Examples of Various Formulations of Optimization Problems

1. Examples of Various Formulations of Optimization Problems

2. Example 1 (bad formulation) A chemical factory produces a chemical from two materials, x and y. x can be purchased for $5 per ton and y can be purchased for $1 per ton. The manufacturer wants to determine the amount of each raw material required to reduce the cost per ton of product to a minimum. Formulate the problem as an optimization problem…

3. Solution (?): Linear problem

4. Solution (?)

5. Example 2 Given the perimeter of a rectangle must be at most 16cm, construct the rectangle with maximum area. Formulate this as an optimization problem.

6. Solution: Nonlinear problem

7. Example 3 Suppose we want to maximize the area of an object, but we have a choice between a square and a circle, where the length of the square is equal to the radius of the circle, and the radius can be at most 4 cm. Formulate this as an optimization problem. Object 1 Object 2

8. Solution: Mixed integer nonlinear problem

9. Parameter Identification Identify the damping, c, and the spring constant, k, of a linear spring by minimizing the difference of a numerical prediction and measured data. Assume that the spring-mass system is set into motion by an initial displacement from equilibrium and measurements of displacement are taken at equally spaced time increments.

10. Parameter Identification continued The motion of an unforced harmonic oscillator satisfies the initial value problem, Formulate this as an optimization problem

11. Nonlinear Least Squares Problem

12. Example: ‘Black-Box’ Formulation Suppose there is a contaminated region of groundwater (a plume) that we wish to keep from moving. We can do this by installing wells in the region and changing the direction of groundwater flow. We would like to do this as cheaply as possible…

13. Hydraulic Capture ModelsGoal: To alter the direction of groundwater flow to control plume migration Possible Decision Variables: • Number of wells • Well rates • Well locations

14. Governing Equations

15. Objective Function: Cost to install and operate wells

16. Implementation: Simulators • MODFLOW for flow equation • USGS code • Cell centered finite differences • FORTRAN 77, serial • MT3DMS for transport equation • EPA code • Links to flow data from MODFLOW • Cell centered finite differences • FORTRAN 90, serial

17. Constraints

18. Capture constraint to keep the plume from spreading “We leave it to the modeler to choose the physical and mathematical representation of the constraint.” 7 + 6 = ?

19. A closer look at FBHC We can consider head differences in adjacent nodes (aligned in the x,y,or z direction) as constraints on the approximate velocity since Dacry’s Law is For example: A finite difference head gradient in the X direction is

20. A closer look at FBHC d? k? Locations?

21. FBHC • Advantages: • Easy to implement • Constraint requires flow info only • Disadvantages • Not constraining the concentration • d? k? locations?

22. Alternate FBHC approach • Is there another way we can use only flow information to capture the plume? • A method for choosing d,k,locations? • Directional derivatives? • Fix well locations easier?

23. Optimization Implementation Issues • Evaluation of J(u) requires a simulation • Parallelism is preferred • Gradient information is unavailable • Removing a well means J(u) discontinuous  Sampling methods look appealing: Optimization is governed by function values