Optimization Methods in Management Science

Fall 2025 | Université de Lausanne

Graduate Level | Teaching Assistant

Course Description

Graduate course covering linear and non-linear optimization, combinatorial methods, and applications in finance.

Resources

• Transshipment Problem: Quick Guide

Course Outline

I. Linear Programming

• Forms: Canonical, standard; transformations between forms
• Tableaus: Bases, basic solutions, feasibility, pivoting
• Simplex Algorithm: Phase I (feasibility), Phase II (optimization), degeneracy, Bland’s rule
• Duality: Weak/strong duality, complementary slackness, dual simplex

II. Graph Theory & Networks

• Fundamentals: Directed/undirected graphs, connectivity, adjacency/incidence matrices
• Shortest Path: Bellman’s principle, Dijkstra’s algorithm
• Transshipment Problem: Network simplex, spanning tree solutions, pricing, ratio test

III. Combinatorial Optimization

• Branch and Bound: Enumeration trees, relaxation, branching, bounding, pruning
• Dynamic Programming: Bellman’s principle, backward induction, applications (knapsack, inventory, shortest path with negative weights)

IV. Non-Linear Optimization

• Unconstrained: Local/global optima, first/second order conditions, convexity, Hessian analysis
• Constrained: Lagrange multipliers, KKT conditions, Slater’s condition, Lagrangian duality

V. Iterative Algorithms

• Descent Methods: Steepest descent, Wolfe conditions, backtracking linesearch
• Newton’s Method: Quadratic approximation, Hessian inversion, Cholesky decomposition
• Quasi-Newton (BFGS): Hessian approximation, Sherman-Morrison formula

VI. Applications

• Portfolio Theory: Mean-variance analysis, diversification, efficient frontier, tangency portfolio
• CAPM: Market portfolio, Security Market Line, beta
• Support Vector Machines: Separating hyperplanes, margin maximization, dual formulation, kernel trick