ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Instructor: Dr. Gautam Das Lecture 10 March 03, 2009 Class notes by Rachit Shah

Overview Linear Programming Integer Programming Approximation Algorithm based on LP relaxation of IP formulation

Linear Programming Problem Instance Set of variables (real variables) x1, x2…xn Restriction – a bunch of linear equations x1 + x2 <= 5 x1 + 2x2 <= 6 x1 >=0 x2 >=0.1

Optimization Goal Optimizing function Maximizing 4x1 + x2 Find a pair (x1, x2) Convexity: Any two points inside the region, connected by a line does not intersect with region boundaries Half planes, when intersect, it maintains convexity

X2 Optimization Function (4x1+x2) x1+x2 <= 5 x1+2x2 <= 6 Convex Region Maximized (4x1+x2) x2 >= .1 X1 x1 >= 0

Algorithm We have to find out the region and maximize 4x1 + x2 in this region Algorithm Input: m equations, Goal function We have m(m-1)/2 corners (intersecting points) Find intersecting points Pij (takes m2) Check Pij is feasible (is it in the common area) If yes, apply function (4x1 + x2)

Analysis In case of 0.0001 x1 + x2 as the optimization function, the optimization point would be different. Why? What if we have x1, x2 and x3 (3 Dimensions) Finding intersecting points take m3 3D – m3 (In case of 3 variables) 4D – m4 (In case of 4 variables) … nD – mn (In case of n variables) Its exponential

Dangzeg Algorithm Greedy Algorithm (Based on Hill Climbing) Because of the Convexity of area, there is only one optimal point/solution Worst case: Exponential Average case: Linear In worst Case, All direction are equally optimistic, except the direction we came in, so we have to visit each and every vertex Example: Transportation problem

Greedy Algorithm (Hill Climbing) Optimization function Optimum location Global Hilltop Climb Climb Start Greedy Algorithm (Hill Climbing) Dangzeg Algorithm

Later Developments Leond Khadyand: Gave proof of polynomial time algorithm using Ellipsoidal method Karmarkar: (ATT) interior point method (which stays in feasible region while doing optimization) There has been no big improvements after

Integer Linear Programming n variables m inequalities Goal function fn The problem is same as Linear Programming except that it is subjected to integer grid IP is NP complete, why? Decision problem: Is there a point where goal funtion >= C

X2 Optimization Function Integer Grid x1+x2 <= 5 x1+2x2 <= 6 Convex Region x2 >= .1 X1 x1 >= 0