This detailed handwritten summary on Optimization covers key concepts and methods from both lecture notes and tutorial notes. It includes topics such as linear optimization models, simplex methods, duality in optimization, sensitivity analysis, integer linear optimization, and dynamic programming. ...
decision variables :
( ,
kn
&n is
these variables are real-valued a
positive integer
·
objective
function :2 e Cnen ,
+ +
a linear function ofa decision variables ; C ..... In are objective constraints
& are real umbers
~
if maximised : max ( , x + Chin +
or minimised min G Chin : + +
·
constraints are :
technology : di, + + dinin Di
all technology constraints are linear combos of the decision variables
nonnegativity Ti
: - 0 Fies1, . . . . n3
>
example : Model Dovetail
Max 311 , + 2 the S t .
.
7 + 72 = 9 (1 .
1)
374 + 1118 (1 2) .
x [ 7 (1 3) .
> = 6(7 4) .
X, 12 8
S
the model is in standard ,
since there are only 2 variables : solve graphically
·
(1 3) is redundant
.
·
the purple area is the feasible region :
X X
the area within we will find our optimal solution
X
·
the points marked X are the vertices .
the intersection of 2 binding constraints
X X
the optimal point is a vertex of the feasible region
* Remarks :
constraint is binding if it holds with equality
if a constraint doesn't hold for c then it is violated ,
.
2 slack variables
slack variables are added to the constraints to transform inequalities
to equalities
~
example Model Dovetail :
x T = 9 +
= ( + +z +
xy = 9
here the slack z is added so that the sumx +
2 +
Ty =
9
· if the inequality was ,
we would add -
>s
, .
3 Lo-models
non-standard
·
5 steps to standardise :
1. min
> max : minz =
-max -
z
.
2 :
multiply both sides by -1
3 .
3
= = :
replace with ,
then apply ·
rule 2
nopositivity nonnega Subw
-Y =
&
<
-
with 1% 1, 0
,
> make sure that (( . = 0 i e
.
. either es = 0 or
< = 0
·
6 equivalent formulations
03 Standard (primal)
3
7. max[cix Ax = b
, x >
max(c Ax = b3
*
.
2 all variables free more away from origin when
.
3 max [C * Ax b, = > 03 Standard ea .
looking for optimal vertex
03 (dual)
3
min Ec Ax b , Standard
↳ * b3
·
· Min [c * Ax more toward origin when
6. minEct Ax = b ,
x 03 looking for optimal vertex
·
abs .
value in objective :
Subxi =
-: with , 0 & ci =
0
convex piecewise f(x)
<3D :3
· : =
5 1 , 212
redefine variables ,
see pg
.
19-21
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