Department of Systems Engineering & Engineering Management

Dynamic continuous-time systems. Examples, modeling, and classification of optimal control problems. Pontryagin's maximum principle: adjoint equation, Hamiltonian system, and sufficient condition of optimality. Bellman's dynamic programming: principle of optimality, Hamilton-Jacobi-Bellman equation, and verification theorem. Linear quadratic control: Riccati equation and linear matrix inequality. Introduction to numerical methods of solving optimal control problems.

Principles of discrete event simulation. Random number generators. Simulation model validation. Input and output analysis. Optimization via simulation. Variance reduction techniques. Introduction of simulation packages and applications to finance, logistics and service systems.

The course covers the underlying theory and fundamental solution methodologies of mathematical programming: linear programming, unconstrained and constrained non-linear optimization. Topics include optimality conditions, search methods, descent methods, Lagrange multipliers, penalty functions. Developments of duality theory are presented. Concepts and issues in global optimization and multi-objective optimization are introduced. Applications are drawn from engineering and financial optimization.

The first part of this course covers underlying theory and fundamental solution methodologies of integer programming: optimality, relaxation, and bounds, complexity and problem reductions, branch and bound, cutting plane algorithms, strong valid inequalities and duality theory. The second part of this course covers some of the recent developments in mathematical programming: Interior point methodology, conic optimization and semidefinite programming. Various applications in engineering, management, and financial economics and discussed.

Poisson process. Birth-and-death process, Markov chain. Martingale. Brownian motion. Renewal and stationary processes. Stochastic integration and Ito's formula. Applications to queueing models, inventory models, and financial investment/hedging models.

The course discusses underlying theory and fundamental solution methodologies for linear and nonlinear integer programming. Theoretical topics include general solution concepts such as relaxation, partition and bounds, submodularity, and duality theory. Solution methods covers partial enumeration methods, dynamic programming methods, branch and bound methods, cutting plane methods, convergent Lagrangian dual methods, convexification methods and global descent methods. These methods can be respectively applied to solve separable/non-separable and convex/non-convex integer programming problems, including nonlinear knapsack problems, quadratic integer programming, and zero-one polynomial integer programming. The course also discusses various applications of integer programming in engineering, management and finance.