linear1 - Linear and Integer Programming 15-853:Algorithms...

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1 15-853 Page1 15-853:Algorithms in the Real World Linear and Integer Programming I –Introduct ion – Geometric Interpretation – Simplex Method 15-853 Page2 Linear and Integer Programming Linear or Integer programming minimize z = c T x cost or objective function subject to Ax = b equalities x 0 inequalities c ∈ℜ n , b ∈ℜ m , A ∈ℜ n x m Linear programming : x ∈ℜ n (polynomial time) Integer programming: x ∈Ζ n (NP-complete) Extremely general framework, especially IP 15-853 Page3 Related Optimization Problems Unconstrained optimization min{f(x) : x ∈ℜ n } Constrained optimization min{f(x) : c i (x) 0, i I, c j (x) = 0, j E} Quadratic programming min{1/2x T Qx + c T x:±a i T x b i , i I, a i T x = b j , j E} Zero-One programming min{c T x:±Ax± = b, x {0,1} n , c R n , b ∈ℜ m } Mixed Integer Programming min{c T x:±Ax± = b, x 0, x i ∈Ζ , i I, x r ∈ℜ , r R} 15-853 Page4 How important is optimization? • 50+ packages available • 1300+ papers just on interior-point methods • 100+ books in the library • 10+ courses at most Universities • 100s of companies • All major airlines, delivery companies, trucking companies, manufacturers, …
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2 15-853 Page5 Linear+Integer Programming Outline Linear Programming – General formulation and geometric interpretation –S imp lex me thod – Ellipsoid method – Interior point methods Integer Programming – Various reductions of NP hard problems – Linear programming approximations – Branch-and-bound + cutting-plane techniques – Case study from Delta Airlines 15-853 Page6 Applications of Linear Programming 1. A substep in most integer and mixed-integer linear programming (MIP) methods 2. Selecting a mix: oil mixtures, portfolio selection 3. Distribution: how much of a commodity should be distributed to different locations. 4. Allocation: how much of a resource should be allocated to different tasks 5. Network Flows 15-853 Page7 Linear Programming for Max-Flow Create two variables per edge : Create one equality per vertex : x 1 + x 2 + x 3 = x 1 ’+ ±x 2 ’+±x 3 and two inequalities per edge : x 1 3, x 1 3 add edge x 0 from out to in maximize x 0 in out 1 2 3 7 6 2 3 5 7 x 1 x 3 x 2 x 1 x 1 15-853 Page8 In Practice In the “real world” most problems involve at least some integral constraints.
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3 15-853 Page9 Algorithms for Linear Programming Simplex (Dantzig 1947) Ellipsoid (Kachian 1979) first algorithm known to be polynomial time Interior Point first practical polynomial-time algorithms Projective method (Karmakar 1984) Affine Method (Dikin 1967) Log-Barrirer Methods (Frisch 1977, Fiacco 1968, Gill et.al. 1986) Many of the interior point methods can be applied to
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linear1 - Linear and Integer Programming 15-853:Algorithms...

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