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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 doesnt 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 algorithms 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 doesnt 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 / lglg n Commonly Occurring Functions Polynomials f ( n ) = k i = a i n 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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 Fall '08
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