Meljun Cortes Algorithm Iterative Improvement

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Iterative Improvement Algorithm Design Technique

Design and Analysis of Algorithms

* Property of STI

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The following are the topics to be discussed under

Iterative Improvement:

Definition of iterative improvement algorithm

design technique

Advantages of iterative improvement algorithm

design technique.

Application of simplex method.

Application of Geometric Interpretation of linear

programming


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a type of algorithm design technique for solving

optimization problems

starts with initial state, and changes it iteratively

for improvement to find the best or the least value

Advantages:

Use very little memory (usually constant amount),because it only stores the current state

Often finds reasonable solutions in large or infinite

(continuous) state


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identified as important in the 1950’s, and a

general solution called the simplex method was

developed

actually name of a branch of applied mathematics

that deals with solving optimization problems of a

particular form


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The generic interpretation of Linear Programming

It must be a maximization problem.

All constraints (except the non-negativityconstraints) must be in the form of liner

equations.

All the variables must be required to be non-

negative.

Thus, the general linear programming problem in

standard form with m constraints and n unknowns

(n ≥ m) is:


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a classic method used for solving linear

programming problems

considered as the one of the most importantalgorithms ever developed

developed by a U.S. mathematician named George

B. Danzig in 1947

generates a sequence of adjacent points of the

problem’s feasible region with improving values of

the objective function until no further

improvement is possible


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Every linear programming problem can be

represented in the following:

The possible solution to be given problem is any

point x, y) that satisfies all the constraints of the

problem’s possible region is the set of all its

feasible points.

Maximize: 3x + 5y

Subject to: x + y ≤ 4

x + 3y ≤ 6

x≥0, y≥0


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It is instructive to visualize on:

A line divides the plane into two half-planes,

whereas for all the points have ax + by < c and on

the other, ax + by > c.

Therefore, the set of points defined by inequality x

+ y ≤ 4 encompasses the points on and below the

line x + 3y = 6.

ax + by = c


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The points of the feasible region must satisfy all

the constraints of the problem.

The feasible region attained by the intersection of

these two half-planes and the first quadrant of the

Cartesian plane defined by non-negative

constraints x ≥ 0, y ≥ 0.

The feasible region for the problem is the convex

polygon with the vertices (0,0), (4,0), (0,2) and

(3,1).


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The last point, which is the point of intersection of

the lines x + y = 4 and x + 3y = 6 is achieved by

solving the system of the two linear equations.

To find the optimal solution, a point in the feasible

region with the largest value of the objective

function: z = 3x = 5y.


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Are there feasible solutions for which the value of

the objective function equals, say, 20?

The points (x,y) for which the objective function, x= 3x + 5y, is equal to 20 form the line, 3x + 5y =

20.


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There are infinitely many feasible points for which

the objective function is equal to, say, 10.

These are the intersection points of the line 3x +

5y = 20 and 3x + 5y = 10 with the feasible region.

To apply the simplex method to a linear

programming problem, it has to be represented in

a special form called the standard form.

The standard form has the following requirements:

It must have a

maximum

problem.

All the constraints

except the non-

negativity

constraints) must

be in the form of

linear equations

.

All the variables

must be required

to be non-

negative


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The general linear programming problem in

standard form with m, constraints and n

unknowns (n≥m) is:

Any linear programming problem can be

transformed into an equivalent problem in

standard form.

If an objective function needs to be minimized, it

can be replaced by the equivalent problem of

maximizing the same objective function with all its

coefficients cj replaced by –cj, j = 12….n.

Maximize: c1x1 + ….+cnxn

Subject to:

ai1x1 + …. + ainxn = bi for i=1,2,….m

x1≥0, ……xn≥0.


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Example: Two inequalities of the given problem

can be transformed, respectively, into the

following equations:

If this is not the case in an initial statement of a

problem, an unconstrained variables x j can be

replaced by the difference between two new

nonnegative variables.

Thus, the given problem in standard form is the

following linear programming problem in fourvariables:

x + y + u = 4 where u ≥ 0and

x + 3y + v = 6 where v ≥ 0

x j = x’ j – x’’j, x’ j ≥0,x’’ j≥0.

Maximize: 3x + 5y + 0u + 0v

Subject to: x + y + u = 4

x + 3y + v = 6

x, y, u, v ≥ 0


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It is easy to see that if we find an optimal

solution (x*, y*, u*, v*) to a given problem, it

can be obtain by simply ignoring its last two

coordinates.

Specifically, we can rewrite the system of

constraint equations of the problem:


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If all the coordinates of a basic solution are

nonnegative, the basic solution is called a basic

feasible solution.

Example: If we set variables x and y to zero and

solve the resulting system u and v, we obtain thebasic feasible solution (0, 0, 4, 6); otherwise, we

obtain the basic solution (0, 4, 0, -6), which is not

feasible.

Each such point can be represented by a simplex

tableau, a table storing the information about thebasic feasible solution corresponding to the

extreme point.

Example: The simplex tableau for (0,0,4,6) is:


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A simplex tableau for a linear programming

problem in standard form with n unknown and m

linear equality constraints ( n ≥ m) has m + 1 rows

and n + 1 columns.

Each of the first m rows of the table contains the

coefficients of a corresponding constraint

equation with the last column’s entry containing

the equations right-hand side.

The columns, except the last one, are labeled bythe name variables.

The rows in the last column are labeled by the

basic variables of this solution.

The columns labeled by the basic variables form n-by-m identity matrix.


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Th e Sim p lex M eth o d

c The last row of a simplex tableau is calledobjective row , which is initialized by thecoefficients of the objective function with theirsigns reversed and the value of the objectivefunction at the initial point.

c The figure below is the outline of the Simplexmethod:


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the problem of finding a feasible flow in an s-t

network with the largest possible flow value for a

given weight function [www.answers.com]

a structured on a network but here the arc

capacities, or upper bounds, are the only relevant

parameters; the problem is to find the maximum

flow possible from some given source node to a

given sink node [www.me.utexas.edu]


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Example 1:

The illustration shows that all arc costs are zero,

but the cost on the arc leaving the sink is set to -1.

Since the goal of the optimization is to minimizecost, the maximum flow possible is delivered to

the sink node.


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Solution:

The figure shows that the maximum flow from

node 1 to node 8 is 30.

The heavy arcs on the figure are called minimal

cut.

The sum of the capacities of the arcs on the

minimal cut equals the maximum flow is a famous

theorem of network theory called the max flow min

cut theorem.


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Example 2:

We will assume that the transportation network

can be represented by a connected weighted

digraph with a scenario where an oil company has

the following pipeline network.

Each pipeline is labeled with its maximum flow

rate (in thousands of gallons per hour).

Question: What is the maximum possible flow rate

from A to G?


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Assumptions of Maximum Flow Problem:


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Some Applications of Maximum Flow Problem:

1. Maximize the flow through a distribution

network.2. Maximize the flow through a company’s

supply network from its vendors to its

processing facilities.

3. Maximize the flow of oil through a system of

pipelines.

4. Maximize the flow of water through a systemof aqueducts.

5. Maximize the flow of vehicles through a

transportation network.

Meljun Cortes Algorithm Iterative Improvement

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