preview: IE411 Linear Optimization Complexity and Ellipsoid Method Efficiency of Algorithms Question: Given a problem of a certain size, how long does it take to solve it? Two kinds of Answers: Average case: how long for a typical problem. Mathematically difficult Empirical studies Worst Case: How long for the hardest problem. Mathematically tractable Limited value LP problem LP problem: {min cTx s.t. Ax = b, x 0} A: mn (A, b, c) integers An instance of LP problem: the specification of m, n, A, b, c defines an instance of a LP problem Measures Measures of size Number of constraints m and/or variables n Number of data elements mn Number of nonzero elements Size, in bytes, of encoding the instance in a computer Measuring time Number of iterations Operations (addition, multiplication, comparison) per iteration Input Length in a Binary Digital Computer How many binary bits required to represent a positive integer or an integer ? Given an instance of LP problem, how many binary bits required to record it? Computational Complexity O(g(m,n,L)) If g(m,n,L) is a polynomial in m, n, and L, the algorithm is said to be of polynomial complexity Other concepts Pseudopolynomial Strongly polynomial Decision problems P, NP P=NP(?) Complexity of Simplex Method On average, it performs quite well However, so far there is no polynomial variant of the simplex method. (Big Open Problem) Klee-Minty's Problem (1972) Klee-Minty's Problem: Transformation Graphical Demonstration What are High-Dimensional Polyhedra Like? What are High-Dimensional Polyhedra Like? Why polynomial? Ellipsoid Method: basic ideas Linear Feasibility Problem Two Important Assumptions Ellipsoid Representation Affine Transformation Cutting Plane New Containing Ellipsoid The Ellipsoid Algorithm Performance of the Ellipsoid Method LP and LFP Integer Data Bounds from L The sliding objective hyperplane method Desired Theoretical Properties ...
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