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Unit of study_

MATH4071: Convex Analysis and Optimal Control

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimisation problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straightforward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

Code MATH4071
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites:
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[A mark of 65 or above in 12cp of (MATH2XXX or STAT2XXX or DATA2X02)] or [12cp of (MATH3XXX or STAT3XXX)]
Corequisites:
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None
Prohibitions:
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MATH3971
Assumed knowledge:
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MATH2X21 and MATH2X23 and STAT2X11

At the completion of this unit, you should be able to:

  • LO1. analyse static optimisation problems with constraints
  • LO2. formulate deterministic and stochastic dynamic optimisation problems, that arise in scientific and engineering applications, as mathematical problems
  • LO3. understand the importance of convexity for optimisation problems, and use convexity to determine whether a solution to a given problem exists and is unique
  • LO4. check if a certain controlled system is controllable, observable, stabilisable
  • LO5. apply the Maximum Principle in order to solve real world control problems
  • LO6. formulate the Hamilton-Jacobi-Bellman equation for solution of dynamic optimisation problems and solve them in special cases
  • LO7. explain the derivations of key theoretical results and discuss the role of mathematical assumptions in these derivations
  • LO8. use game theory to formulate optimisation problems with many competing payers
  • LO9. identify important solvable classes of optimisation problems arising in finance, economics, engineering and insurance and provide solutions.

Unit outlines

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