LP - Linear Programming(based on Coremen et al LP in...

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Linear Programming (based on Coremen et al.) LP in Standard form Find values for n variables x 1, x 2, …, x n : Maximize ( j=1 n c j x j ) (Maximize the objective function) Subject to: j=1 n a ij x j b i , for i = 1, … m (subject to m linear constraints, and) for all j = 1,…n, x j 0 (all variables are non-negative) The constant coefficients are c j , a ij , and b i . Problem size: (n, m). Example 1: Maximize (2x1 –3x2 +3x3) Three constraints: x1 +x2 –x3 7, -x1 –x2 +x3 -7, x1 –2x2 +2x3 4, x1, x2, x3 0 Converting some non-standard form to equivalent standard form Note: In the standard for the inequalities must be strict (‘ ,’ rather than ‘>’), equations are linear (power of x is 0 or 1) 1) Objective function minimizes as opposed to maximizes Example 2.1: minimize (-2x1 + 3x2) Action: Change the signs of coefficients Example 2.1: maximize (2x1 –3x2) 2) Some variables need not be non-negative Example 2.2: Constraints: x1 +x2 = 7, x1 –2x2 4, x1 0, no constraint on x2. Action: replace occurrence of any unconstrained variable xj, with a new expression (xj’ – xj’’), and add two new constraints xj’, xj’’ 0. Example 2.2: Constraints: x1 +x2’ –x2’’ = 7, x1 –2x2’ +2x2’’ 4, x1, x2’, x2’’ 0. The number of variables in the problem may at most be doubled, from n to 2n, a polynomial-time increase. 3) There may be equality linear constraints Example 2.3:
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Constraint: x1 +x2 –x3 = 7 Action: replace each equality-linear constraint with two new constraint with , and , and same left and right hand sides. Example 2.3: Two new constraints: x1 +x2 –x3 7, and x1 +x2 –x3 7. Total number of constraints may at most be doubled, from m to 2m, a polynomial-time increase. 4) There may be linear constraints involving ‘ ’ rather than ‘ ’ as required by the standard form. Example 2.4: Constraint: x1 +x2 –x3 7 Action: Change the sign of the coefficients (as in the case 1). Example 2.4: New constraint replacing the old one: -x1 –x2 +x3 -7. (Note that now the example 2 is the same as example 1) Note that equivalent forms of an LP have the same solution as the original one. Simplex algorithm, Khachien’s algorithm, Karmarkar’s algorithm solves LP. First one is worst-case exp-time algorithm, other two are poly-time algorithms. Simplex: A non-null region in the Cartesian space over the variables, such that any point within the region satisfies the constraints. When the constraints are unsatisfiable the simplex does not exist or it is a null region. The optimum value of the optimizing function exists at some boundary (a corner point, or
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This note was uploaded on 02/10/2012 for the course CSE 5211 taught by Professor Dmitra during the Spring '12 term at FIT.

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LP - Linear Programming(based on Coremen et al LP in...

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