17. Intro to nonconvex models Overview Discrete models - - PowerPoint PPT Presentation

CS/ECE/ISyE 524 Introduction to Optimization Spring 2017–18

• 17. Intro to nonconvex models

❼ Overview ❼ Discrete models ❼ Mixed-integer programming ❼ Examples

Laurent Lessard (www.laurentlessard.com)

Convex programs

❼ We saw: LP, QP, QCQP, SOCP, SDP ❼ Can be efficiently solved ❼ Optimal cost can be bounded above and below ❼ Local optimum is global

• 1

1 2 3 4 x 1 2 3 4 f(x) 17-2

Nonconvex programs

❼ In general, cannot be efficiently solved ❼ Cost cannot be bounded easily ❼ Usually we can only guarantee local optimality ❼ Difficulty depends strongly on the instance

• 1

1 2 3 4 x 1 2 3 4 f(x) 1 2 3 4 5 x 1 2 3 4 f(x) 17-3

Outline of the remainder of the course

❼ Integer (linear) programs

◮ it’s an LP where some or all variables are discrete

(boolean, integer, or general discrete-valued)

◮ If all variables are integers, it’s called IP or ILP ◮ If variables are mixed, it’s called MIP or MILP

❼ Nonconvex nonlinear programs

◮ If continuous, it’s called NLP ◮ If discrete, it’s called MINLP

❼ Approximation and relaxation

◮ Can we solve solve a convex problem instead? ◮ If not, can we approximate?

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Discrete variables

Why are discrete variables sometimes necessary?

• 1. A decision variable is fundamentally discrete

❼ Whether a particular power plant is used or not {0, 1} ❼ Number of automobiles produced {0, 1, 2, ...} ❼ Dollar bill amount {$1, $5, $10, $20, $50, $100}

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Discrete variables

Why are discrete variables sometimes necessary?

• 2. Used to represent a logic constraint algebraically.

❼ “At most two of the three machines can run at once.” z1 + z2 + z3 ≤ 2 (zi is 1 if machine i is running) ❼ “If machine 1 is running, so is machine 2.” z1 ≤ z2 ❼ Goal: (logic constraint) ⇐ ⇒ (LP with extra boolean variables)

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Return to Top Brass

250 500 750 1,000 250 500 750 1,000 1,250 1,500

(650, 1100)

footballs (f ) soccer balls (s) max

f , s

12f + 9s s.t. 4f + 2s ≤ 4800 f + s ≤ 1750 0 ≤ f ≤ 1000 0 ≤ s ≤ 1500

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Return to Top Brass

250 500 750 1,000 250 500 750 1,000 1,250 1,500

(650, 1100)

footballs (f ) soccer balls (s) max

f , s

12f + 9s s.t. 4f + 2s ≤ 4800 f + s ≤ 1750 0 ≤ f ≤ 1000 0 ≤ s ≤ 1500 f and s are multiples of 50

Same solution!

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Return to Top Brass

250 500 750 1,000 250 500 750 1,000 1,250 1,500

(800, 800)

footballs (f ) soccer balls (s) max

f , s

12f + 9s s.t. 4f + 2s ≤ 4800 f + s ≤ 1750 0 ≤ f ≤ 1000 0 ≤ s ≤ 1500 f and s are multiples of 200

Boundary solution!

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Return to Top Brass

250 500 750 1,000 250 500 750 1,000 1,250 1,500

(675, 900)

footballs (f ) soccer balls (s) max

f , s

12f + 9s s.t. 4f + 2s ≤ 4800 f + s ≤ 1750 0 ≤ f ≤ 1000 0 ≤ s ≤ 1500 f and s are multiples of 225

Interior solution!

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Mixed-integer programs

maximize

x

cTx subject to: Ax ≤ b x ≥ 0 xi ∈ Si where Si can be:

❼ The real numbers, R ❼ The integers, Z ❼ Boolean, {0, 1} ❼ A discrete set, {v1, v2, . . . , vk}

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Mixed-integer programs

maximize

x

cTx subject to: Ax ≤ b x ≥ 0 xi ∈ Si The solution can be

❼ Same as the LP version ❼ On a boundary ❼ In the interior (but not too far)

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Common examples

❼ Facility location

◮ locating warehouses, services, etc.

❼ Scheduling/sequencing

◮ scheduling airline crews

❼ Multicommodity flows

◮ transporting many different goods across a network

❼ Traveling salesman problems

◮ routing deliveries

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Knapsack problem

My knapsack holds at most 15 kg. I have the following items: item number 1 2 3 4 5 weight 12 kg 2 kg 4 kg 1 kg 1 kg value ✩4 ✩2 ✩10 ✩2 ✩1 How can I maximize the value of the items in my knapsack? Let zi =

• 1

knapsack contains item i

• therwise

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Knapsack problem

My knapsack holds at most 15 kg. I have the following items: item number 1 2 3 4 5 weight 12 kg 2 kg 4 kg 1 kg 1 kg value ✩4 ✩2 ✩10 ✩2 ✩1 How can I maximize the value of the items in my knapsack? maximize

z

4z1 + 2z2 + 10z3 + 2z4 + z5 subject to: 12z1 + 2z2 + 4z3 + z4 + z5 ≤ 15 zi ∈ {0, 1} for all i notebook: Knapsack.ipynb

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General (0,1) knapsack

❼ weights w1, . . . , wn and limit W . ❼ values v1, . . . , vn ❼ decision variables z1, . . . , zn

maximize

z n

• i=1

vizi subject to:

n

• i=1

wizi ≤ W zi ∈ {0, 1} for i = 1, . . . , n

17-16