Matlab Optimization Toolbox
By YANG, Haiqin
Outline
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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/