# Tagged Questions

Questions on linear programming, the optimization of a linear function subject to linear constraints.

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### Feasible Condition with a single constraint

A linear program with a single constraint minimize $z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$ subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$ (a) ...
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### 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 ...
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### Set of optimal solutions for a linear programs

Consider the linear program: minimize $z = x_{1} - x_{2}$, $x_{1}, x_{2}\geq 0$ subject to: $-x_{1} + x_{2}\leq 1$ , $x_{1} - 2x_{2}\leq 2$ Derive an ...
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### 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 ...
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### how to use linear programming for Heaviside Step function and L1 norm?

I want to find a hyperplane that can divide my sets of points into 2 groups that have nearly equal size. If the hyperplane is $w$, there is a scalar offset $b$. I have $N$ points that are $d$-...
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### 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?)
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### Linear programming, how to do the opposite transformation?

This is from a pdf file I found linked to from this site: They first define the primal dual problem like this: Then they have another representation of (p), and show that it's dual is: Now here ...
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### Affine and Linear programming

Can someone give a simple explanation as to why the feasible region of a set of linear program/equations is affine?
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### 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 ...
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### 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 $n+1$ ...
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### Find the minimum value of C subject to the given constraints.

C=2x+5y Constraints: x+y>=2 2x-3y<=-6 3x-2y>=6 A-42 B-4 C-49 D-10 I encountered this question while doing the Systems of Linear Equations and Inequalities test at http://www.classzone.com/books/...
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### The meaning of initial value in linear programming

I am new to LPP. I would like to know what is meant by setting an initial value(IV) to a variable. For example I was solving a problem where objective function(OF) is non-negative. When I give some IV ...
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