lecture 1: introduction · lecture 1: introduction shu-cherng fang north carolina state university...
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LECTURE 1: INTRODUCTION
Shu-Cherng FangNorth Carolina State University – Fall 2017
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Outline• What is Linear Programming? • Why to study Linear programming?• How to study Linear Programming?• History of Linear Programming• How to solve an LP problem?• Where to go?
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What is Linear Programming (LP)?• Optimize a linear objective function of decision variables subject to a set of linear constraints.
• Example
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Graphic representation
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General form
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Why study LP ?• Wide applications:
- One of the most widely applied methodologies.- 85 % of Fortune 500 companies had used LP models.
• Passage to advanced subjects:- Nonlinear Programming- Network Flows- Integer Programming- Conic Programming- Semidefinite Programming- Robust Optimization
Reference: Operations Research, Vol. 50, No. 1, 2002
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Example – manpower allocation
• = man-hours of skill i working on project j
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How to study LP ?• Geometric intuition• Algebraic manipulation• Computer programming
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History of Linear Programming• Conceived by G. B. Dantzig (1947).
– mechanized planning tool for a deployment, training,and logistical supply program in USAF.
• Named by T. C. Koopmans & G. B. Dantzig (1948).• Simplex Method proposed by G. B. Dantzig (1948).
- L. V. Kantorovich worked on a restricted class in 1939.• Kantorovich & Koopmans received the Nobel Prize inEconomic Science (1975).
– Theory of optimum allocation of resources.
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History of Linear Programming• Ellipsoid Method proposed by L. G.
Khachian (1979).– First polynomial-time algorithm for LP.
• Interior-Point Method proposed by N.Karmarkar (1984).
– First “good” polynomial-time algorithmfor LP.
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How to solve an LP problem?• What’s special about LP?
- linear constraints shape the feasible domain as aconvex polyhedral set with a finite number of vertices.
- linear objective function provides a linear contour of each fixed value.
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Fundamental theorem of LP
•For a linear programming problem, if its feasible domain is not empty, then its optimum is either unbounded or is attained at least at one vertex of the feasible domain.
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How to find an optimal solution?• Fact:
• Question: How to find ?
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Different solution methods• Enumeration Method
– Find all vertices and choose the optimal one bycomparison.
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Simplex Method• Basic ideas: (walking on the boundary)
Step 1: Start at a vertex.Step 2: If current vertex is optimal, STOP! Otherwise,Step 3: Move to a better neighboring vertex. Go To
Step 2.
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Is Simplex Method Good?
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Is there a polynomial-time algorithm for LP?• Yes! - Ellipsoid Method by L. G. Khachian 1979.
Basic ideas:
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Ellipsoid method
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Ellipsoid method
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Is ellipsoid method good?
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Is there a good polynomial-time algorithm for LP?• Yes! - Interior Point Method by Karmarkar in 1984.• Basic ideas: (walk through interior)
Step 1: Start with an interior solution.Step 2: If current solution is good enough, STOP. Otherwise,Step 3: Check all directions for improvement and move to a better interior solution. Goto Step 2.
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Is interior point method good?
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Where to go?• Integration of Interior-point methods to develop hybrid
algorithms for solving very large size LP for real-world applications by exploring:- special structure.- sparsity.- decomposition.- parallel computation.
• Nonlinear optimization with linear constraints.• Positive semidefinite programming.• Linear conic programming.• Robust optimization