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Advanced Linear Programming Techniques in Engineering and Management

This course, offered by Ashish Goel from the Department of Management Science and Engineering at Stanford University, delves into the formulation and application of linear programming (LP), covering both theory and practical engineering problems. The curriculum includes discussions on convex and nonconvex optimization, allowing students to undertake self-selected projects. Collaboration and group study are encouraged, with resources available on Piazza for questions. Lectures are based on Yinyu Ye's slides, providing foundational knowledge in optimization and algorithmic principles.

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Advanced Linear Programming Techniques in Engineering and Management

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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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