ZEIT4700 – S1, 2015Mathematical 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

minimumGlobal

Minimum

(unconstrained)

Constraint 2 (active)

Constraint 1Global

Minimum

(constrained)

Optimization - basics

x1

x2

f1

f2

Variable space Objective 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 : 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)

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 http://seit.unsw.adfa.edu.au/research/sites/mdo/Hemant/design-2.html