Optimization - Day II

Today, we continued our discussion of simple optimization. In our first encounter with a simple problem, we realized how vital the initial modelling of our problem is. In the end, we came up with 5 simple steps to solving problems involving optimization:

1. Declare your variables.
-They don't have to be clever--as long as you don't mix them up with other variables.

2. Determine (and write down) your objective function.
-Your objective function is simply the function you are trying to optimize the result of. For example, if you are trying to optimize volume, your objective function would be V=xyz, assuming x, y, and z are your variables.

3. Determine your constraints.
-There are always constraints for optimization problems--otherwise, there would be nothing to base your optimization off of, making the entire problem pointless. Recognize these constraints and use them to solve the problem.

4. Take the objective function and find the first partial derivatives, set each
(fx and fy) equal to zero and solve for your critical values.
-Keep an open mind during this step, as you may find minor tricks to make your solution simpler.

5. Take the first partial derivatives and evaluate the second partial
derivatives at the critical point you extracted in Step 4. Using your second
derivative values (fxx, fyy, and fxy), determine d by using formula.


If d > 0 and fxx > 0, then your critical point is at a minimum.
If d > 0 and fxx < 0, then your critical point is at a maximum.
If d < 0, then your critical point is at a saddle point (both max and min, depending on your perspective).
If d = 0, prepare for Armaggedon.

Happy optimizing!