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LP Examples

LP Examples. Ashish Goel Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A . http://www.stanford.edu/~ashishg http://www.stanford.edu/class/msande211 /. Further Class Information.

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LP Examples

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  1. LP Examples Ashish Goel Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/~ashishg http://www.stanford.edu/class/msande211/ Lecture #2; Based on slides by Yinyu Ye

  2. Further Class Information • We cover formulation/application as well as theory (difference with 111 and 311) • We cover engineering problems as well as managerial • We cover convex optimization as well as nonconvex • The project is implicit (you make request additional projects if you like) Also, you may choose 3-4 units independently of project choice. • Please use Piazza for all non-personal questions, such as to make a project team or study group. • HWs: discussion allowed, but you must write and submit separately • Ask your neighbor if they want to meet up for study-group • Discussion section starts next week • Attend if you can; don’t worry if you have class conflict since you can watch it later • Read Chapter 1, 2 , and 3 in Optimization and Algorithms Lecture #2; Based on slides by Yinyu Ye

  3. Abstract Model • x11 • 1 • 1 • 2 • 2 • 3 • 3 • x34 • 4 LP Example 2: Transportation Lecture #1; Based on slides by Yinyu Ye

  4. LP Example 3: Support Vector Machine • ai • bj {y: yTx + x0 = 0} x is the normal direction or slope vector and x0 is the intersect Lecture #1; Based on slides by Yinyu Ye

  5. LP Example 3: Is Strict Separation Possible Are there x and x0 such that the following (open) inequalities are all satisfied Are there x and x0 such that the following inequalities are all satisfied for arbitrarily small ε. Divide x and x0 by ε., the problem can equivalently reformulated. This is a special LP, called linear feasibility problem. Lecture #1; Based on slides by Yinyu Ye

  6. LP Example 4: Electric Vehicle Charging Schedule Lecture #1; Based on slides by Yinyu Ye

  7. LP Example 4: When Discharge is Allowed Lecture #1; Based on slides by Yinyu Ye

  8. Linear Programming Abstraction Lecture #1; Based on slides by Yinyu Ye

  9. Abstract Linear Programming Model Lecture #1; Based on slides by Yinyu Ye

  10. Coefficient matrix • Obj. vector decision vector • RHS vector LP in Compact Matrix Form Lecture #1; Based on slides by Yinyu Ye

  11. Some Facts of Linear Programming • Adding a constant to the objective function does not change the optimality • Scaling the objective coefficients does not change the optimality • Scaling the right-hand-side coefficients does not change the optimality but the solution gets scaled accordingly • Reordering the decision variables (together with their corresponding objective and constraint coefficients) does not change the optimality • Reordering the constraints (together with their right-hand-side coefficients) does not change the optimality • Multiplying both sides of an equality constraint by a constant does not change the optimality • Pre-multiplying both sides of all equality constraints by a non-singular matrix does not change the optimality Lecture #1; Based on slides by Yinyu Ye

  12. Hidden LPs • Today • Supporting Vector Machine when strict separation may not be possible • Air traffic landing time control • Later: • Financial Big-Data analysis • Combinatorial auction for information market Lecture #2; Based on slides by Yinyu Ye

  13. ai • bj Supporting Vector Machine Revisited Lecture #2; Based on slides by Yinyu Ye

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