1
vote
0answers
54 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
49 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
28 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
26 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
14 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
29 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
11 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
26 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
24 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
32 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
33 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
32 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 ...
3
votes
0answers
28 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
18 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
35 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
30 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
18 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
36 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
33 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
33 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
0answers
26 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
10 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
75 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
27 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
13 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
23 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
23 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
14 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 ...
1
vote
2answers
40 views

What is wrong with this linear program $\max z = 3x_1+2x_2$?

I solved a linear program. It is wrong. The answer is that $(x_1,x_2) = (50,75)$ and the maximum value is $300$, but instead I am getting $(x_1,x_2) = (50,100)$ and the maximum being $350$. Why is ...
0
votes
0answers
13 views

Solving $\max z (3x_1 - x_2)$ linear program

I am trying to solve this linear programming exercise using the Simplex method. First of all, I detail every step, so it is pretty long, but the real question (the part where I'm stuck at) is at the ...
0
votes
0answers
18 views

Robust LP question using box uncertainty model

I am trying to solve this robust LP problem by writing it as a QP $$\min_x x^TSx : \mu \leq r^T x , Ax \leq b$$ Under Box uncertainty model: $$R = \{r : \| r - \hat{r}\|_\infty \leq \rho\}$$ Here ...
0
votes
0answers
17 views

Why do I have to solve a minimisation for an expanded maximisation linear program?

I'm learning about artifical variables with the Simplex method. There is an example where I'm getting a bit stuck at some point: Solving $$\max z = -2x_1 +x_2+1$$ subject to $$\begin{cases} ...
0
votes
1answer
36 views

How can I verify my linear program solutions?

I started solving linear programs with the Simplex algorithm, however it is unclear to me how can I verify my solutions. I have heard about geometrical solutions easy to check visually, but I'd ...
1
vote
1answer
18 views

What to do if the righthand of a constraint is a variable when solving Linear programs?

I'm starting to solve linear program exercises with the Simplex method. $$max \ (3x_1-x_2)$$ $$\begin{cases} x_1-x_2 \le 3\\ 2x_1\le x_2\\ x_1+x_2\ge 12\\ x_2 \le 10\\ x_1,x_2 \ge 0 \end{cases}$$ I ...
1
vote
1answer
39 views

Explicit solution of linear program: minimize $c^T x$ subject to $Ax = b$

This is the given question in a textbook I am following. I will paste the solution below which I do not understand: I am a little hazy on linear algebra theory, so I don't fully understand how the ...
2
votes
2answers
45 views

Which mathematical programming method is good for solving the described problem?

I need to solve the following problem. Let's say that there are 3 clients with different time windows. For simplicity let's say that travel distance is always 10 minutes and service time is 30 ...
0
votes
1answer
10 views

Finding the corresponding constrained subspace under a over determined mapping Ax=b, where b is constrained

Suppose that $A$ is a $m\times n$ matrix with m>n, $Ax=b$, and $b$ is constrained in every component, $b_\min^i<b_i<b_\max^i$ for $i=1,\dots,m$. There should be a similar set of constraints for ...
1
vote
0answers
31 views

How to use the simplex method for linear programs?

I believe to be missing something important in the Simplex algorithm, because I can't get it to work. From what I gather, there are three steps per iteration, given a matrix for a linear program in ...
0
votes
5answers
122 views

Ok, I know what does linear independence mean but why should I care?

I understand that for a set of vectors to be linearly independent, none of the vectors in the set should be a linear combination of some other vectors in that set. But why on earth should I care about ...
0
votes
0answers
18 views

How to practically perform reinversion in PFI (Product Form of Inverse) Simplex.

While doing Revised Simplex using Product Form of Inverse. We have product of set of Eta Vectors(Elementary Matrix) to describe the Basis Inverse after certain number of steps in simplex. ...
0
votes
0answers
31 views

Mixed Interprogramm remodeling

for example i have the following problem min z 5 x_1a + 6 x_1b - 3 x_2a + 0 x_2b <= z -3 x_1a + 0 x_1b - 1 x_2a + 2 x_2b <= z x_1a + x_1b = 1 (Constraint say of this group only one variable ...
0
votes
1answer
24 views

About canonical linear programs.

Starting out with linear programming, I'm having some questions about canonical linear programs: Do all linear programs have a canonical form? So far I couldn't figure an example stating otherwise. ...
0
votes
0answers
19 views
0
votes
0answers
37 views

Linearization of multiple normal functions

I have noticed that it takes a very long time to perform non-linear least squares fitting on datasets similar to this: where there are multiple Gaussian distributions to be fit to experimental ...
0
votes
1answer
72 views

A question about rational number.

Denote $M$ as a $m\times n$ matrix whose components are all nonnegative integers (actually 0 or 1) and $1$ as the $m$ dimensional vector $(1,1,\cdots,1)$. Show that: There is a vector $x_0$ ...
0
votes
1answer
23 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
0
votes
1answer
23 views

condition for having a positive solution to a linear equation.

Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, ...
0
votes
1answer
41 views

orthogonal triangular decomposition and ordinary least squares

I have just come across orthogonal triangular decomposition whilst looking at ordinary least squares regression. I'm not quite sure how this is being used though to find a solution. In my example I ...
2
votes
0answers
47 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...