ZEIT4700 – S1, 2015 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 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).

Optimization - basics Maximization or minimization of an objective function, possibly subject to constraints x F(x) Local minimum Global Minimum (unconstrained) Constraint 2 (active) Constraint 1 Global Minimum (constrained)

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

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

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

Classical optimization techniques  Region elimination (one variable)  Gradient based  Linear Programming  Quadratic programming  Simplex Drawbacks 1.Assumptions on continuity/ derivability 2.Limitation on variables 3.In general find Local optimum only 4.Constraint handling 5.Multiple objectives Newton’s Method (Image source : Nelder Mead simplex method (Image source :

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

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.

Project 1 Nature optimizes both living and nonliving objects. Identify an object that has been optimized; Develop the mathematical formulation of what has been minimized/maximised and present results to justify why it has taken the form. (Due April 09, 2015)

Resources Course material and suggested reading can be accessed at Hemant/design-2.html Hemant/design-2.html