# lec11 - IE521 Advanced Optimization Lecture 11 Dr. Zeliha...

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IE521 Advanced Optimization Lecture 11 Dr. Zeliha Akc ¸a December 2011 1 / 28

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Integer Programming Models I The optimal solution is a solution that gives the best objective function value. I If minimization problem, a feasible but not optimal solution is an upper bound for the optimal solution. I If maximization problem, a feasible but not optimal solution is an lower bound for the optimal solution. I Recall that LP relaxation solution is a lower bound for the optimal solution for minimization problems (for maximization problem, it is an upper bound). 2 / 28
Solving Integer Programming Models I Unlike linear programs, it is very difﬁcult to solve integer programming problems. I There is no efﬁcient algorithm known to solve integer programs. I In order to ﬁnd the optimal solution, we need to enumerate the feasible solutions. I We can categorize the solution algorithms into three groups: I Exact algorithms provide the optimal solution. However, it may take an exponential number of iterations. Cutting Plane, branch and bound, branch and cut, dynamic programming are examples. I Heuristic Algorithms: provides a solution to the problem without any guarantee about how close the solution to the optimal solution. It is not guaranteed to terminate quickly but generally heuristics provides a solution quickly. Search algorithms, meta-heuristics (tabu search, simulated annealing, etc) 3 / 28

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Overview I Consider integer programming problem: (IP) min c > x s.t. Ax = b x 0 x integer I Consider the linear relaxation of (IP): (LP) min c > x s.t. Ax = b x 0 I Let x LP be an optimal solution to the (LP) and z LP is the optimal solution value. I Let x IP be an optimal solution to the (IP) and z IP is the optimal solution value. 4 / 28
Overview, Cont. I z LP z IP I If x LP is integer, then x LP = x LP and z LP = z IP Figure: Feasible Points for IP Figure: Feasible region for LP relaxation of IP 5 / 28

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Cutting Plane Algorithm I A solution algorithm designed for integer programming models (IP). I The main idea is to solve the IP by solving a sequence of linear programs (LP). I We ﬁrst solve (LP) and ﬁnd x LP . I If x LP is integer, then it is the optimal solution for (IP), x LP = x IP . I If not, we ﬁnd an inequality of the form d > x f to (LP): I that is satisﬁed by all integer solutions of (IP), I but that is not satisﬁed by x LP ( dx LP > f ). I The LP relaxation with the new inequality results in a better lower bound for (IP). I Continue to add inequalities until we obtain an integer solution feasible for (IP). 6 / 28
Generic Cutting Plane Algorithm Let (IP) be the integer programming model and (LP) is its LP relaxation (given in slide 3) 1. Solve the linear programming relaxation (LP). Let x LP be the optimal solution. 2.

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## This note was uploaded on 01/09/2012 for the course IE 521 taught by Professor Zelihaakça during the Fall '11 term at Fatih Üniversitesi.

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lec11 - IE521 Advanced Optimization Lecture 11 Dr. Zeliha...

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