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Optimization Methods


Credits: 6

Objectives: The goal of this course is to provide analysis of mathematical methods of optimization. Applications are mainly devoted to networks and equilibrium problems. Algorithms and optimization software are analyzed.

Description: Linear and integer optimization (20 hours). Introduction and overview of linear programming. Problems typical of embedded systems and algorithms: bin packing problem and algorithms; matching problem and algorithms; scheduling problems and algorithms; multi-commodity problem and algorithms.

Nonlinear optimization (15 hours). Unconstrained optimization problems: existence of global optima, optimality conditions. Gradient methods, Newton method, quasi-Newton methods, trust region methods. Constrained optimization problems: existence of global optima, optimality conditions. Penalization methods. Introduction to global optimization methods.

Multiobjective optimization (10 hours). Partial orders. Pareto optimal solutions. Existence of optimal solutions. First order optimality conditions. Scalarization approach. Solution methods: no-preference methods, a-posteriori methods, a-priori methods, interactive methods.

Non-cooperative game theory (10 hours). Nash equilibrium. Two-person matrix games: pure and mixed strategies, existence of equilibria, linear programming based solution method. Two-person bimatrix games: existence of Nash equilibria, solution methods. N-person games: existence of equilibria, optimality conditions, optimization reformulations, solution methods. Generalized games: the jointly convex case, existence of equilibria, optimality conditions, optimization reformulations, solution methods.

Optimization software (5 hours). Modeling languages: introduction to AMPL. Optimization solvers: CPLEX, LPSOLVE, KNITRO, MINOS, SNOPT, etc. Example of linear and nonlinear optimization problems solved by different solvers.

Assessment: Oral examination.

Prerequisites: Basic calculus. Basic linear programming.

Modules: 1

Module 1 (6 Credits): Optimization Methods

 

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