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Linear Programming: Feasibility and Optimality, Exercises of Linear Algebra

University of Toronto Linear Algebra

Key concepts in linear programming, focusing on feasibility and optimality. It delves into the relationship between primal and dual problems, demonstrating how to determine if a solution is feasible and optimal. The document also provides examples and explanations of how to solve linear programming problems using the simplex method.

Typology: Exercises

2023/2024

Uploaded on 11/27/2024

zhiqi-fu 🇨🇦

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PA jso

aif xep then Ax bX0there exist some

PERM P'As0p'b 0PAX P'b 0

Since we know that P'A 70 so Xj need to be

0to hare PA Xj 0

bsince xj oforallxep.to make sure that Ax b

still valid according to Farka'slemma thereexists avec to

psueh that P'A 0P'b 0and pAIpo Imeans

that here is aredundancy in te constraints of Pthat

force xj to be zero

CSince Xi is not null variable from part bhe can

know that we can not find avector pwith pA 0pino

and IP'Aj70 therefore for any pwith PAs0P'bo

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O

PA j so

a

if xep^ then (^) Ax b X 0 there^ exist some PERM P'As (^0) p'b 0 PAX (^) P'b 0 Since we^ know

that P'A 70 so

Xj need (^) to (^) be

0 to hare^

PA Xj^ 0

b since

xj oforallxep.to^ make (^) sure (^) that Ax b still valid^ according^ to Farka'slemma^ there^ exists^ avec^ to psueh that P'A (^0) P'b 0 and^ pAIpo I means that here^ is a^ redundancy (^) in te constraints^ of P^ that

force

xj

to (^) be zero C Since Xi is (^) not null variable^ from

part

b he can

know (^) that we can (^) not find a^ vector

p

with

pA 0 pino and (^) I P'A therefore^ for with^ (^0) P'bo

ngl we need^ to have^ pA i (^0) Let x yep where yio so (^) xj y (^70) now consider^ xt A'p.in ith component he have^ A'p i

x

A'p

j y^ i^

t o^ yi so^ sine

yi

so he^ conclude^ that xtltpsoinj

thomponent.ca

lbI Assume^ cal is True every vector (^) x that (^) satisfies Ax and (^) xs.no (^) must hate (^) xi ⼆^0 and (^) suppose b (^) is false which (^) Meansthat there is (^) no

p

that P'A (^) o (^) p o^ and^ pA co

ten

Frka's lemma^ there exist some wihx.no

tt (^) satisfies Ax 0 Contradicts^ a^ Thus (^) he (^) assumption that (^) Ibl is^ False leads^ to contradiction

b1 a Assume^ D is True here exists a vector

p such that p'A 0 psoandpA.co het x be^ any vector that Ax o^ and

x

(^0) according assumption P'Ax would be (^0) PAX P'AiXi^ 章 P'Aj^ Xj since PA.co x so (^) then (^) P'Aix would^ be negative.he

ptt

satisfies (^) p'P

p

he dual^

of

the

pnblemiminoyst.IP IJ.gs (^1) here is no

y

can (^) satisfy this inequality According^ to^ the^ duality^ heorem^ if

he primal is

feasible and^ bounded^ the dual must also be feasible

if he

prinal is^ feasible^ and^

unbounded then the dual^ mustbe infeasible In this^ case^ the prinal is (^) feasible implies that a^ nonzero P

exists that satisfies

p

P I^ o^ which^ means^

TP

p

T a Let^ P^ 9xER^ IAxb了^ Q^ xER (^) ICxd了^ A^ b^ C^ d^ are

givenmatrix^

vectors (^) that (^) specify the constraints (^) on PandQ To decides if P^

Q we

need (^) to check for every^ x that satisfies Axeb^ must also^ satisfies Cxed An

algorithm to do this is to construct^ a feasibility problem

Cxd (^) Cxd 0 Thus (^) we can set up an (^) optimization problem (^) that maximize^ the Cxd max Cix^ di St (^) Ax b

the optimal^ value^

of

cix (^) di is (^) greater than zero for any i (^) then there exists^ a^

point

in (^) Ptt violate Cx d.co^ Cxed^ which^ implies that (^) P (^) Q if the optimal value for all^ such^

linear programs are o ten

PEQ b Let P 9

⾔nixi Ojwi Ti 0

Oi 0 喜⼊i (^1 ) extremepoint i extreme rays Q (^) 倍Migi (^) ⾔Ujz

J

Mizo Ui 0 意Mi 1

extremepoints extreme rays To decide if PEQ we^ need (^) to check if

each

xiofp can (^) beexpressed^ as a combination^ of yi and (^) zi

of Q

xi 蒸

Miy

it

Ujz with^ Miso (^) Vj 0 ⾔^ Mit (^) bysolving (^) this ⼭ (^) for each^ extreme point

in P

if

such combinationdo (^) not exist PIEQ^ We also^ need^ to^ check^ if each (^) wi

of

P

can be expressed^ as a^ combination^

of

(^25) of Q^ meaning that wI^ must^ lie witin the^ cone^ generated^ by the^ extreme^ rays of Q W (^) Vjzi with (^) Ui 0 again this^ can be^ checked by solving his^ Lp (^) for extreme^

ray ofP^ if

both and^ are

satisfied (^) PEQ if any^ or fail P4Q

which 12

38201 0 and^ siti 0 C optimal cost^ CisB181^ b 15.11.38 11 品 ssi 8 In section (^) 5.2 FIb^ is defined^ as^ the^ optimal cost of

LP

problem where^ b^ is^ the^ right

hand side

veetor.by scaling

b to^ 7b where^770 the^ primal problem

min C'x^ changes^ to^ Min C'x

Ax b^ Do^ Ax ⼊b^0

Then dual

wax (^) by wax^ ⼊ by Aig

c

A'y c If y^ is an (^) optimal solution^ to^ the^ duel^ then ily ⽜ would

yield the^ optimal^ objective^

valve for deal^ wit ⼊b By strong^ duality^ Flb^ bye^ ⼊D'y ⼊FIb