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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
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if xep^ then (^) Ax b X 0 there^ exist some PERM P'As (^0) p'b 0 PAX (^) P'b 0 Since we^ know
Xj need (^) to (^) be
PA Xj^ 0
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
to (^) be zero C Since Xi is (^) not null variable^ from
know (^) that we can (^) not find a^ vector
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
j
j y^ i^
so he^ conclude^ that xtltpsoinj
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
that P'A (^) o (^) p o^ and^ pA co
by
tt (^) satisfies Ax 0 Contradicts^ a^ Thus (^) he (^) assumption that (^) Ibl is^ False leads^ to contradiction
p such that p'A 0 psoandpA.co het x be^ any vector that Ax o^ and
(^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
satisfies (^) p'P
pnblemiminoyst.IP IJ.gs (^1) here is no
can (^) satisfy this inequality According^ to^ the^ duality^ heorem^ if
if he
unbounded then the dual^ mustbe infeasible In this^ case^ the prinal is (^) feasible implies that a^ nonzero P
p
T a Let^ P^ 9xER^ IAxb了^ Q^ xER (^) ICxd了^ A^ b^ C^ d^ are
vectors (^) that (^) specify the constraints (^) on PandQ To decides if P^
need (^) to check for every^ x that satisfies Axeb^ must also^ satisfies Cxed An
Cxd (^) Cxd 0 Thus (^) we can set up an (^) optimization problem (^) that maximize^ the Cxd max Cix^ di St (^) Ax b
if
cix (^) di is (^) greater than zero for any i (^) then there exists^ a^
in (^) Ptt violate Cx d.co^ Cxed^ which^ implies that (^) P (^) Q if the optimal value for all^ such^
PEQ b Let P 9
Oi 0 喜⼊i (^1 ) extremepoint i extreme rays Q (^) 倍Migi (^) ⾔Ujz
extremepoints extreme rays To decide if PEQ we^ need (^) to check if
xiofp can (^) beexpressed^ as a combination^ of yi and (^) zi
xi 蒸
Ujz with^ Miso (^) Vj 0 ⾔^ Mit (^) bysolving (^) this ⼭ (^) for each^ extreme point
such combinationdo (^) not exist PIEQ^ We also^ need^ to^ check^ if each (^) wi
(^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^
satisfied (^) PEQ if any^ or fail P4Q
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
problem where^ b^ is^ the^ right
b to^ 7b where^770 the^ primal problem
Ax b^ Do^ Ax ⼊b^0
wax (^) by wax^ ⼊ by Aig
A'y c If y^ is an (^) optimal solution^ to^ the^ duel^ then ily ⽜ would
valve for deal^ wit ⼊b By strong^ duality^ Flb^ bye^ ⼊D'y ⼊FIb
