Linear programs with strict inequalities and supremum objectives

Linear programming can only solve problems with low inequalities, such as "maximize $ c x $ such as $ A x leq b $This makes sense because strict inequality problems often have no solution, for example, "maximize $ x $ such as $ x <$ 5"There is no solution.

But suppose we seek to find the supremum instead of the maximum. In this case, the above program has a solution – the supremum is $ 5 $.

With a linear program with strict inequalities and a supreme objective or infimum, is it possible to solve it by reducing to a standard linear program?

What is the eigenvalue and eigenvector of a composition of two arbitrary linear transformations

Let $ S $ and $ T $ to be two linear transformation on $ mathbb {R} ^ n $. One can find the eigenvalue and the eigenvector of $ S $ and $ T $. I'm trying to discover the reality between the value and the eigan vector of $ S $, $ T $ and $ S circ T $. Is there a relationship like this?

Thank you.

Linear algebra – Scope of the null space of A

I apologize for the brief nature of this question, but I do not think it has been clarified in a previous post on this topic – Find a Covering Set for Null Space.

When we say that the null space of the matrix $ A $ is equal to the duration of a set of vectors:

$ N (A) = span ( v with v_1, vec v_2, $ $ … vec v_n) $

Are we actually saying that:

$ A ( vec v_1) + A ( vec v_2) + $ $ … A ( vec v_n) = vec 0 $

reference request – ODE almost linear

Let $ A, B $ be $ n times n $ matrices. I am interested in the following ODE in $ mathbb {R} ^ n $

$$ frac {dx_t} {dt} = Ax_t + Bx ^ + _ t $$

or $ x ^ + = (x ^ + _ {1, t}, …, x ^ + _ {n, t}) $ and $ ( cdot) ^ + $ is the rectifier: $ r ^ + = max {0, r }. $

Does this type of EDE have a name? And are there any known stability criteria? Has it been studied by anyone in general?

The closest I've found is the "linear threshold networks" studied here for example. I appreciate any reference similar to this system.

linear dependence and independence [on hold]

What does it mean if we say that the functions e ^ x, e ^ 2x, e ^ -x are linearly dependent? What is the intuition of linear dependence and independence?

linear transformations – Understanding the linearity of derivatives

I am a little confused as to the assertion that the derivative of a function $ f $ (whether it is a partial derivative, the total derivative or simply a derivative in $ mathbb {R} $) is a linear transformation.

1.) By definition, the derivative is always defined at a given moment, for example $ a $, so let $ Df (a) $ to be the derivative at this point $ a $. But if you consider the derivative according to any point $ x $ from the domain of $ Df then $ Df (x) $ is not necessarily linear.

2.) If you consider a derivative at a given time $ a $, $ Df (a) $ and then you plug in another variable or vector $ v $ then $ Df (a) (v) $ is a linear transformation with respect to $ v $.

Are these thoughts about derivatives correct?

linear algebra – Retrieves the translational component of a particular matrix multiplication

Let's say that I have a rotation matrix R 4x and a 4×4 translation matrix T. If I multiply the matrices in this order T * R, the translation component of T will not be affected. But if you multiply them in the R * T order, it will be affected by R.'s rotation values.

How can I describe the position vector of the resulting matrix from the multiplication R * T, in terms of the rotational components of R and the translational component of the matrix T? Can I describe it as the scalar product of the respective orientation component of the matrix R and the translation component of the matrix T?

Linear algebra – Conditions necessary for the consistency of a system of inequalities.

I was trying to find the answer to the next question I had when solving a more complex problem:

What are the conditions necessary for the coherence of a system of inequalities, that is to say that there exists at least one set of values ​​for the unknowns which satisfies all the inequalities of the system?

If the above question needs to be answered in a general way, I would make an exception to the answer to the following more specific question:

What are the necessary conditions for the following
begin {align}
a_j> f_ {i, j} (x) a_i forall x text {and} forall i, j = 1, …, m, i not = j
end {align}

or $ f_ {i, j} $ is a smooth continuous function, and $ a_i, a_j $ are constants, to be consistent (there is at least one set of constants $ a_1, …, a_m $ which satisfy each inequality).

If you have any questions, do not hesitate to ask them.

Edge coloring in dense linear hypergraphs

Let $ H = (V, E) $ to be a hypergraph. Yes $ kappa $ is a cardinal, we say that a map $ c: E to kappa $ is a edge coloring if every time $ e_1, e_2 in E $ with $ e_1 cape_2 neq emptyset $ then $ c (e_1) neq c (e_2) $. The smallest cardinal $ kappa $ such as there is a coloring of the edges $ c: E to kappa $ is called the chromatic number of edge of $ H $, noted by $ chi_e (H) $.

We say that $ H = (V, E) $ is a dense linear hypergraph if

  1. $ bigcup E = V $,
  2. does not matter when $ e_1 neq e_2 in E $ then $ | e_1 cap e_2 | $ 1, and
  3. given $ a neq b in V $ there is $ e in E $ with $ {a, b } in e $.

Given a positive integer $ k, is there a dense linear hypergraph $ H = (V, E) $ with $ V $ finished and $ chi_e (H) <1 / k cdot | V | $?

linear programming – The simplex method does not solve the problem of assignment?

The problem:

I'm trying to solve http://acm.timus.ru/problem.aspx?space=1&num=1076, it's an online judge for programming problems. The problem could be solved by simply applying an assignment problem solving algorithm such as mincost-maxflow or a Hungarian algorithm. The dice are small (150 out of 150) and I try to put forward a solution with the simplex method – it does not exceed the time limit and I do not think it's my fault.

Problem of assignment

$ min sum c_ {i, j} x_ {i, j} \
sum_ {i} x_ {i, j} = 1 \
sum_ {j} x_ {i, j} = 1 \
x_ {i, j} geq 0
$

What I did:

  • I added the cycling check – indeed, cycling was present. I've used some schemes to eliminate cycling: Bland's rule, lexicographic rule and Zadeh's rule. None of them triggered the cycle check, but the solution still has not exceeded the time limit.
  • I've removed all the artificial variables and pants. My initialization is $ x_ {i, i} = $ 1 and the other n-1 basic variables are 0 (I use variables that do not have a non-zero value after a Gaussian elimination with $ x_ {i, i} $)

My hypothesis

I am not an expert in the areas of linear programming, but here are some of my thoughts: I think the problem is very degenerate, the rank matrix of constraints is $ 2n-1 $ with n variables having a value equal to 1 and rest n-1 having a value equal to 0. During a step of the algorithm, we enter a situation in which many variables with value 0 are exchanged, which makes the simplex method exponential. I print the cost of the function and the discount ceases after a while, although the cost reduction indicates that an improvement is still possible (that is, the criterion of the ## EQU1 ## Stop of the simplex method is not satisfied).

What I do not want

  • I have an accepted verdict on this problem with mincost-maxflow, so please do not suggest any Hungarian algorithm or similar, I want to know if simplex method can solve this problem
  • Please do not suggest changes for the simplex method that make assumptions about the graphical model of the problem. suggestions such as "make the edges of value 0 an alternative path". I want to know if the purely simplex method can solve the problem (with possible modifications, but that makes no assumptions about the problem).

What I want

  • I want to know if my hypothesis is correct and that the simplex method simply does not apply to this kind of problem.
  • If not, what kind of system can be used to solve this problem.