Global Optimization by Bound Contraction

Global Optimization by Bound Contraction -...

Info iconThis preview shows pages 1–16. Sign up to view the full content.

View Full Document Right Arrow Icon
Global Optimization by Bound Contraction
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Function to Optimize: minimize O= 4X Y Subject To: 0 X 4 0 Y 8 X Y 4 UB : -11.6 The optimum can 0.64X Y eop u ca be visually seen when O= -4X-Y is graphed for varying alues of O values of O. From the graph, the upper bound can be found to be at -11.6
Background image of page 2
Discretization of a Variable: iscreti ation of a Variable: X
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ow to deal with the issue of Non onvexity Now to deal with the issue of Non Convexity… One of the slices, from x [2,3] Y Z X
Background image of page 4
ow to deal with the issue of Non onvexity Now to deal with the issue of Non Convexity… These terms are still non linear Z 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
From the previous equations the feasible region when
Background image of page 6
xing the integer ntinuous variable nonlinearity Fixing the integer × continuous variable nonlinearity Defining: We get the following equations: W d =0 V d =0 Trivial V d =1 Trivial W d =Y
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ewritten Problem Statement Rewritten Problem Statement inimize O= X Subject To: minimize O 4X Y 0 X 4 0 Y 8 Z 4
Background image of page 8
ethod to solve for the Method to solve for the Optimized function: minimize O( 4X Y ) 0 X 4 First solve for a lower bound using our method 0 Y 8 X=3 Y=2 L.B=-14
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ow disregard the section field that was just used and solve again Now disregard the section field that was just used and solve again X=1 Y=8 L.B.=-12
Background image of page 10
•Now we compare our new found lower bounds with an upper bound that has been previously und. found. •If the second L.B is larger than the U.B, then we can disregard everything but the field that ontained LB1 contained LB1. •However, if we have a lower bound that is between, than we have two choices: •Branch and Bound on continuous domain •Cut the domain into more subunits and repeat the discretization method
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Continuing to Discretize the Variable more: X=2.5 Y=1.78 B= 1 78 X=2.25 Y=2 L.B=-11 L.B.=-11.78
Background image of page 12
Since the New L.B. 2 > U.B, than we can disregard all the rest of the fields outside the location of L.B.1
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
olution: =2 5 Solution: X=2.5 Y=1.6 L.B.=-11.6 *When the LB>UB- ε than you have found the global Optimum
Background image of page 14