LINEAR PROGRAMMING PROBLEM Definition and Examples.
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Transcript of LINEAR PROGRAMMING PROBLEM Definition and Examples.
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LINEAR PROGRAMMING PROBLEMDefinition and Examples
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Linear ProgramDecision Variables
Objective Function
Constraints
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Linear Program (General Form)Objective Function
Constraints
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The Linear Programming Model• Standard form
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Linear Programming Problem
(6,11)
(6,3)
(3,11)
(3,0)(0,0)
(0,2)
Not drawn to scale
x1+2 x2 = 0
x1+2 x2 = 35x1+2 x2 = 28
Solution: x1 = 6, x2 = 11Optimal Objective Value: 28
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Overview• Solving a Linear Program.• Visualizing Linear Programs.• What does solving a Linear Program mean?
• Algorithms for Linear Programming.• Simplex.• Ellipsoidal Methods.• Interior Point Methods.
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VISUALIZING LINEAR PROGRAMS
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Linear Program (General Form)Objective Function
Constraints
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Feasible Region
Feasible Region: Polyhedron(n dimensional)
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Optimization
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Optimization• Will the optimal solution always be at a vertex? • Prove it.
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Solving Linear Programs
• Outcome #1: Optimal Solution(s) exists.
• Outcome #2: Objective Function is unbounded.
• Outcome #3: Feasible Region is empty.
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Unbounded Problem (Example)
y
x
Feasible Region
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Infeasible Problem• Issue: Constraints contradict each other.
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Solving Linear Programs1. Find which of the three cases are applicable.• Infeasible?• Unbounded?• Feasible + Bounded = Optimal?
2. If Optimal, find optimal solution.• Note multiple optimal solutions possible.
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LINEAR PROGRAMMING ALGORITHMS
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Linear Programming• Solving systems of Linear Inequalities.• Early work by Fourier (Fourier-Motzkin Elimination Algorithm).• In symbolic logic, this is called “Linear Arithmetic”.
• World War II: Optimal allocation of resources.• Advent of electronic/mechanical calculating machines.• L.V. Kantorovich in USSR (1940) and G.B. Dantzig et al. in the USA
(1947).
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SIMPLEX• Simplex: algorithm for solving LPs.
• First Published by George B. Dantzig
• Prof. Dantzig contributed numerous seminal ideas to this field.
G.B Dantzig: Maximization of a linear function of variables subject to linear inequalities, 1947. Photo credit:
Stanford University
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Visualizing the Simplex Algorithm
(6,11)
(6,3)
(3,11)
(3,0)(0,0)
(0,2)
Not drawn to scale
Solution: x1 = 6, x2 = 11pt. Objective Value: 28
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Linear Programming Theory• Duality: John Von Neumann• Early work by Lagrange.• Connections to game theory.
• Generalized to Karush-Kuhn-Tucker Conditions.
• Complexity of Simplex: • Exponential time in the worst case (Klee + Minty).• Polynomial time in the “average case”.• Much remains to be understood.
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Polynomial Time Algorithms• Leonid Khachiyan’s ellipsoidal algorithm [Kachiyan’1980]• First polynomial time algorithm.
• Interior Point Methods• Ideas go back to Isaac Newton (Newton-Raphson).• First algorithms for Linear Programs by Narendra Karmarkar
[Karmarkar’1984]• Interior point methods are useful for non-linear programming (Cf.
Nocedal + Wright textbook).
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Applications of Linear Programming Theory• Too numerous to list exhaustively…
• Major application areas:• Operations Research.• Optimal allocation of resources.• Decision making.
• Computer Science • Algorithms, Machine Learning, Automated Reasoning, Robotics.
• Engineering • Control Theory
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INTEGER LINEAR PROGRAMMINGReal vs. Integer Variables
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Feasible Region
Feasible Region: Polyhedron(n dimensional)
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Linear vs. Integer Linear Programs
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Integer Linear Programming
Feasible Region: Z-Polyhedron(n dimensional)
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Linear vs. Integer Linear Programs (Complexity)
Polynomial TimeLinear
Programming (Reals)
Linear Programming(Integers)
NondeterministicPolynomial Time(NP)
Million Dollar Question: Can Integer Linear Programs be solved in polynomial time?
( P =?= NP)
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Example #1
(6,11)
(6,3)
(3,11)
(3,0)(0,0)
(0,2)
Not drawn to scale
Solution: x1 = 6, x2 = 11 Objective Value: 28
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Example #2
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LINEAR PROGRAMMINGFormulating a Linear Program
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Example (H&L)• Example 1: Design of radiation therapy for cancer treatment• Goal: select best combination of beams and their intensities to
generate best possible dose distribution• Dose is measured in kilorads
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Example 1: Radiation Therapy Design32
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Example 1: Radiation Therapy Design• Linear programming model• Using data from Table 3.7
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Example 1: Radiation Therapy Design• A type of cost-benefit
tradeoff problem
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