I had to create a problem for my Optimization by Vector Space Methods class. It was suggested to me that I post my problem here. I certaintly don't expect that anyone will try to solve this, but I think that some people may be entertained by it.

Joe Shmo is a lawyer who often drives from Santa Barbara to the San Francisco Bay Area, a distance of 300 miles, 100 of which has a speed limit of 70 mph. The remaining 200 miles has a speed limit of 65 mph. In the 70 mph zone, he estimates that if he drives at speed x, the number of speeding tickets he can expect to get, if x is above the speed limit, is t₁(x)=.01(x-70). He estimates that he can expect to get t₂(x)=.005(x-65) tickets if he drives at speed x for the remaining 200 miles. He also knows that each ticket will cost him $10 for each mph over the speed limit he is driving. He estimates that for each ticket he gets, it will cost him 15 minutes at the side of the road. He knows that the number of gallons of gasoline it takes him to go 100 miles is a function of his speed x, G(x)=7.3-.13x+.001x², and that gasoline costs him $1.50 per gallon.

Joe wants to do some optimization with regard to his frequent trips, but unfortunately he has never taken ECE 271c, so he needs some help. He wants to minimize how much his trip costs him. During his trip he talks to his clients on his cell phone and charges them $100/hour. So, when he is pulled over at the side of the road getting a ticket, this costs him $100/hour in addition to the cost of the ticket. He also insists that his average speed be at least 100 miles per hour, but he insists that this be calculated (incorrectly) using the following equation: avgspeed=(x₁+2x₂)/3 where x₁ is his speed during the 70 mph stretch and x₂ is his speed during the 65 mph stretch. Help Joe solve his optimization problem. Do the problem two different ways, first using Lagrange duality, and then using Fenchel duality.