1. Vector Optimization Study Guide for ES205 Yu-Chi Ho Jonathan T. Lee Jan. 12, 2001

2. 2 Outline  Problem Statement  Motivation  Pareto Optimality  Scalarization  Nonconvex S’  Numerical Methods

3. 3 Problem Statement  where S is the feasible set in  n and J is the objective function, J: S   m and, S’ is the feasible performance region. N J1J1 J2J2 S’S’

4. 4 Motivation  Many real world problems require to optimize multiple criteria at the same time. Find the optimal path of an airplane flight while minimizing both the time it takes and the fuel consumption. Buying a car with the best quality while spending the lowest amount of money. N

5. 5 Pareto Optimality  x’  S is a Pareto optimum if there is no x  S such that J i (x)  J i (x’) for all i = 1, …, m and J i (x) > J i (x’) for some i.  The “best that could be achieved without disadvantaging at least one group.” (Allan Schick, in Louis C. Gawthrop, 1970, p.32) N

6. 6 Pareto Optimality (cont.)  m = 2 Pareto frontier: the set of Pareto optimum N J 1 (x)  J 1 (x’) J 2 (x)  J 2 (x’) x’x’ J1J1 J2J2 S’S’

7. 7 Scalarization  for i > 0, i = 1, …, m.  To “summarize” the multiple criteria into a single criterion — scalar- valued optimization problem.

8. 8 Scalarization (cont.)  m = 2 J1J1 J2J2 x’x’ 1 J 1 + 2 J 2 N S’S’

9. 9 Nonconvex S’  Only two of the optimal points could be identify through scalarization J1J1 J2J2 S’S’ N

10. 10 Constraint Optimization Pb. max J 1 (x) s.t. J 2 (x)   for different  N  J1J1 J2J2 S’S’

11. 11 Nonconvex S’ (cont.) J1J1 max min [J 1, J 2 ] N J2J2 S’S’ x’x’

12. 12 Nonconvex S’ (cont.) N J1J1 J2J2 S’S’ x: Aspiration Point x’x’

13. 13 Numerical Methods  Linear Programming: both the objective function and the constraints are linear  Non-linear programming

14. 14 References: Heylighen, F., Web Dictionary of Cybernetics and Systems, http://pespmc1.vub.ac.be/ASC/. Ho, Y.-C., “Optimization – A Many-Splendored Thing –,” slides presented at IFAC World Congress, 1999. Jahn, J., “Theory of Vector Maximization: Various Concepts of Efficient Solutions,” in Chapter 2 of Multicriteria Decision making – Advances in MCDM Models, Algorithms, Theory, and Applications by T. Gal, T.J. Stewart and T. Hanne, Kkuwer, 1999. Mas-Colell, A., M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, 1995.