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Unformatted text preview: Lecture 6 Lecture 6 Introduction to Optimization Introduction to Optimization Prof. Prof. Ismail Chabini Ismail Chabini and Prof. and Prof. Amedeo Amedeo R. R. Odoni Odoni 1.225 1.225 J (ESD 205) Transportation Flow Systems J (ESD 205) Transportation Flow Systems 1.225, 11/19/02 Lecture 6, Page 2 Lecture 6 Outline Lecture 6 Outline Mathematical programs (MPs) Formulation of shortest path problems as MPs Formulation of U.O. traffic assignment as an MP Relationship between U.O. and S.O. traffic assignment Solving S.O. traffic assignment by hand Lecture summary 1 1.225, 11/19/02 Lecture 6, Page 3 Optimization: Mathematical Programs Optimization: Mathematical Programs General formulation ( n variables, m constraints) : : decision variables : A constraint Notes: ) ,..., , ( min 2 1 n x x x z m n m n n b x x x g b x x x g b x x x g ) ,..., , ( ) ,..., , ( ) ,..., , ( 2 1 2 2 1 2 1 2 1 1 M b x x x g b x x x g b x x x g b x x x g b x x x g n n n n n = ) ,..., , ( and ) ,..., , ( ) ,..., , ( ) ,..., , ( ) ,..., , ( 2 1 2 1 2 1 2 1 2 1 Objective function Feasible set Subject to (s.t.): ) ,..., , ( 2 1 n x x x j n j b x x x g ) ,..., , ( 2 1 ) ,..., , ( z(x) ) ,..., , ( ax 2 1 2 1 n n x x x f Min x x x f M = = 1.225, 11/19/02 Lecture 6, Page 4 Types of Mathematical Programs (MPs) Types of Mathematical Programs (MPs) Linear programs (LPs) : objective function is linear, and constraints are linear Non-linear programs (NLPs) : objective function is linear. (constraints are usually linear. Otherwise, there might be more than one optimal solution (finding such a solution can be a very time consuming task)) If decision variables are further constrained to take integer values, a linear program is an integer program If decision variables are constrained to take 0/1 values: an integer program is an 0/1 integer program If some, but not all, variables are constrained to take integer values: a linear program is called a...
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This note was uploaded on 12/06/2011 for the course ESD 1.225j taught by Professor Ismailchabini during the Fall '02 term at MIT.
- Fall '02