0
votes
0answers
10 views

Maximization over linear surjective mapping of polyhedron

I am reading this paper and confused about the derivation of equation (11) (page 3, bottom of column 2). I will rephrase it in this question. Let $\mathcal{P}_r = \{ x \in \mathbb{R}^n : P_r x \leq ...
1
vote
1answer
37 views

books on the application of linear algebra on statistics/finance/machine learning

I am reading "linear algebra done right" by Axler and like it a lot. One thing though, in the end I would like to put these theory to use and as a math textbook it doesn't cover much application. ...
1
vote
1answer
26 views

Taking the dual of this non-standard linear program

I am just beginning to learn linear programming have a question about taking the dual of a non-standard LP specifically the one below: $\min M\\ 2x_1 + 3x_2 + 4x_3 \leq M \\ 2x_1 - x_2 + x_3 \leq M\\ ...
0
votes
1answer
52 views

Linear Programming and Geometry Question

I have a question that involves some linear programming and linear algebra, and I really don't have a clue how to approach this question. Could someone give me some hints and ideas as to how to attack ...
0
votes
2answers
33 views

References for Linear Algebra needed for Differential Equations and Linear Programming

I am in need of learning the Linear Algebraic theory behind the following Applied disciplines. Could someone please recommend Linear Algebra books for: Differential Equations: Specifically learning ...
1
vote
0answers
23 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
0
votes
0answers
25 views

optimisation problem with linear constraint

optimisation problem with linear constraint I have an optimisation problem. I wish to maximise a function subject to a constraint. It is the constraint that is causing me problems. I am using an ...
0
votes
0answers
17 views

How to I see that $n-1$ linearly independent constraints $a_i^Tx \ge b, i \in \{1,\dots, n-1\}$ define a line in $\mathbb R^n$?

How to I see that $n-1$ linearly independent constraints $a_i^Tx \ge b, i \in \{1,\dots, n-1\}$ define a line in $\mathbb R^n$ ? ($b, a_i, x \in \mathbb R^n$) If I can find a vector $p$ that ...
3
votes
0answers
32 views

Easy elementary proof of Farkas Lemma?

Is there any elementary proof of Farkas lemma which does not use convex analysis and hyperplane separation theorem? What about special case below: If the Matrix $A$ is invertible, then there is ...
0
votes
0answers
20 views

How to justify that a basic feasible solution to a Linear Program corresponds to an extreme point of the feasible region?

Say we have an LP Problem in standard form. That is, $$\text{Maximise} \;\; C^T X $$ $$ \text{subject to:} \;\;\; AX = B --(1) \;\;\;\; \text{where $A$ is an $m \times n$ matrix }$$ I read ...
0
votes
0answers
15 views

Case of affine hull and linear hull possibly being euqal

Let C be a set in $\mathbb{R}^n$. Let $aff(X)$ denote the affine hull of $X$, and $lin(X)$ denote the linear hull of $X$. Suppose $x \in aff(C)$. Then, is it true that $aff(C-x)=lin(C-x)$? The ...
0
votes
0answers
27 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
1
vote
0answers
14 views

Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
0
votes
1answer
54 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
0
votes
0answers
11 views

How to prove that the solution in every next iteration of Simplex algorithm is a BFS

Let Ax = b x >= 0 be the feasible region. Let A be mxn matrix with m <= n and rank(A) = m. Simplex algorithm starts with a Basic Feasible solution. In each step it moves to a new solution. How do ...
0
votes
1answer
32 views

what is the dual of the following linear program over a convex set?

Let $\mathbf{x}=[x_0,x_1,\dots,x_N]^T$ be a $(N+1)\times 1$vector. Let $\mathcal{S}$ be a bounded, compact convex set in strictly positive quadrant of $\mathbb{R}^{N+1}$. Consider the following ...
0
votes
0answers
26 views

Linear Algebra - find basis without reduced echelon

For the start of a simplex solver I'm building in Python, I need to find a basic feasible solution. To do that, I need to find a basic solution/find the basis of the constraint matrix. Here's my ...
0
votes
2answers
36 views

Maximum cardinality affinely independent subset of $\mathbb{R}^n$

Let $S \subset \mathbb{R}^n$ such that $S$ is affinely independent. Then $$|S| \le n + 1.$$ Why? (e.g. does anybody know some place where this is proven?)
0
votes
0answers
20 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
0
votes
1answer
30 views

Can there be a unique natural number vector solution to $Ax =b$ where $A$ is not a specific type of square matrix?

Let $A$ be $(n-1) \times n$ matrix that is of the following form: $$\left( \begin{array}{ccc} n-1 & 1 & 0 &.... & ....\\ 0 & n-2 & 2 & .... & ....\\ 0 & 0 & n-3 ...
2
votes
1answer
57 views

Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
1
vote
0answers
58 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
1answer
56 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
0answers
54 views

Proof of Strong Duality via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
0
votes
1answer
27 views

shortest point on a line segment from point out side the line

from the above pic I found the value x from line (p1,p2) and point a using y=mx+b and imaginary red line which is perpendicular to black line having slope -1/m and the intersecting point x. the ...
0
votes
0answers
16 views

Variable bounds of under-determined linear system

If I have a non-negative, under-determined linear system $\mathbf{Ax}=\mathbf{b}$ $\mathbf{x}\geq\mathbf{0}$ is there a fast way to compute the upper and lower bounds on values of each element of ...
0
votes
1answer
34 views

Minimze min max (A*x)

has this example matrix A some special propertries, which might be useful? $$ \left[\begin{array}{rrrrrr} 3 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 4 & 3 & 0 & ...
1
vote
0answers
13 views

How to import matrix from sif document

I want to make some computation (with scilab, scipy or other) over the matrix A of linear problems (in inequational form). Those problems are in .sif format (from the netlib library in fact) and I ...
0
votes
1answer
26 views

Under-determined linear problem

To compute all solution of following under-determined linear problem in matrix form $ Ax = y $ we can use Pseudo inverse of A and the solution would be : $ x = A^{PI}y + [I - A^{PI}A]w $ I couldn't ...
0
votes
1answer
27 views

Linear Constraints Solution Existence

how can one decide if $$A*t\ge b$$ $A$ is a Matrix with integer Entries and $t$ is a Vector with integer Entries, $b$ is a fixed Vector with integer Entries exists?
2
votes
0answers
29 views

Finding optimal hyperplane

I have a set of vectors $\{V_i\}$ in $n$-dimensional space. There is a number corresponded to each vector $\alpha_i = f(V_i)$ ($\alpha_i$ can be negative). I want to find a hyperplane which would ...
0
votes
1answer
34 views

Pseudoinverse system of linear equation

$Ax = b$ describes a convex polyhedron, where $A$ is a real matrix and $b$ is a real vector. Now assume $A$ has less rows than columns. If you take a look here: ...
0
votes
1answer
41 views

Simplying linear equation to get quartic in q using Maple and then using Descarte’s rule of sign

Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots. \begin{equation} ...
2
votes
1answer
40 views

Does the Duality Theorem of Linear Programming hold only in closed convex cones

I've just read the the Duality Theorem of Linear Programming. Here is the proof from my book (and my questions after it): Duality Theorem of Linear Programming: If the primal or dual linear ...
2
votes
0answers
30 views

Rayleigh Quotient variant?

If $A$ is a covariance matrix and I want to get $\max X^TAX$ where each value of $x$ is between -1 and 1. Is there a closed-form solution for this? I understand when $X^TX = 1$ this becomes the ...
1
vote
0answers
26 views

Check feasibility of a system of integer linear equations

I'm currently working on a very large integer linear programme which cannot be solved within any reasonable time. The initial set of linear equations S={Ax<=b) is feasible. I want to add more ...
1
vote
1answer
36 views

Why $(1,\textbf{0}) \not\in \{(r,\textbf{w}): r=tz_0-\textbf{c}^T\textbf{x}, \textbf{w}=t\textbf{b}-\textbf{Ax}, \;\textbf{x}\geq\textbf{0}, t\geq0\}$

I'm learning about linear and nonlinear programming and on the chapter about duality I have the following statement and proof I can't understand: minimize $\textbf{c}^T\textbf{x}$ subject to ...
0
votes
0answers
31 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
0
votes
0answers
22 views

Name search for special Linear Mixed Integer Programm

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
0
votes
0answers
44 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
0
votes
1answer
39 views

Projection onto Polyeder

I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 ...
0
votes
1answer
78 views

Can someone explain the effects of degenerate basic feasible solutions in the simplex algorithm?

I was given this on an assignment sheet, and am now using it to revise from...I cannot remember the issues that arise from degeneracy of basic feasible solutions... Let $P$ =$\{x\in \mathbb{R}^n ...
1
vote
1answer
33 views

Why is this simplex procedure not working? $\min z = y - x + 1$

I have read of two ways to solve this program with the Simplex algorithm. One worked and the other didn't. The only difference is that, in the one that didn't work, I rewrote the function. I will ...
0
votes
1answer
12 views

Is there relations between earth mover's distance and vector norms?

Say I have two vectors $a$ and $b$. Can I estimate $\mbox{EMD}(a,b)$ via some combination of things like $\|a-b\|_p$ and such?
2
votes
1answer
82 views

Linear Algebra 101 - Optimizing inequalities

I am considering the region contained in $\mathbb{R}^2$ consisting of all the points that satisfy all the following inequality: $-4 \leq y < 4 \\ -9 \leq 2x + y \leq 9 \\ -9 \leq x + 2y \leq 9 \\ ...
0
votes
0answers
34 views

Union of all sets of optimal solutions to a perturbed linear programming problem

Please let me know if you have some ideas on how to approach this proof? I got stuck part-way through. The following linear program is a function of $\theta$, $ \begin{array}{ll} \min & c^\top x ...
1
vote
1answer
14 views

the differences and relationship between linear independent and affinely independent

When learning optimization, I heard the two related concepts on linear algebra: linearly independent and affinely independent. ...
1
vote
1answer
24 views

Linear Optimization Study Material

I've recently enrolled in a linear optimization course, and it's been a while since I've taken linear algebra. I do not yet have access to the book for the course or I would skim it to see what I need ...
0
votes
1answer
25 views

What's wrong with this linear program formulation?

You have two item factories, $A$ and $B$, and there are two clients that buy such item. Each client has a demand - the first one needs $400$, and the second $300$. Each factory has a ...
0
votes
0answers
16 views

Total Least Squares problem when some columns of data matrix have no error

I'm reading through Golub and Van Loan and they mention that to solve the total least-squares problem $(A + E)x = b + r$, where the first $s$ columns of E are zero, then we can solve the problem by ...