Optimization & Optimism
Michael Bartholomew-Biggs looks at links between mathematics & poetry
On seeing the words “optimal” and “control” appearing together in the June 2010 issue of London Grip (In Control – in an Optimal Way by David H. Jacobson), I wondered whether I had mistakenly opened a specialist technical journal.
Of course, it is perfectly reasonable for a review of an applied mathematics book to appear alongside articles on art, politics & literature in a cultural magazine for general readers; and yet, as a professional mathematician, I am surprised when it happens. That surprise now prompts me to share some of my own reflections (in rather less austere terms than Jacobson’s) on the practice of optimization and on its practitioners or optimizers, among whom I number myself.
Optimization is a mathematical discipline closely related to optimal control. Optimal control deals with changing situations whereas optimization handles static ones; but this distinction need not trouble us here. Both make use of mathematics to work out how to “do one’s best”, it being assumed that best is measurable – a job done in minimum time or a product made for maximum profit
It has been claimed that everything is an optimization problem. Many systems in nature have evolved (or been created) to function with minimum expenditure of energy. Human beings often pursue this same goal when planning a day’s tasks – although most of us do it by guesswork rather than precise mathematics. (But even rough-and-ready efforts to optimize our lives might be more successful were it not for restrictions or constraintsimposed by other people.)
Mathematical optimization comes into its own in science and engineering applications such as designing an aircraft wing to be as light as possible for a given strength or shaping a spacecraft’s trajectory to complete a mission using minimum fuel. Optimization works well in such situations, where well-founded mathematical models enable us to predict quite accurately what will happen if some variable is adjusted. Optimization can be less effective – to put it mildly – when human behaviour is involved. Our “optimal” financial decisions can let us down if formulae for modelling risk prove much less trustworthy than Newton’s laws for describing flight dynamics.
Optimization employs iterative computational algorithms – computer programs which seek the optimal solution via a sequence of steadily improving approximations. Progress of an algorithm can be viewed as a systematic exploration of a mathematical landscape (think of a flat contour map extended into many hundreds of dimensions). This exploration searches for the highest or lowest point; and to be purposeful and efficient, rather than random and haphazard, it must make and check sophisticated inferences about the shape of the terrain as it goes along. The search may also be complicated by the presence of constraints like engineering safety limits and the laws of physics. When such limitations intrude on our hyper-landscape it is as if a capricious hyper-landowner has erected fences that cannot be crossed and pathways that must be followed.
Many ingenious methods have been suggested for searching a hyperspace quickly and efficiently. Optimizers seem to have quite competitive natures and so they are often “done-to as they do” – that is, they find themselves under continual scrutiny aimed at establishing whose optimization algorithm is currently the best. Sections of the technical literature resemble arenas for numerical duels to try and determine which algorithm can solve the most difficult problems most quickly.
More detailed reflections on all the above appear in my own textbooks:Nonlinear optimization with financial applications (Kluwer, 2005) andNonlinear optimization with engineering applications (Springer, 2008). Excellent though these books are, neither of them is hot off the press and so the mere announcement of their existence would hardly justify an article in London Grip. What does make them worth mentioning is the fact that their content is embellished by intriguing cover art and enhanced by poetry.
Howard Fritz’s cover paintings were not made specifically for the books; but they interact well with the contents. The financial book’s cover shows a queue of people ascending a mountain which is topped by a rocket-like tower – perhaps symbolizing the profit motive – while studiously ignoring an empty set of burning overalls beside the path. Does this illustrate a human tendency to go along with the crowd and ignore the hazards? The picture on the engineering volume is of a sad and solitary lady on a platform overlooking a shipyard. Has she come to launch a new vessel? But the yard contains only half-finished and abandoned hulks; perhaps it has been closed down for allegedly optimal economic reasons.
My poetic chapter-endings mostly play with the picture of optimization as a geographical search. The steepest descent method can be understood by imagining ourselves walking on a hillside in thick mist. We can only see the ground at our feet; and to find the bottom of the valley we must make controlled moves down lines of greatest slope:
Still the fog persists.
Let the incline have its way
and set your compass.
Keep taking footsteps
until that first suspicion
of an uphill slope
then turn left or right,
just one of many zig-zags.
Will it ever end?
The need to comply with several simultaneous constraints suggests a zen-like observation:
To walk a tightrope
is hard. So how much harder
to walk several.
Spiders manage it
spinning sticky contour plots
which aren’t safety nets.
A particularly trying difficulty is that we can rarely be certain whether an optimization search has located Everest or Snowdon – that is, the overall peak (global solution) rather than merely a local high point. We can say of a global solution that
It’s the only place
to be – if you can find it.
(If not, you won’t know.)
Juxtapositions of mathematics and poetry are more common than many people seem to expect: googling “poetry + mathematics” produces some three-and-a-half million hits; and a recent anthology Strange Attractors (A. K. Peters, 2008) contains about 250 pages of poems linking mathematics with love. Joan Margarit, the Spanish poet who is also an architect and structural engineer, has remarked that mathematicians and poets are both concerned with wider truths that are stirred up by particular observations. Both seek to abstract away from the solid and specific in order to say something more elusive, comprehensive and beautiful. Thus
Poets show, don’t tell:
build metaphors from concrete
and specific bricks.
abstract and general is
our bread and butter.
But optimizers and poets may be kindred spirits not only in aspects of technique but also in what drives them. Being human, both are liable to pride: rivalries over algorithm performance and reputation may be as keen as any which exist between parallel contenders for the T.S. Eliot Prize. But, in their nobler selves, optimizers and poets share a belief that they can show how the world can be improved. They also share a wish that the world would listen to them. For poets at least, that wish is commonly ungranted; hence they remain the unacknowledged control engineers of the world.
Reader Emeritus in Computational Mathematics University of Hertfordshire
Author of Nonlinear optimization with financial applications (Kluwer, 2005)
and Nonlinear optimization with engineering applications (Springer, 2008)
Poems from his two text books have an independent existence in
Uneasy Relations (Hearing Eye, 2007) which includes explanatory diagrams and notes.