mthsc810-lecture11-2x2

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 )
Is Linear Programming “easy”?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern