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Incremental Linear Programming. Linear programming involves finding a solution to the constraints, one that maximizes the given linear function of variables. D = number of variables or dimensions. Objective function is the function to be maximized.

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incremental linear programming
Incremental Linear Programming
  • Linear programming involves finding a solution to the constraints, one that maximizes the given linear function of variables.
    • D = number of variables or dimensions.
    • Objective function is the function to be maximized.
    • Linear program is the set of constraints together with the objective function.
    • Feasible region is the intersection of the half-spaces, which is the set of points that satisfy all the constraints.
      • Feasible region can be bounded, unbounded, empty. If empty problem is infeasible

Maximize C1X1 + C2X2 + … + CdXd

Subject to A1,1X1 + … + A1,dXd ≤ b1

A2,1X1 + … + A2,dXd≤ b2

An,1X1 + … + An,dXd ≤ bn

linear programming
Linear Programming
  • Operations Research has developed many algorithms to solve linear programs that perform well in practice.
  • Our LP has N linear constants in 2 variables.
  • Most OR applications have high-dimension

(# constraints and variables) and do not work well in low dimensions (# of variables).

  • Computational Geometry algorithms can do better in low dimensions
linear program h c
Linear Program (H,c)
  • H is set of n two-dimensional constraints
  • gives objective function
  • GOAL: find so that and is maximized
  • Let C denote feasible region for (H, c)
linear program
Linear Program
  • Four possible cases
  • Convention: to give unique solution for 3rd example, choose lexicographically smallest point.

C

p

P

e

Unbounded

Return ray EC

Infeasible

No solution

Non-Unique Solution

Unique (vertex) solution

incremental 2 dimensional linear programming
Incremental 2-dimensional linear programming
  • Add constraints one by one
  • Maintain optimal vertex of intermediate feasible region.
  • Slight problem, requires that solution exists!
    • Not true for unbounded linear program
    • We will use subroutine for this
unbounded lp h c
Unbounded LP (H,c)
  • If (H,c) unbounded return ray in C
  • else return so that is bounded.

(h1 and h2 are certificates)

certificates

linear programming7
Linear Programming
  • Let (H,c) be bounded linear program
    • h1 and h2 are certificates returned by UnboundedLP(H,c)
    • Number remaining halfplanes h3,h4,…,hn

Let Ci = feasible region with respect to halfplanes h1-hi =

Note:

Fact: ci = Ø then cj = Ø for all j≥i (and LP is infeasible)

how does optimal vertex change as we add h i
How does optimal vertex change as we add hi?
  • Vi is optimal vertex for Ci
  • Li is line bounding hi
slide9

Lemma 4.5 :Let ci and vi be defined as before(i)If vi-1 Îhi, then vi = vi-1 (ii)If vi-1 Ï hi, then either ci = f or viÎ li .

Proof :Let vi-1 Î hi

(1) ci = ci-1 Ç hi implies ciÍ ci-1

(2) vi-1 Î ci-1 and viÎ hi implies vi-1 Î ci

Note that the optimal point in ci

cannot be better than optimal point

in ci-1 (smaller) implies vi-1 is

optimal in ci

let v i 1 h i suppose c i f and v i l i contradiction
Let vi-1 ÏhiSuppose ci ¹f and viÏ li (contradiction)

(1) Consider segment vector vi-1 vi

-by definition vi-1Îci-1

-since ciÌ ci-1 ,viÎ ci-1

-since ci-1 is convex this implies

the vector vi-1viÌ ci-1

(2) since vi-1 is optimal for ci-1 and

fc is linear this implies fc(p)

increases monotonically along the

vector vi1vi as p moves from vi to

vi-1. 

proof continued
(Proof continued)

(3) Consider intersection point q

of vector vi-1vi and li

- q exists since vi-1Ïhi and viÎc i

Since vector vi-1vi Îci-1 , q must

be in ci but value of the objective

function increases along the vector

vi-1vi so fc(q) > fc(vi) which is a

contradiction to the definition of

vi

to update optimal point
To update optimal point :

(1) If vi-1Î hi then we are done (vi = vi-1)

   (2) If vi-1Ïhi we need to find vi on li but this is just a one dimensional LP

One-Dimensional LP : Find p on li that maximizes fc(p) subject to constraints p Î h j , 1£ j £ i.

Without loss of generality, assume li is x-axis and let xj = liÇ hj.

We will now see how to solve one dimesional LP

to solve one dimensional lp
To solve One-Dimensional LP :

x left = max 1£ j < i {x j | li Ç hj is bounded to left }

x right = min 1£ j < i {x j | li Ç hj is bounded to right}

The interval [x left, x right] is a feasible region

- LP is infeasible if x left > x right

- Otherwise, optimal point is x left or x right

Running time of One-Dimensional LP : O(n)

slide14

Algorithm: Two Dimensional LP(H,c)

Input :LP(H,c)

Output :Infeasible, Unbounded (and ray in c), or solution point p

maximizing fc(p)

1. Run UNBOUNDEDLP(H,c)

report if (H,c) is infeasible or unbounded (and ray in c)

2. Let h1 and h2 be certificates returned by UNBOUNDEDLP(H,c)

letv2 = h1Ç h2 and let h3,h4, ...,hn be half planes in H

for i = 3 to n

if vi-1 Î hi

then vi := vi-1

else vi := 1DLP({h1, h2, ... , hi-1}, c)

if vi doesn’t exist report infeasible.

endif

end for

return vn

end algorithm

slide15

Running time :

  • Unbounded LP implies O(n) (We will see later)
  • - Each iteration O(i) implies å O(i) = O(n2)
  • Therefore O(n2) in total.
  • Correctness :
  • Follows from Lemma 4.5 (each iteration have correct)
  • But this algorithm is worse than the one for constructing entire convex region.
incremental lp
Incremental LP
  • Nice and simple.
  • But…takes O(n2) time in worst case, which is worse than the previous algorithm that computed the entire feasible region!
is our analysis too crude i e is algorithm actually better than we thought
Is our analysis too crude?i.e. is algorithm actually better than we thought?
  • Algorithm has n-2 stages, (each time add a half plane)
  • We said stage i takes O(i) time, the time for 1D-LP with i half-planes.
  • Note however: stage i takes:
    • O(i) time if optimal vertex changes  do 1D-LP (previous optimal is not in hi).
    • O(1) time if optimal vertex does not change (previous optimal is in hi, so still optimal).
question how many times can optimal vertex change
Question: how many times can optimal vertex change?
  • Idea: if we can show it changes only say k times, than we can bound running time at O(k•n).
  • Unfortunately: there are cases in which optimal vertex can change every time…
question how many times can optimal vertex change19
Question: how many times can optimal vertex change?
  • Thus, if we consider the planes (in this order), then the optimal vertex changes every time, and we have to do a 1D-LP each time! Running time O(n2) !!
  • Notice however, that if we had been lucky and added the vertices in the reverse order then the optimum would never change!
  • Hmm… can we determine the right order in which to add the planes?
randomization
Randomization
  • Unfortunately, we can not really determine the exact best order without a lot of work…
  • Answer: Randomization

Choose a random permutation of the planes and add them in that order.

    • We could have bad luck and pick a bad order that gives O(n2) running time.
    • But most orders are not bad (as we’ll see) and so usually we do pretty well.
changes to algorithm
Changes to Algorithm
  • Before start adding half-planes, randomly permute them. The running time is O(n).
  • RandomPermutation(A)

input: A[1…n]

output: A[1…n] --- permuted randomly

for i = n downto 2

random_index = Random(i)

swap(A[i], A[random_index])

endfor

randomized incremental algorithm
Randomized incremental algorithm
  • Algorithm is now randomized algorithm.random choices made in permutation subroutine
  • What is running time of randomized incremental algorithm?
    • Depends on permutation, and there are (n-2)! of them…
    • We’ll study the expected running time.
      • Each permutation of input is equally likely and doesn’t depend on the input planes
      • No assumptions made on input and so expectation is w.r.t. random order in which half-planes are treated and holds for any set of half-planes.
expected running time
Expected running time
  • Theorem 4.8: The 2D-LP with n constraints can be solved in O(n) expected time using a randomized incremental algorithm.
  • Proof:
    • Running time of RandomPermutation() and UnboundedLP() are O(n). We’ll see the latter one.
    • Need to consider time for adding n-2 half-planes.
expected running time24
Expected running time
  • Adding a half-plane takes
    • Constant time if the optimum doesn’t change
    • O(i) time if does change (ith half-plane with 1D-LP).
  • We will bound time for all 1D-LPs.
  • Let Xi be random variable:
expected running time25
Expected running time
  • If Xi = 1, then 1D-LP takes O(i) time. Otherwise, adding hi takes O(1) time. Total time adding all half-planes (with 1D-LP) is:
  • We bound this sum using linearity of expectation: expected value of sum of RV’s is sum of the expected values.
expected running time26
Expected running time
  • What is E[Xi]?

Probability that vi-1 hi.

  • “Backward analysis”
    • Algorithm done, vn is optimum vertex and vertex of Cn.
    • Is it a vertex of Cn-1?

The answer is no only if hn is one of half-plane defining vn. How likely is this…? Only at most 2/(n-2).

expected running time27
Expected running time
  • And in general, to bound E[Xi] we
    • Fix subset of first i half-planes (determine Ci).
    • Compute a new optimum when adding hi if hi was one of two half-planes defining new optimum.

E[Xi] = 2/(i-2) .

  • So total bound:
expected running time28
Expected running time

Randomized Incremental Algorithm takes O(n) expected time.

Important:

Expectation is only with respect to random permutation and applies to any input set.

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