mthsc810-lecture11-2x2

mthsc810-lecture11-2x2 - How fast is the simplex method...

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MthSc 810: Mathematical Programming Lecture 11 Pietro Belotti Dept. of Mathematical Sciences Clemson University October 6, 2011 Reading for today: Sections 3.7 and 4.1-4.2, textbook Reading for Oct. 11: Sections 4.3-4.4, 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 worst-casecomplexity 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 > 0 such that f ( n ) cg ( n ) n n 0 } If f O ( g ) , then we say that “ f is big-O of g It means that f grows no faster than g
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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 = 0 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 0 1 ×10 5 2 ×10 5 3 ×10 5 4 ×10 5 5 ×10 5 6 ×10 5 10 0 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 (worst-case) running time of the best-known algorithm . Problems for which there is an algorithm with polynomial running time (or better) are called polynomially solvable . Generally, these problems are considered easy . They define the complexity class P . In general, if a problem admits a polynomial-time complexity algorithm, i.e., if P ∈ P , that problem is “easy” (even if the corresponding complexity is f ( n ) = n 500 )
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Is Linear Programming “easy”?
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