Interior point method

This is the nonlinear optimization problem in Primal-dual interior point method.

minimize f(x), subject to ci(x)≧0 for i=1,2,..,m , x∈Rn, f:Rn→R, ci∈Rn→R

x is a Lagrange multiplier.

ci(x)λ=μ, ∀i=1,2,..,m , λ∈Rm

F(X0)≧F(x1)≧F(X2)≧...

F is minimal and the bottom.

Duality Theory

Duality Theory is an upper bound to the optimal value of the primal problem in every linear programming problem.

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P

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D

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P and D come as a pair implying that the dual to D should be P. x∈Rn is feasible for P, and y∈Rm is feasible for D. You see the direction x→ and y←.

∴

This is Weak Duality Theorem.

Moreover, when the finiteness of the optimal value implies the existence of a solution, this is called The Strong Duality Theorem.

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The alphabetical order don't mean anything. It is the direction, min ←.

Hn-approximation algorithm

Hk is the kth harmonic number.

Hk=1+1/2+1/3+．．．+1/k

Harmony is the concept. I often read articles about brain computers. This is the new era of the computer science, and it is beyond calculation. I don't usually calculate anything at all even in paying my bills because of automation. Math was just copying the formula in my school days. The concept was philosophy. Your brain may be just algorism of repeating. I hope your existence must be more than that.

We know that an optimal solution covers all elements with solution weight OPT.

The average weight ≦ OPT/n

When k elements are covered, the optimal solution have n-k elements.

Remaining uncovered elements with an average weigh ≦ OPT/n-k

The greedy algorithm ≦ OPT/n-k+1

∴

Then we formalize the intuition.

This is expansion of uncovered elements.

wj is the covered weight.

∴

The ellipsoid method

The ellipsoid is symmetry.

The relaxation of linear programing must be in it.

∴

Cj≧rj+pj, ∀j∈N (N={1,2,…,n}), ∀S⊆N

This is the whole universe, so you need to minimize it.

wj is weight.

You can also rewrite this.

This is the violation, so you remove the job k from S.

The knapsack problem

The greedy is corruption, so you can't exceed your capacity. This is called the knapsack problem. You add the value in your napsack, not items.

S⊆I

S is a subject, and I is items.

You maximize the value of your items.

B is the capacity, so your items are restricted.

i∈I

You use dynamic programing for optimization.

A(j) is the array. j=1,・・・,n.

(t,w) is in A(j). t≦B and w is the value.

(t',w') is t≦t' and w≧w'. (t,w) can't expand the space in your napsack, but the value is increased.

If (t, w) dominates (t', w') and (t', w') dominates (t'', w''), then (t, w) also dominates (t'', w'').

A(j) is (tk, wk), so (B+1, V+1) is possible because of integers.

A(1)={(0,0), (s1,w1)}

A(j)←A(j-1), (j=2,．．．,n )

(t, w)∈A(j-1), (t+sj, w+wj)∈A(j), t+sj≦B

You remove A(j), so A(j-1) and (sj, wj) are remained.

Then you return (t, w) and A(n).

max(t,w)∈A(n)^w

This is not polynomial time because of the binary.

Θ(nB)

This would be polynomial because of vagueness. Both t and all B may be valuable.

(1-ε) is approximation algorithm for maximization.

Natural Greedy Heuristic

Recently quantum annealing is well known as Traveling salesman problem.

It must be P=NP.

The symmetric matrix means that the cost of traveling from city i to city j is equal to the cost of traveling from j to i. This is Zero, and Hamiltonian cycle is undirected complete graph with a cost associated with each edge.

You can choose any start points, which is Φ. It must be P≠NP.

This is natural greedy heuristic, because j is added to the cycle.

cik≤cij+cjk

i∈S and j∉S (S is the subset)

G=(V,E)

This is the undirected graph.

F⊆E

F is the minimal subset of edges, and each pair of nodes in G is connected.

S={v}, v∈V, T=(S,F), F=Φ

T is a tree, and v is an arbitrary node.

The metric traveling salesman problem is a 2-approximation algorithm.

The cost of the tour on the first two nodes i and j is exactly 2cij.

△ijk is the triangle, and the perfect matching on Θ has cost at most OPT/2. It is not the odd number.

Skipping over previously visited cities is sometimes called shortcutting.

This is Φ=0, which is corruption and wide open.

Computer Viruses

There are no optimal solutions without P=NP.

P=Polynomial time, NP=Nondeterministic Polynomial time

Maximization problems are expressed as α<1. 1/2 approximation algorithm for a maximization problem is P because the solution is half of optimal value. NP hard is the increasing numbers of optimal value. Specific solutions are optimal than promised by the performance guarantee.

E={e1, . . . , en}

This is elements.

S1,S2,・・・Sm are subjects of E.

Sj⊆E, wj≧0

wj is weight for for each subset Sj.

The goal is to find a minimum-weight collection of subsets that covers all of E, which is P=NP.

I⊆{1, . . . , m}

If wj= 1 for each subset j, the problem is called the unweighted set cover problem, which is P≠NP.

Then you detect computer viruses.

G=(V,E)

This is the undirected graph.

wi≧0

wi is weight for each vertex i∈V.

The goal is to find a minimum-weight subset of vertices C⊆V.
(i, j)∈E, and i∈C or j∈C

If wi= 1 for each vertex i, the problem is an unweighted vertex cover problem.

Computer viruses are everywhere.