A First Course in Optimization

1. Introduction: Basic definition, problem formulation and illustrative examples. Existence of minimum, Weierstrass's Theorem, Necessary conditions for unconstrained minimization, Sufficiency conditions for unconstrained minimization. 2. Convex Analysis: convex sets, closest point theorem, Theorem of alternatives, Farka's lemma, Gordan's theorem, convex functions, minima and maxima of convex functions, generalizations. 3. Linear programming: Motivation, formulation, optimality conditions, simplex method, dual formulation and optimality conditions. 4. Constrained minimization: role of constraints, linear and non-linear constraints, equality and inequality constraints, optimality conditions, Fritz John optimality conditions, Karush Kuhn Tucker optimality conditions, necessary conditions and sufficiency conditions. 5. Application of optimization theory: Application to networks and economics. 6. Quadratic programming: formulation, optimality conditions and algorithms, applications. 7. Algorithms for unconstrained minimization: Univariate Line search methods, Multidimensional search methods, steepest descent, Newton's, Quasi newton and trust region approaches, method of conjugate directions, CG method.