OR Chapter 4. How fast is the simplex method

OR  Running time : worst case view point  An algorithm is considered efficient if the worst case running time is bounded from above by a polynomial function of problem size (for large instances only, small instances can be solved fast anyway). (called polynomial time algorithm) Size of problem encoding Running time

OR  sec400  sec2500  sec3600  sec 1000  sec1 sec35.7 years 366 centuries

OR

5

6  Geometric view largest increase rule

OR  Alternative rules : Largest increase rule : counter example by Jeroslow (1973) Bland’s rule : counter example by Avis and Chvatal (1978) Hence, until now, simplex method is not a polynomial time algorithm theoretically. But it performs very well practically.  First polynomial time algorithm for LP : ellipsoid method (1979) by L. G. Khachian. But, practically much inferior to simplex. However, it has some important theoretical implications in determining computational complexity of some optimization problems.  Another polynomial time algorithm : Interior point method by L. Karmarkar (1984) : Better than simplex in many cases practically. Many versions. Based on ideas from nonlinear programming.

OR  Interior point method will be briefly mentioned when we study complementary slackness theorem.  Comparing simplex and interior point method: Interior point method is generally fast, especially for large problems. But simplex is competitive for some problems and recent developments in dual simplex algorithm makes solving LP of large size manageable. In addition, simplex is effective when we solve the LP again after making some changes in data (reoptimization). Such capability is quite important when we solve integer programming problems. But little progress has been made for interior point algorithm in this respect. Recently, interior point method is used for some nonlinear programming problems (convex programs) with successful results.