Optimization, the biggest problem of our generation
Ever been in calculus class slaving away at implicit differentiations and wondering so what? Beyond the wall painting and paper folding questions of MCV4U, there lies a cutting-edge field of mathematics, computer science, and physics.
At the heart of all these advancements? Multi-variable calculus.
Calculus can be looked at as a way of analyzing the output of a function by its input. In other words, it is the relationship between different variables of a function through derivatives, integrals, etc…
For example, the concept of the derivative is the instantaneous rate of change of some variable in respect to another. For the purpose of this article, it may be more helpful to think of this concept as the sensitivity of one variable to another:
if my x value changes by some amount, how much does my y and z variables change?
In this sense, it is ideal for analyzing optimization, or problems in which one must maximize one variable, dependent on other variables. In math class, you may have seen the example of maximizing the area of a shape by its perimeter, or the revenue from a product by its pricing. These problems generally only have two variables, and are simplifications of many complex and puzzling problems.
Although the idea of calculus has been with us for nearly 400 years, this field of study still drives scientific inquiry and advancements to this day. If you think about it, many problems in our life can be boiled down to a matter of optimizing many complex data structures. In fact, this is what our brains seek to do when making decisions: optimize tasks by time, optimize choices by potential happiness, so on and so forth. So why don’t we discuss some examples?
It turns out that a great application for optimization is in materials science!
In finding new materials, there are an abundance of factors to consider: conductivity, melting point, strength, structure, etc… In regular alloys such as steel, there are many ways for faults to be created in their crystal structures. You can read about all these potential defects here. With this many variables, small differences in structures can create a whole different material with whole new behaviours.
There’s an infinite number of potential alloys with different ratios of different metals. Nonetheless, optimization is employed to accelerate the process of materials discovery and testing.
This is an extensive topic in information science and computational methods and research. Although the number of materials discovered so far by projects such as The Materials Genome Project funded by the US Government are few and far between. It is an exciting and interesting field of research and innovation with applications in engineering new medical tools, producing cheap and durable solutions for construction projects in developing countries, and sustainable straws for the special needs. Optimization in this sense is still awaiting discovery and breakthroughs!
In finance, every decision is a trade-off or balance between risk and reward. It’s good to note at this point that almost anything that is a trade-off can be thought of as an optimization problem but in general financial optimization deals with maximizing return for a given amount of risk.
Individuals who are investing seek to maximize profit. But an individual who is retiring in 5 years will most likely be investing in something different than an individual who is retiring in 50. This concept is related to risk, where an individual with less time will invest in a less risky investment so as to avoid the case where they lose all their money and is not able to gain it back within the 5 years and are then forced to retire with a bank balance of $0.
This is where the idea of diversification comes into play. By spreading their investment among many different types with different returns, in different sectors and different locations, if one investment under-performs, the appreciation from the others are able to balance it out. For an example of a balanced collection of investments, or portfolio, click here.
For portfolio managers, balancing and diversifying the portfolios of their clients relies heavily on their personal situation and risk tolerance. Some factors include: time horizon, liquidity, age, other investments, and their goals. It is easy to see with the number of factors affecting risk, plus the number of factors affecting the return of various investments with different levels of risk, return, maturity, volatility, etc… this balance and optimization problem can become rather complex rather quickly.
Many times, individuals will invest in different types of mutual funds of generally managed portfolios for categories of risk or investment styles. By optimizing and customizing based on more personal and specific metrics, a better and more catered solution can be found.
Again, this optimization process is rather complex. Different algorithms and simulations have been suggested and are currently being used to simplify the problem and create a more automated and personalized experience for personal finance.
For a tougher read, here is an excellent Medium article on machine learning and its applications in portfolio management.
Earth-like Exoplanet Discovery
So, I’ll admit that this one is really a crackpot theory. Exoplanets are planet sized objects orbiting stars very very far away. Planets are small and dark and hard to detect from far away. As such scientists have found many wacky and clever ways to detect them.
Radial velocity is one such example. It turns out that stars aren’t completely still (shocker, I agree). Nonetheless, it tracks an ellipse or circle corresponding to the gravitational pull of the smaller planets. This movement actually alters the wavelength of the star’s colour spectrum so that when it is farther away the light becomes red-shifted (or redder) and blue-shifted when it is closer. Scientists can use spectroscopes to detect these regular patterns of red and blue to find these planets. However, this detection system cannot give the actual mass of the planet. In fact the mass detected could very well just be a small star. Additionally, if a star was positioned just correctly, most of its wobble or motion could actually be perpendicular to our line of vision, resulting in planets that are undetected.
In addition to that, there are also techniques such as Transit Photometry (which looks at the dimming of a star when a planet passes by) or Microlensing (which looks at the bending of light around massive objects). Each of these techniques have different advantages and scopes in terms of distance and information generated.
There are various aspects and factors of exoplanets that make them inhabitable or similar enough to earth to colonize. It seems like soon enough this should be a good application for optimization! I don’t know exactly how or why but I guess that’s just a problem for our future scientists to solve!
The problem is that complex systems require exponentially more computational powers, some problems which are theoretically solvable, may not be feasible within our life time. Deep Learning and AI has made great strides in this aspect. Machine learning uses optimization to accomplish techniques of classification, regression, clustering and dimensionality reduction.
This is also a place where high hopes for quantum computing motivates tons of research and funding. Everyone’s hoping that eventually the quatum properties embedded in qubits will be able to be harnessed to create a more powerful and efficient computer which may be used for optimization. Although this technology seems years away because of quantum noise and other barriers, google’s recent announcement gives hope for many, it is still a new and emergent field.
Just kidding, more math is beyond the scope of this article and can be left as a example for the reader to do as practice. Haha… In any case, it turns out that calculus alone may not be sufficient in modeling our very tricky problems. Although it is the basis on which other concepts are predicated, we can later open a more intermediate mathematical toolbox filled with things like linear algebra, graph theory, and so much more just in undergrad.
These key mathematical disciplines are fundamental in understanding and describing these complex systems. In fact, some of math’s most difficult problems have to do with this and optimization. Such as the case of the travelling salesman!
Nonetheless, whether it’s chemistry or business, math is always math. And it is arguably the most important tool for describing and solving problems now and forever.
One of my friends recently asked me “what’s so cool about theoretical science” and I found it hard to answer. It seemed like quite frankly a waste of time. All it seems that the physicists of the world are doing is shooting atoms at each other really fast while engineers are crafting solutions to all of society’s greatest problems.
The thing is, theory is important for application. Even when it is completely unrelated! The fact that these seemingly unrelated disciplines (alloys, finance, space, and nerdy math) have a link to each other, where discovery and advancement in one could lead to a break through in another, makes my love for theory even greater (and somewhat justified). After all, the world is a giant complex system, probably nearly impossible to optimize, but by entertaining our own nerdy curiosities, we are able to discover and describe the world with such rich and deep meaning that really any application can be derived.
So, application is nothing without theory and theory is meaningless without application. How should be allocate our resources between the two? That’s really an optimization problem for you!