Optimization and Decision Models
PART A: Mathematical Programming
Art of Modeling: more than just mathematics. Introduction to Linear Programming. Linear Programming Applications (case studies in marketing, financial, business and management, etc.). The Simplex Method. Sensitivity Analysis. Duality and Post-Optimal Analysis. Other Algorithms for Linear Programming. Transportation Model and Its Variants (transshipment problem, assignment problem). Network Optimization Models (the shortest-path problem, the minimum spanning tree problem, the maximum flow problem, the minimum cost flow problem, project management with PERT/CPM, the network Simplex method). Goal Programming, Data Envelopment Analysis. Integer Linear Programming (types of Integer Linear Programming Models, modeling flexibility provided by 0-1 integer variables, the Branch-and-Bound Technique, the Cutting-Plane algorithm). Deterministic Dynamic Programming (recursive nature of Dynamic Programming computations, the shortest-path problem, Knapsack model, Equipment Replacement model, Inventory models, Workforce Size model, Traveling Salesman problem). Inventory Theory (static Economic-Order-Quantity models). Decision Analysis and Games (Utility Theory, Nash equilibrium, Cooperative Games, the bargaining set and related concepts, Algorithmic and Evolutionary Game Theory).
PART B: Numerical methods for non-linear unconstrained optimization
The problem of non-linear optimization: mathematical formulation, method categories, local and global optimum, mathematical background. No free lunch theorems for optimization. Conditions for existence of a minimum point. Iterative process, termination criteria. Line Search Methods. Step length determination strategies (exact and inexact). Inexact linear search strategies: Armijo, curvature, Wolfe, Strong Wolfe and Goldstein conditions. Backtracking line search. Methods: Steepest Descent, Newton, Line search Newton, Conjugate Gradient, Quasi Newton. Applications.