Matlab Optimization Toolbox By YANG, Haiqin Outline • • • • Mathematical Programming Linear Programming Quadratic Programming Unconstrained Non-linear Programming Mathematical Programming • Framework • Linear Programming • Quadratic Programming • Applications • Applications – Network flow – Transporation – Least square approximation – Portfolio optimization Mathematical Programming • Non-linear Programming • Semi-indefinite Programming Mathematical Programming • Multiple-Objective Optimization • Applications – Financial applications: maximize return, minimize risk – etc. Minimization Algorithm Minimization Algorithm (Cont.) Equation Solving Algorithms Least-Squares Algorithms LP • Formulation • Calling Implementation of LP • Algorithms – Simplex: medium-scale – Active set: medium-scale – Primal-dual interior point: large-scale • LP standard form – Primal − Dual Optimal Condition • Karush-Kuhn-Tucker (KKT) condition Complemetarity conditions Interior Point for LP • Central path Algorithm QP • Formulation • Calling Implementation of QP • Algorithms – Medium-scale: active set – Large-scale: an interior reflective Newton method coupled with a trust region method. Active Set Method • Transformation • Optimal condition Unconstrained Non-linear Programming • Formulation • Calling Implementation Ideas • Formulation • A general descent algorithm Quasi-Newton Algorithm • Idea • The BFGS - update (Broyde-Fletcher-Goldfarb-Shanon) • The inverse DFP - update (Davidon-Fletcher-Powell) Line Search • Find step-length • Wolfe conditions • Goldstein conditions Trust Region Method • Formulation • Algorithm Other Toolboxes • CVX – http://cvxr.com/cvx/ • Mosek – http://www.mosek.com/ • Yalmip – http://users.isy.liu.se/johanl/yalmip/