Operations Research, Extended Course

Aim

Optimization deals with mathematical theory and methods aiming at analyzing and solving decision problems that arise in technology, economy, medicine, etc. The course gives a broad orientation of the field of optimization, with emphasis on basic theory and methods for discrete optimization problems in finite dimension, and it also gives some insight into its use for analyzing practical optimization problems. After the course, the student shall:

• be able to explain important classes of optimization problems and to be able to classify them according to their properties, into, for example, linear and discrete problems
• be able to model mathematical models of simple optimization problems
• be able to explain basic concepts, such as, for example, optimal conditions, valid inequalities, weak and strong duality
• have knowledge about and be able to apply basic theory for some common types of optimization problems, such as, for example, duality theory for linear optimization problems, and have knowledge about and be able to use optimality conditions, such as, for example, Bellman-conditions, to determine the optimality of a given solution
• be able to explain and to apply basic principles for solving some common types of optimization problems, such as, for example, the branch-and-bound method for discrete problems
• be able to use relaxations, and especially Lagrangian duality, to approximate optimization problems, and be able to estimate the optimal objective value through lower and upper bounds
• be able to use commonly available software for solving optimization problems of standard type
• have some knowledge of practical applications of optimization

Content

• Network optimization: Shortest path problems, maximum flow problems, minimum cost network flow problems, the network simplex method, integer problems with graph structure.
• Integer programming: Model formulation, branch-and-bound methods, cutting plane methods, applications to special structured integer problems.
• Dynamic programming: Formulation of deterministic problems, the principle of optimality, applications to network, inventory and resource allocation problems