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Unformatted text preview: MthSc 810: Mathematical Programming Lecture 11 Pietro Belotti Dept. of Mathematical Sciences Clemson University October 6, 2011 Reading for today: Sections 3.7 and 4.14.2, textbook Reading for Oct. 11: Sections 4.34.4, textbook Homework #7 is out! Due October 13, 2011 Midterm 2: November 1, 2011, 2pm How fast is the simplex method? Consider an LP in standard form identified by ( n , m , c , A , b ) . ◮ How much time does it take to solve this problem? ◮ Time is hard to measure (due to computer speed etc.). ◮ However, the sequence of iterations doesn’t change 1 ⇒ Better question: in how many simple steps (sums/products/etc.) do we solve an LP? ◮ It could be a function f ( n , m , c , A , b ) , still too complicated. Q. Given m , n , what is the maximum number of steps to solve an LP problem with n variables and m constraints? T ( n , m ) := max { f ( n , m , c , A , b ) : A ∈ Q m × n , c ∈ Q n , b ∈ Q m } ◮ This is called the worstcase complexity of an algorithm. 1 Assuming we are given the same instructions. Measuring an algorithm’s complexity ◮ We are interested in performance for large input sizes . ⇒ We need only compare the asymptotic growth rates . ◮ Consider algorithm A with running time given by f and algorithm B with running time given by g . ◮ We are interested in L = lim n →∞ f ( n ) g ( n ) ◮ What are the four possibilities? ◮ L = 0: g grows faster than f ◮ L = ∞ : f grows faster than g ◮ L = c : f and g grow at the same rate. ◮ The limit doesn’t exist. Big O Notation ◮ We now define the set of functions O ( g ) = { f : ∃ c , n > 0 such that f ( n ) ≤ cg ( n ) ∀ n ≥ n } ◮ If f ∈ O ( g ) , then we say that “ f is bigO of ” g ◮ It means that f grows no faster than g Examples Some Functions in O ( n 2 ) ◮ n 2 ◮ n 2 + n ◮ n 2 + 1000 n ◮ 1000 n 2 + 1000 n ◮ n ◮ n 1 . 9999 ◮ n 2 / lg lg n Commonly Occurring Functions Polynomials ◮ f ( n ) = ∑ k i = a in i is a polynomial of degree k ◮ Polynomials f of degree k are in O ( n k ) . Exponentials ◮ A function in which n appears as an exponent on a constant is an exponential function , i.e., 2 n . ◮ For all positive constants a and b , lim n →∞ n a b n = 0. ⇒ All exponential functions grow faster than polynomials Comparing Algorithms 1 ×10 5 2 ×10 5 3 ×10 5 4 ×10 5 5 ×10 5 6 ×10 5 10 10 50 10 100 10 150 10 200 10 250 10 300 n 50 e n 100 Measuring the difficulty of problems ◮ The difficulty of a problem can be judged by the (worstcase) running time of the bestknown algorithm . ◮ Problems for which there is an algorithm with polynomial running time (or better) are called polynomially solvable ....
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This note was uploaded on 03/14/2012 for the course MTHSC 810 taught by Professor Staff during the Fall '08 term at Clemson.
 Fall '08
 Staff
 Math

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