Archives for December 2015

Conversations with a Founder: Constrained Optimization

Rural System is committed to:

  • Enhancing the history and beauty of the rural region,
  • Generating meaningful work with salaries and benefits,
  • Stabilizing and building the strength of small rural communities (first in western Virginia), and
  • Improving lasting natural resource management.

In speaking with the founder of Rural System, Dr. Robert H. Giles, just yesterday, I discovered that Rural System has at least three different kinds of objectives. “The first level of objectives,” he explained, such as those listed above, “is grandiose and stuff that you find in legal documents that start off the business. The second level is still very general, but sort of why you’re doing it in more specific terms. The third calls in constraints.”

With every type one or type two objective that Rural System sets, there will be type three objectives that set conditions or boundaries on how the first two types of objectives may be achieved. Rural System’s valuable operation is to use prescriptive software, drawing on data organized spatially, to find optimal solutions to type one and type two objectives within the constraints of type three objectives.

Constrained optimization is a type of mathematical operation used in a variety of fields to try to solve complex, realistic problems in the best possible way. Alex Edelman, a Ph.D. student of condensed matter physics at the University of Chicago, provided this succinct explanation of how constrained optimization works.

Dr. Giles elaborated with a helpful example as well, “When somebody says that my objective is to go to Roanoke, I have other objectives of not going too fast or too slow, not being damaged, so these are limits to every first objective. I must not speed. I must not injure any dogs or cats.”

Constrained optimization is a principle that acts on every level of Rural System. The best example of its application, perhaps, is in how Rural System approaches profits. “Rural System works on the premise that we’re not going to maximize profit, because if we maximize profit we will do exactly what has caused problems for land management in past years,” Dr. Giles stated. He elaborated that the traditional rural business approach in Virginia and elsewhere in the United States was to try to maximize profits, only to find that conditions change and soon after a maximum is reached, profits bottom out. This traditional boom and bust model makes farming extremely risky, and in fact requires a great deal of government bailout in order to support.

“We have to get away from this traditional rural approach,” Dr. Giles emphasized, “and so what Rural System is designed to do, its strength, is to set boundaries on profits according to current, ongoing ecological research in order to sustain profits in the long term. The only way that society is going to achieve success and stability by 2050 is for us to implement an agricultural system, all of us, in our nation, to provide stable profits.”

Dr. Giles likened his business model to a common strategy used by stock brokers. Brokers set the bounds for how far they will tolerate the ups and the downs of different stocks. As long as they stay within the bounds, they know they will not lose too much. They are risk intolerant. In this way, constrained optimization is just a form of risk management.

The administrative group of Rural System, System Central, would be comprised of a board of businessmen and women who would set the bounds on their projected profits, based on information given to them by Rural System ecologists.

Dr. Giles sketched it out, speaking as he drew, “So we know, here’s constraint number one here, here’s the other constraint here, and as long as our curves stay up in here, we’re ok.”

Rural System profits optimized within constraints

Constrained optimization is clearly one of the most important keys to Rural System’s anticipated success. The approach is Dr. Giles’ original answer to a problem that has plagued land managers for decades. Do we conserve natural resources, or should we use them freely? Dr. Giles insists that there will be no lasting motivation to use natural resources wisely without some profit incentive, but if it can be shown that maximizing profits is not possible without periodic ruin, perhaps a crucial balance may be found. Without such a balance, our limited natural resources are surely doomed.

“We don’t know what life is, but this is life,” Dr. Giles drew a circle. “If this is all possible life then a person’s life can be talked about in terms of constraints,” he drew lines cutting through the edges of the circle. “We can talk about how fuzzy these boundaries are and whether I can attack these constraints. Before you know it, if you see these as limitations, if you recognize these as real constraints or as constraints that can be modified, if that’s all that can exist, then in one sense life is just a bunch of constraints. If you see then that life is constraints based, it may be a thing to accept or it may be an opportunity.”

Sketch by Robert H. Giles showing life as constraints-based

Perhaps we won’t work through our lives so mathematically, but the truth here cannot be denied: when we choose how to live our lives, we need to be able to recognize the constraints we can push and those we must accept.

A Mountain View of Constrained Optimization

Written by Alex Edelman. Illustrated by Laurel Sindewald.

You are dropped on an alien planet and told to climb the highest mountain.

Constrained Optimization on an Alien Planet

This is a perfectly good optimization problem. You are given a so-called objective function – in this case your elevation, as a function of latitude and longitude – and your problem is to maximize it. In principle there is an obvious way to solve this problem: just survey every possible latitude and longitude and once you’ve covered the whole planet, name the one that was highest. For that matter you can imagine solving any optimization problem this way, by trying out all possible combinations of input parameters and choosing those that win.

You may protest that this is a bad strategy, and your intuition is right. Implicit in your mountain-climbing mission is the desire to solve the problem efficiently, meaning that your oxygen tank should not have to get exponentially large with the size of the problem. Now in general there is no foolproof trick to be efficient. Instead you must exploit the particular simplifications that your problem allows.

For one thing your objective function, elevation, is fairly simple, and generally goes up and down smoothly as you move around the planet. If you permit me to make this planet somewhat abstract, an objective like this has a simple enough mathematical structure that it can be optimized by calculus alone, without ever leaving your armchair. But the more difficult optimization problems that we often care about usually have a more complicated objective function – for instance, find the place on the planet with the best view. This is pretty hopeless to predict without actually stepping out, but there are nevertheless some regularities. If you’re somewhere on a hill, for example, and the view is good, it’s probably a good idea to walk a bit up the slope and see if it’s any better.

In this case our armchair solution also supposes that our problem is unconstrained, that is, you are allowed to go anywhere on the planet that you wish. Suppose, though, that you want to find the best view within a five-minute walk of your spaceship. Realistic optimization problems are also subject to such constraints: it is usually just not possible to push a parameter up to arbitrary values. With constraints, a lot of naive strategies, like simply climbing up the nearest hill, fail. For some problems, with simple mathematical structures, the constraint can be turned into a sort of third parameter – latitude, longitude, distance-from-spaceship-tude – while for others more exotic strategies must be adopted.

In practice there are relatively few optimization problems that can be solved efficiently. In many cases we can only hope to find a good-enough local optimum in finite search time, trading off between climbing hills in one place and looking for places that might have better hills to climb. An algorithm can be used to quantify such trade-offs, for instance the simulated annealing algorithm was inspired by a metallurgical technique, which alternates heating, to dislodge kinks, with cooling, to let the material settle down around kinks that remain, without any guarantee that the final product is completely kink-free.

There is a second trade-off, this one in modeling, between constructing a mathematical optimization problem that captures the full richness of reality though impossible to solve efficiently, and making enough simplifying assumptions to get a solvable model – better yet, a model simple enough to give interpretable statements about how reality works.