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ZEIT4700 – S1, 2014

ZEIT4700 – S1, 2014. Mathematical Modeling and Optimization. School of Engineering and Information Technology. Optimization - basics. Maximization or minimization of given objective function(s), possibly subject to constraints, in a given search space.

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ZEIT4700 – S1, 2014

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  1. ZEIT4700 – S1, 2014 Mathematical Modeling and Optimization School of Engineering and Information Technology

  2. Optimization - basics Maximization or minimization of given objective function(s), possibly subject to constraints, in a given search space Minimize f1(x), . . . , fk(x) (objectives) Subject to gj(x) < 0, i = 1, . . . ,m (inequality constraints) hj(x) = 0, j = 1, . . . , p (equality constraints) Xmin1 ≤ x1 ≤ Xmax1 (variable / search space) Xmin2 ≤ x2 ≤ Xmax2 (discrete/continuous/mixed) . .

  3. Optimization - basics Maximization or minimization of an objective function, possibly subject to constraints Constraint2 (active) F(x) Local minimum Global Minimum (unconstrained) Constraint 1 Global Minimum (constrained) x

  4. Optimization - basics Linear / Non-linear / “Black-box” f2 x2 x1 f1 Variable space Objective space

  5. Some considerations while formulating the problem Objective function(s) -- Should be conflicting if more than 1 (else one or more of them may become redundant). Variables – Choose as few as possible that could completely define the problem. Constraints – do not over-constrain the problem. Avoid equality constraints where you can (consider variable substitution / tolerance limits). f2 f1

  6. Example Design a cylindrical can with minimum surface area, which can hold at least 300cc liquid.

  7. Classical optimization techniques Region elimination (one variable) Gradient based Linear Programming Quadratic programming Simplex Drawbacks Assumptions on continuity/ derivability Limitation on variables In general find Local optimum only Constraint handling Multiple objectives Newton’s Method (Image source : http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif) Nelder Mead simplex method (Image source : http://upload.wikimedia.org/wikipedia/commons/9/96/Nelder_Mead2.gif)

  8. Optimization – types / classification Single-objective / multi-objective Unimodal / multi-modal Single / multi - variable Discrete / continuous / mixed variables Constrained / unconstrained Deterministic / Robust Single / multi-disciplinary

  9. Optimization - methods Classical • Region elimination (one variable) • Gradient based • Linear Programming • Quadratic programming • Simplex Heuristic / metaheuristics • Evolutionary Algorithms • Simulated Annealing • Ant Colony Optimization • Particle Swarm Optimization . .

  10. Resources Course material and suggested reading can be accessed at http://seit.unsw.adfa.edu.au/research/sites/mdo/Hemant/design-2.htm

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