1© 2014 The MathWorks, Inc.

Best Practices for Optimisation

Dr Jasmina Lazić

Application Engineer, MathWorks Inc.

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Agenda

What’s new?

Choose your solver

How to set up optimisation problems

Cutting-edge solvers in action: demonstrating the new

algorithms

Accelerate your optimisation by:

– Using parallel computing

– Tuning solver options

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Demo: Global Optimization

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What’s New?

Particle Swarm Optimization (particleswarm) R2014b

Dual-simplex algorithm in linear programming solver (linprog) R2014b

Interior-point algorithm in linear least-squares solver (lsqlin) R2014b

Mixed Integer Linear Programming solver (intlinprog) R2014a

– Plot functions and output functions for monitoring progress of intlinprog solver R2014b

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Optimization Problems

• Managing Portfolios

• Supply Chain

• Engine Calibration

• Power Generation

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Anatomy of an Optimization Problem

General Form for MATLAB Solvers

Objective Function

uxl

eqbxeqA

bAx

)(min xfx

Typically a linear or nonlinear function

Linear constraints

• inequalities

• equalities

• bounds

Nonlinear constraints

• inequalities

• equalities

Decision variables

0)(

0)(

xeq

c

xc

Subject to Constraints

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OptimizationFind optimal solutions, analyze tradeoffs

• General (nonlinear)

• Linear

• Quadratic

• Least-squares

• Multiobjective

min𝑥

fcn(𝑥)

min𝑥

𝑓′𝑥

min𝑥

1

2𝑥′𝐻𝑥 + 𝑐′𝑥

min𝑥

𝐹(𝑥) 22

min𝑥

[𝐹1 𝑥 , 𝐹2 𝑥 ,… ]

Objective Types

• Unconstrained

• General (nonlinear)

• Bound

• Linear

• Discrete (integer)

Constraint Types

𝐿𝐵 ≤ 𝑥 ≤ 𝑈𝐵

𝐴𝑥 ≤ 𝑏, 𝐴𝑒𝑞𝑥 = 𝑏𝑒𝑞

𝑐(𝑥) ≤ 0, 𝑐𝑒𝑞 (𝑥) ≤ 0

𝑥𝑖 ∈ ℤ

Global Algorithms

• Multistart

• Direct Search

• Particle Swarm

• Genetic Algorithm

• Interior Point

• Trust Region

• Simplex

• Levenberg Marquardt

• Sequential Quadratic Programming

Gradient-based Algorithms

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Item Nuts Bolts

Gadget 5 2 3.00$

Widget 3 8 10.00$

Requirements Revenue

per Item

How to Set Up an Optimisation Problem: Modelling Example

Overview

Challenge

– Current inventory 28 nuts / 30 bolts

– How many gadgets / widgets to make to maximize revenue?

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How to Set Up an Optimisation Problem

Variables:

𝑥 = number of gadgets

𝑦 = number of widgets

Objective - maximise revenue:

max𝑥,𝑦

3𝑥 + 10𝑦

Subject to constraints:

5𝑥 + 3𝑦 ≤ 28 (nuts total)

2𝑥 + 8𝑦 ≤ 30 (bolts total)

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Demo: Barge Container Ship Routing

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Shipping Company Objective

Profit RevenueTransport

Costs

Empty container handling

Fuel Costs

Other Costs…Service of

loaded containers

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Ship Routing: Problem Characteristics

Decision Variables Given Constants

𝑛: number of ports

𝑙: distance between ending ports

𝑣1, 𝑣2: upstream/downstream speed

𝑠𝑐𝑓, 𝑠𝑐𝑙: fuel/lubricant consumption

𝐶: ship carrying capacity

𝑑𝑐𝑐: daily time charter cost

𝑝𝑒𝑐𝑖: entry cost at port 𝑖

𝑡𝑚𝑖𝑛, 𝑡𝑚𝑎𝑥: min/max turnaround time

𝑧𝑟𝑖𝑗: expected number of loaded

containers between ports 𝑖 and 𝑗

𝑟𝑖𝑗: freight rate per container from port 𝑖

to port 𝑗

𝑢𝑓𝑡𝑖 , 𝑙𝑓𝑡𝑖: avg. loading/unloading time

per container at port 𝑖

…

𝑥𝑖𝑗 = 1, if ports 𝑖 and 𝑗 are directly

connected in the route0, otherwise

𝑤𝑖𝑗 = the number of empty containers

transported from port 𝑖 to port 𝑗

𝑧𝑖𝑗 = the number of loaded containers

transported from port 𝑖 to port 𝑗

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Ship Routing: Key Constraints

The ship must visit the ending

ports

Can transport from 𝑖 to 𝑗 only if

both ports are on the route

Cannot transport more than

the expected number of

containers

Total number of containers on-

board cannot exceed ship

carrying capacity

…

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Problem Characteristics and Solution Times

#Ports

𝒏#Variables

𝟑𝒏𝟐 + 𝟓𝒏 + 𝟗#Constraints

𝟑𝒏𝟐 + 𝟗𝒏 + 𝟖

10 359 398

15 759 722

20 1309 1388

Minimum

solution time

Maximum

solution time

1 sec 6 sec

3 min 23 min

> 5 h > 5 h

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How Can Parallel Computing Speed

Optimisation Up?

Computing the gradient numerically requires the objective

function and non-linear constraints to be evaluated at

N + 1 points each iteration (N = number of dimensions)

Compute function evaluations in parallel by setting

‘UseParallel’ option to be ‘always’

2-Dimensional Problem (peaks)

x

xfxxfxf

)()()( forward

difference

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Demo: Running an Optimization Algorithm

in Parallel

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Summary

Optimisation workflow in MATLAB:

Optimise, post-process and visualise results on the same platform

What is new in R2014a and R2014b:

– Mixed Integer Linear Programming

– Particle Swarm

Boost the solver

performance

Choose the right solver

Set your problem up

NEW !!

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For more information…

Documentation:

Optimization Toolbox

Webinar: Getting Started

Using Optimization with

MATLAB

Video: Mathematical

Modeling with

Optimization, Part 1

Video: Mathematical

Modeling with

Optimization, Part 2

MATLAB Digest: Using Symbolic

Gradients for Optimization

MATLAB Digest: Improving

Optimization Performance with

Parallel Computing