Introduction: Mathematical optimization, Convex optimization, Least-squares and linear programming, Simplex method, Two phase method, Integer linear programming, Nonlinear optimization.

Convex sets: Affine and convex sets. Some important examples. Operations that preserve convexity.

Generalized inequalities. Separating and supporting hyperplanes. Dual cones and generalized inequalities.

Convex functions: Basic properties and examples. Operations that preserve convexity. The conjugate function. Quasiconvex functions. Log-concave and log-convex functions. Convexity with respect to generalized inequalities.

Convex optimization problems. Optimization problems. Convex optimization. Linear optimization problems. Quadratic optimization problems. Geometric programming. Generalized inequality constraints. Vector optimization.

Duality: The Lagrange dual function. The Lagrange dual problem. Geometric interpretation.

Saddle-point interpretation. Optimality conditions. Perturbation and sensitivity analysis. Theorems of alternatives. Generalized inequalities.

Text Book

1. Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2009.

References

1. 1. Dimitri P. Bertsekas, Convex Optimization Theory, University Press, 2016.
   2. Hamdy A. Taha, “Operations Research-An Introduction”, Prentice Hall, 9th Edition, 2010.
   3. Edwin K.P. Chong and Stanislaw H. Zak, “An Introduction to Optimization”, Second Edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, 2004.