Decision-making under uncertainty poses huge risks that could spell out the difference between a company’s—or even a military action or government’s—success or failure.
To address these uncertainties, mathematicians are developing stochastic optimization models and methods, which consider random variables and statistical models to account for the uncertain parameters or data and to maximize the chances of success under these uncertain circumstances. This work is an example of applied math—that is, math that is applied to real-world scenarios.
Darinka Dentcheva, professor and chair of the Department of Mathematical Sciences at Stevens Institute of Technology, researches and develops stochastic optimization models with a variety of applications. In the last 12 years, she has received five grants from the National Science Foundation supporting the development of various aspects in the theory of risk-averse optimization
The field of stochastic optimization was born more than 60 years ago, but, until recent years, it focused on average performance. Out of a need to solve problems and make robust predictions in finance, a new area in stochastic optimization has emerged: how to model and quantify risk. There are some rare events that can have devastating impacts, and in these cases careful consideration of risk models is critical. Since its beginnings, this area has broadened to inform diverse business and engineering applications, data management, and even environmental conservation, among many other specialized areas. In any problem where there are risks and uncertainties, stochastic optimization can help researchers and industry leaders make the best decisions possible.
“In most practical applications in our world, if you want to create a model for some system, and facilitate optimal decisions, then you discover that most of the parameters or elements that have to go into the model are not known,” Dentcheva explained. “A good model is not completely deterministic; there are some random elements in mathematical terms.”
As an example, Dentcheva relates her work at Vereinigte Energiewerke AG (VEAG), a large power company in Germany that has about 40 plants, where she spent two years developing an optimized model for their power production. Her model needed to meet the uncertain demand of the industry and population. While Dentcheva was responsible for developing the model using the theory and methods of stochastic optimization, the industry was responsible for ensuring that the model was practical enough to address their needs.
“We have to know what our goals are, and according to this, determine what the best course of action is,” she said. “In dynamical systems, we consider not only the current course of action and short-term goals, but we also look to balance it with the strategic course of action and the long-term goals.”
Stochastic optimization models are fundamental in machine learning and, more generally, artificial intelligence, Dentcheva explained. Stochastic risk-sensitive models can produce robust estimations and predictions, as they take into account variations and unknowns in the data. “In that sense, I expect that those models will play a role for new methodologies in machine learning,” she said.
“My area encompasses very theoretical and highly sophisticated mathematical tools, the development of efficient numerical methods based on the theory, and their application and implementation of real-life systems. One can verify a theory and see its impact in reality.”
She added, “This is an area in which you really need to have an imagination, so I like that.”
In search of clear and elegant answers
Dentcheva spent her high school years devoted to mathematics, where she had a competitive side. Her home country of Bulgaria, she explained, has a tradition of math competitions. There she competed with her high school’s team, winning the first prize in a national competition. That same year, she qualified for the Bulgarian team for an international olympiad. “So I was automatically accepted to study math,” she said of her higher education at Humboldt University in Berlin, Germany.
“I like math because it is always clear whether [the answer] is right or not,” she said. “I have diverse interests. I like literature, I had music lessons, and I like biology also—but for me, literature and art are always a matter of subjective preferences. In mathematics, it is much easier to agree on a beautiful result.”
Dentcheva delights in the creativity, complexities, and possibilities of math. “Mathematics is indispensable in advancing technology,” she said, adding, “Isn’t it beautiful, mathematics? It is beautiful.”
Learn more about applied mathematics at Stevens: