Sample Optimization Applications

• In radiotherapy for cancer treatment, the placement of radioactive seeds in tumors and the targeting of X-rays aims to deliver a specified radiation dose to the tumor while minimizing radiation exposure in surrounding healthy tissue.
• In computational biology, the folded conformation adopted by a protein in the presence of water determines its function. It is widely agreed that this conformation can be found by minimizing a function that defines the potential energy of the molecule: a global optimization problem.
• Optimization theory and algorithms are used in areas such as machine learning and data mining.
• To predict the flow of traffic in a congested road network, we can apply an equilibrium principle that says that each driver will choose the fastest route between his or her origin and destination, and that in the flow pattern resulting from these individual decisions, all routes actually used between an origin and destination will take the same time to travel. (This is an example of an equilibrium problem in which, in general, there is no objective function, but certain elements of the system have to be balanced at the solution.)
• Design optimization may be used to design structures such as vehicle and aircraft parts to meet standards of strength and performance at minimal weight.
• In process control, the aim is to maximize the productivity of chemical or manufacturing processes while respecting the physical limits of the equipment and applying safety standards.
• In operations research, production schedules are designed to harness personnel, raw materials, and transportation systems in a way that optimizes productivity and profit.
• In the design of telecommunications networks, capacity should be added in a way that maximizes the expected future performance of the network while meeting budgetary limitations.
• Scheduling of flight crews in an airline company should be done in a way that covers all scheduled routes, while minimizing the number of crews required and satisfying legal and logistical requirements.

Many of these problems can be formulated as one of the standard paradigms in optimization, such as linear programming, network optimization, nonlinear programming, stochastic optimization, or complementarity or variational inequality. In their research, optimization specialists develop algorithms for these and other types of optimization problems, study their mathematical properties and practical performance, implement them in high-quality software, and apply them to practical problems, including some of the applications mentioned above.