Linear programming maximize x 1 + x 2 x 1 + 3x 2 3 3x 1 + x 2 5 x 1 0 x 2 0.

Linear programming

maximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

Linear programmingmaximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

x1

x2

Linear programmingmaximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

x1

x2

Linear programmingmaximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

x1

x2

feasible solutions

Linear programmingmaximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

x1

x2

optimal solution x1=1/2, x2=3/2

Can you prove it is optimal ?maximize x1 + x2

x1 + 3x2 33x1 + x2 5x1 0x2 0

x1

x2

optimal solution x1=1/2, x2=3/2

Can you prove it is optimal ?maximize x1 + x2

x1 + 3x2 33x1 + x2 5

4x1 + 4x2 8

x1

x2

optimal solution x1=1/2, x2=3/2

Can you prove it is optimal ?maximize x1 + x2

x1 + 3x2 33x1 + x2 5

x1+x2 2

x1

x2

optimal solution x1=1/2, x2=3/2

Another linear program

maximize x1 + x2

x1 + 2x2 34x1 + x2 5x1 0x2 0

Another linear program

maximize x1 + x2

x1 + 2x2 34x1 + x2 5x1 0x2 0x1=1, x2=1, optimal ?

Another linear program

maximize x1 + x2

x1 + 2x2 3 *34x1 + x2 5 *1x1 0x2 0

x1=1, x2=1, optimal !

7x1 + 7x2 14

Systematic search for the proof of optimality

maximize x1 + x2

x1 + 2x2 3 * y1

4x1 + x2 5 * y2

x1 0x2 0

Systematic search for the proof of optimality

maximize x1 + x2

x1 + 2x2 3 * y1

4x1 + x2 5 * y2

x1 0x2 0

y1 0y2 0

Systematic search for the proof of optimality

maximize x1 + x2

x1 + 2x2 3 * y1

4x1 + x2 5 * y2

x1 0x2 0

y1 0y2 0

min 3y1+5y2

y1 + 4y2 12y1+y2 1

Systematic search for the proof of optimality

max x1+x2

x1 + 2x2 3 4x1 + x2 5x1 0x2 0

y1 0y2 0

min 3y1+5y2

y1 + 4y2 12y1+y2 1

dual linear programs

Systematic search for the proof of optimality

max x1+x2

x1 + 2x2 3 4x1 + x2 5x1 0x2 0

y1 0y2 0

min 3y1+5y2

y1 + 4y2 12y1+y2 1

dual linear programs

Linear programming duality

max x1+x2

x1 + 2x2 3 4x1 + x2 5x1 0x2 0

y1 0y2 0

min 3y1+5y2

y1 + 4y2 12y1+y2 1

Linear programs

variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn

linear constraint: equality

a1x1 + a2x2 + ... + anxn = b inequality

a1x1 + a2x2 + ... + anxn b

Linear programs variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn

linear constraint: equality a1x1 + a2x2 + ... + anxn = b inequality a1x1 + a2x2 + ... + anxn b

max/min of a linear functionsubject to collection of linear constraints

Linear programs variables: x1,x2,...,xn linear function: a1x1 + a2x2 + ... + anxn

linear constraint: equality a1x1 + a2x2 + ... + anxn = b inequality a1x1 + a2x2 + ... + anxn b

max/min of a linear functionsubject to collection of linear constraints

Goal: find the optimal solution

(i.e., a feasible solution with themaximum value of the objective)

Linear programs

one of the most important modeling tools oil industry manufacturing marketing circuit design

very important in theory as well

Shortest path

s

t

5

61

3

2

4

u

v

w

Shortest path

s

t

5

61

3

2

4

u

v

wds = 0du ds + 5dv ds + 6dw du + 3dw dv + 1dt dw + 2dt dv + 4

max dt

Max-Flow

FLOW CONSERVATION

CAPACITY CONSTRAINTS

fu,v = 0vV

fu,v c(u,v)

SKEW SYMMETRY

fu,v = - fv,u

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

objective = ?

us,t:

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

max fs,vvV

us,t:

Linear programming dualitymaximize minimize

constraint variable equality unrestricted non-negative

variable constraint unrestricted equality non-negative

Linear programming duality

maximize minimize

constraint variable equality unrestricted non-negative

variable constraint unrestricted equality non-negative

max x1+x2

x1+x2+x3+x4=1x1+2x3 1x2+2x4 2x1 0x4 0

Linear programming duality

maximize minimize

constraint variable equality unrestricted non-negative

variable constraint unrestricted equality non-negative

max x1+x2

x1+x2+x3+x4=1x1+2x3 1x2+2x4 2x1 0x4 0

y1

y2 0y3 0

DONE

Linear programming duality

maximize minimize

constraint variable equality unrestricted non-negative

variable constraint unrestricted equality non-negative

max x1+x2

x1+x2+x3+x4=1x1+2x3 1x2+2x4 2x1 0x4 0

y1

y2 0y3 0

min y1 + y2 + 2 y3

DONE

DONE

Linear programming duality

maximize minimize

constraint variable equality unrestricted non-negative

variable constraint unrestricted equality non-negative

max x1+x2

x1+x2+x3+x4=1x1+2x3 1x2+2x4 2x1 0x4 0

y1

y2 0y3 0

min y1 + y2 + 2 y3

DONE

DONE

y1 + y2 1y1 + y3 = 1y1 + 2y2 = 0y1 + 2y3 0 DONE

Linear programming duality

max x1+x2

x1+x2+x3+x4=1x1+2x3 1x2+2x4 2x1 0x4 0

y2 0y3 0

min y1 + y2 + 2 y3

y1 + y2 1y1 + y3 = 1y1 + 2y2 = 0y1 + 2y3 0

a1 x1 + ... + an xn b

a1 x1 + ... + an xn b + y, y 0

a1 x1 + ... + an xn – y b, y 0

“” “=” and non-negativity

a1 x1 + ... + an xn b

a1 x1 + ... + an xn b a1 x1 + ... + an xn b

“” “”

a1 x1 + ... + an xn b -a1 x1 - ... - an xn -b

optimization feasibility

max a1x1+...+anxn

a1x1+...+anxn P

+ binary search on P

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

max fs,vvV

us,t:

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

max fs,vvV

yu

zu,v 0

w{u,v}

us,t:

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

max fs,vvV

yu

zu,v

w{u,v}

min c(u,v)zu,vu,v

us,t:

zu,v 0

Max-Flow

fu,v = 0vV

fu,v c(u,v)

fu,v + fv,u=0

max fs,vvV

yu

zu,v

w{u,v}

min c(u,v)zu,vu,v

+

+=0

us,tus,t:

zu,v 0

Max-Flow min c(u,v)zu,vu,v

us,tyu + zu,v + w{u,v} =0

zs,v + w{s,v} =1

zt,v + w{t,v} =0

zu,v 0

ys = -1

yt = 0

Max-Flow min c(u,v)zu,vu,v

yu + zu,v + w{u,v} =0

zu,v 0

ys = -1

yt = 0

Max-Flow min c(u,v)zu,vu,v

yu + zu,v + w{u,v} =0

zu,v 0

ys = -1

yt = 0

yv + zv,u + w{u,v} =0

Max-Flow min c(u,v)zu,vu,v

yu + zu,v + w{u,v} =0

zu,v 0

ys = -1

yt = 0

yv + zv,u + w{u,v} =0

yu - yv = zv,u - zu,v

Max-Flow

min c(u,v)zu,vu,v

zu,v 0

ys = -1

yt = 0yu - yv = zv,u - zu,v

Max-Flow

min c(u,v) max{0,yu-yv}u,v

zu,v 0

ys = -1

yt = 0yu - yv = zv,u - zu,v

Max-Flow

min c(u,v) max{0,yu-yv}u,v

ys = -1

yt = 0

Max-Flow = Min-Cut

min c(u,v) max{0,yu-yv}u,v

ys = -1yt = 0

min c(u,v) u S,v SCS,s S

tSC

one more trickachieves yu {-1,0}