One-dimensional optimisation, Victorian style
For many years, I taught an undergraduate course on nonlinear optimisation. It was a subject close to my heart, as it formed a significant part of my PhD thesis. Nonlinear optimisation is concerned with finding the location (parameter values) and objective function value of a function which is not a linear expression. School calculus teaches you to differentiate the function and find where the gradient is zero. All very well, until the function is only found by a computer evaluation of a series of expressions, or by experiment, or the solution of the derivative equation(s) is difficult/impossible. Then, the only way forward is to search.
At the time, the 1970s, there was a great deal of interest in the numerical analysis of algorithms to be efficient in terms of both the number of evaluations of the objective and the computer storage needed for the algorithm (How times have changed!). Many methods relied on a series of linear searches - I used to illustrate this with the aid of a transistor radio with built in antenna. First, turn the dial to pick up the station. Then rotate the radio to improve reception. Then turn the dial again for more improvement. Then a further rotation. This was a situation where the derivative of my objective was unknown, so we used search methods.
On holiday, Tina and I visited the Tall Ship, Glenlee, moored on the River Clyde in Glasgow. (That's her figurehead, above.) A fascinating tour of this 1895 ship, restored with great care. A comment in the book about the ship was about optimisation. Something which happened with all cargo-carrying tall ships, was the decision about the trim of the ship. "Most captains preferred the vessel to be trimmed by the stern, often down between two and eight inches. Glenlee seemed to sail best when down by six inches" So here was an optimisation problem - find the best trim, within a given range. How much experimentation went into the search for the best trim for the ship? And how much difference does it make? And if this happened in Victorian times, did it happen earlier in history?
One-dimensional optimisation like this is relatively straightforward - the captains would know which way to change the trim - but it is still a practical problem, with a numerical answer - good O.R.!
At the time, the 1970s, there was a great deal of interest in the numerical analysis of algorithms to be efficient in terms of both the number of evaluations of the objective and the computer storage needed for the algorithm (How times have changed!). Many methods relied on a series of linear searches - I used to illustrate this with the aid of a transistor radio with built in antenna. First, turn the dial to pick up the station. Then rotate the radio to improve reception. Then turn the dial again for more improvement. Then a further rotation. This was a situation where the derivative of my objective was unknown, so we used search methods.
On holiday, Tina and I visited the Tall Ship, Glenlee, moored on the River Clyde in Glasgow. (That's her figurehead, above.) A fascinating tour of this 1895 ship, restored with great care. A comment in the book about the ship was about optimisation. Something which happened with all cargo-carrying tall ships, was the decision about the trim of the ship. "Most captains preferred the vessel to be trimmed by the stern, often down between two and eight inches. Glenlee seemed to sail best when down by six inches" So here was an optimisation problem - find the best trim, within a given range. How much experimentation went into the search for the best trim for the ship? And how much difference does it make? And if this happened in Victorian times, did it happen earlier in history?
One-dimensional optimisation like this is relatively straightforward - the captains would know which way to change the trim - but it is still a practical problem, with a numerical answer - good O.R.!
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