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Optimization

Optimization. A Recap. Local Optimization and Absolute Optimization - How Different Are They? . Absolute Optimization The domain is constrained to a closed and bounded region most of the time. Closed and bounded regions are guaranteed to have absolute extremes.

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Optimization

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  1. Optimization A Recap

  2. Local Optimization and Absolute Optimization - How Different Are They? • Absolute Optimization • The domain is constrained to a closed and bounded region most of the time. • Closed and bounded regions are guaranteed to have absolute extremes. • We analyze the partial derivatives for extremes in the region only. • We analyze all paths. • We analyze all corner points. • We evaluate everything we found. • The max is the highest z-value; • The min is the lowest. • Discard anything in between. Local Optimization The domain is unconstrained. We analyze the partial derivatives alone for stationary (critical) points. We use the determinant to test stationary points. We are concerned about saddles since they look like extremes if not correctly analyzed.

  3. Looking Inside the Boundaries

  4. Looking Inside the Boundaries Find the first partials. Solve the system.. Keep all points in or on the feasibility region. Discard points outside the feasibility region.

  5. Optimizing on the Boundaries No extremes here! • Analyze each edge of the rectangle for any extremes between the end points (corners). Keep all points in or on the feasibility region. Discard points outside the feasibility region.

  6. The Corners • Find all corners points of the region. • The rectangle has four corners at • If it were more difficult than this, we solve each pair of constraints for the intersections.

  7. The Corners • Evaluate every critical point and corner points. • The absolute maximum is the highest z-value. • The absolute minimum is the lowest z-value.

  8. A New Constraint • Actually there are two new constraints. They are parallel lines. • They eliminate the origin from consideration and replace it with two new corner points.

  9. The Last Constraint • This one slices the region into upper left and lower right. • We need to make sure we choose the right region!

  10. Linear Optimization Suppose we had two machines (x and y). The value of each machine is $1,000 per unit of machine x use and $3,000 units of machine y use. We have set a limit of machine use for x and y of 4,000 units each. Similar we insist that use machine x and 2 y must be at least 1,000 units and no more than 9,000 units. Last we want the ratio of machine y to machine x to be 2 to 1. Read through the constraints carefully and you will see that except for scale (x and y in thousands of units)they do create our linear optimization problem.

  11. The Endof the Optimization Block! Reread your on-line lessons.

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