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

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1answer
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Lexicographic Pivoting Rule: Why does that work?

On purpose, I do not write details about it, because I am interested in the intuition and not the details. But if someone asks for it, I can do that. So my question: Could anyone help me with the ...
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1answer
24 views

Prove that $b_{\perp}^{T}b_{\parallel}=0$

If $A \in \mathbb{R}^{mxn}$ then the unique expansion of every $b \in \mathbb{R^{m}}$ is $b =b_{\perp}+b_{\parallel} $. Prove that $b_{\perp}^{T}b_{\parallel}$. Comment: Saying that they are ...
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15 views

linear programming graphical problem [on hold]

I am new in linear programming. I am trying to understand graphical method for solving linear programming problems. so please solve this problem. $$\begin{align}&\max\ Z = 6X_1 + ...
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0answers
28 views

Linear program for way optimization with unusual constraints

I've just finished the lecture "Introduction to Optimization", so I know something about linear programming, and came across the following game on lumosity.com: The goal of the game is to pick up ...
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0answers
11 views

refomulation of an optimization problem

I have written a program for optimizing a set of generators. And I need to reformulate this problem, to include additional generators and constraints. I have hourly price and cost data and need to ...
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9 views

Can anyone help me solve this problem about linear programming? [on hold]

Sailcraft, Inc., is a builder of sailboats in New Bern, North Carolina. The company currently offers three models: the Adventurer (32 feet), the Explorer (42 feet), and the World Cruiser (50 feet). ...
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0answers
14 views

Positive/Negative Definite Bordered Hessian?

I understand how to check a function for concavity and convexity using the Hessian matrix and the rules for the determinants of the leading principal minors. I understand if these rules are violated, ...
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0answers
15 views

maintaining monotonicity in an optimization problem

I have written a program for optimizing a set of generators. I have hourly price and cost data and need to figure out when a generator should run or just stay off. I describe the problem in more ...
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0answers
16 views

Why does the simplexmethod 'break up' - unbounded, LP program, very basic problem

I've calculated a very, very basic LP problem: with >= "greater or alike" and <= "smaller or alike" max x + 2y 4x + 3y >= 12 x <= 4 ...
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0answers
11 views

Scaling linear constraints to give increased weight to values closer to 0

I am working on devising a integer linear programming model for a gene copy number problem. The general gist of it is that a ILP variable is created for each window for which we have data values. ...
0
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1answer
28 views

How to find on each face of a polyhedron one point?

We have a polyhedron in $\mathbb R^n$ generated by the intersection of a collection of finete hyperplanes or the convex hull of the set of vertices. My question is: Is there any algorithm for ...
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42 views

Show both relaxations of boolean LP give equal lower bounds

Given the boolean LP: $$\text{Minimize}\;\; c^Tx$$ $$\text{Subject to}\;\; Ax \leq b$$ $$\hspace{57mm} x_i(1-x_i)=0\;\; i=1,...,n$$ Show that the LP relaxation: $$\text{Minimize}\;\; c^Tx$$ ...
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1answer
14 views

Finding the intersection between 2 lines using matrices

My professor uploaded some notes, and there's a step in his explanation of a Linear Programming Problem which I do not understand. He takes 2 lines and converts them into matrices in order to find the ...
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0answers
16 views

Linearizing the sum of product of a binary and a continuous variable

Now suppose we have this expression $z = \sum_jA[j] * x[j]$, where $A[j]$ is a continuous variable and $x[j]$ is binary variable. $A[j]$ is bounded below by zero and above by $AM[j]$ , so I know how ...
1
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1answer
22 views

Primal of Dual of LP problem

Given that the following relation holds: $$\begin{align*} &\textbf{Primal problem} \\ &\max Z = c^Tx \\ &s.t. \\ &Ax \leq b \\ & x \geq 0\end{align*}$$ $\Longrightarrow$ ...
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0answers
14 views

Duality of Linear Program

Formulate the dual of the following linear problem: Min $\sum$ $\sum$ $c_{ij}$ $x_{ij}$ s.to $\sum$ $x_{ij}$ - $\sum$ $x_{ji}$ = 0, $\hspace{5mm}$ $x_{ij}$ >= $l_{ij}$ and $x_{ij}$ $\leq$ $u_{ij}$ ...
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0answers
8 views

Showing that every extreme point of the set of solutions of the standard form of constraints of any L.p.p. is a basic feasible solution

Let $\vec y$ be an extreme point of the convex set of solutions of $A \vec x=\vec b $ where only the solutions of $\vec x(\in \mathbb R^n)$ with all components non-negative are taken ; then I want ...
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2answers
36 views

Represent if-else or OR condition in a linear equation (optimisation with simplex algorithm)

I would like to write some linear equations and inequations to state that the sum of all possive x - C is smaller than L. As my ...
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0answers
31 views

A basic question related with the solutions of linear programming problems

I have to select one option from the problem statement given below. Which of the following statements is true in case of linear programming. $1$: An optimal solution exists at extreme points of a ...
2
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0answers
45 views

Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...
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1answer
18 views

Does optimal solution always occur at a vertex?

Is it true that if LP $ \text{max} \{c^Tx \ | \ Ax \leq b \}$ has an optimal solution, then $\exists$ a vertex which is simultaneously an optimal solution for LP? I know this works for LP of a ...
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0answers
32 views

LP in standard form

I don't know how to properly named this question but here it goes: Let $x, c \in \Bbb{R}^n$, $b \in\Bbb{R}^m$, $A \in \Bbb{R}^{m \times n}$. Consider LP in the form: min $\{c^tx : Ax = b, x \ge 0\}$ ...
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1answer
28 views

Non-negative solution to a underdetermined linear system

I have an underdetermined linear system (more unknown that equations) Ax=b where both b and x represent probabilities. Im currently using ALGLIB (rmatrixsolvels) to find a least square solution but ...
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0answers
18 views

Find all optimal solutions by Simplex

Let "stable operation" be an operation on a simplex tableau such that the entering variable has a reduced cost of 0. Recall that a pivoting operation will not change the objective value if either the ...
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0answers
19 views

Simplex minimum ratio test fails on bounded problem

Consider the linear program $\max 3x_1 + x_2 \ @ \\ 3x_1+2x_2 -s_1 = 1 \\ 2x_1+x_2 +s_2 = 2 \\ x_1 \geq 0$ Leting $x_2 = x_2^+ + x_2^-$, introducing slack and solving phase 1 gives $\textbf{x}_b ...
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1answer
31 views

Linear regression with constrained weights

I have a set of $n$ linear combinations, each with $m$ parameters and desired value $b$. I want to find the set of weights $w$ which minimizes the total equations distances (e.g. the sum of distances ...
0
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1answer
15 views

Optimization relaxtion quesiton

I have the following LP relaxation of an integer programme (the programme formed from the set cover problem) minimize $\sum_{j=1}^{m} w_{j}x_{j}$ subject to $\sum_{j:e_{i} \in S_{j}} x_{j} \geq 1$ ...
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0answers
14 views

Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
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1answer
24 views

Dual part of complementary slackness

The proof of the complementary slackness of P: min $c^Tx $ @ $Ax = b, x \geq 0$ D: max $b^Ty $ @ $A^Ty \leq c$ Goes something like $c^Tx = b^T y = y^TAx \Leftrightarrow c^Tx-y^TAx = 0 ...
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1answer
26 views

Linear Optimization: Objective function value, basic feasible solutions and reduced cost

For the system $$Ax=b, x \geq 0$$ for $A \in \mathbb{R}^{m \times n}$, $m \leq n$, we call a set $B \subseteq \{1, \dotsc, n\}$, $|B|=m$ a basis for $A$, if $A_B$ is invertible, where $A_B \in ...
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11 views

Bicritiera combinatorial/linear optimization problem with an exponential number of non-dominated extreme point

In [Ruhe 1988] an instance of a bicriterial combinatorial optimization problem is constructed such that the number of non-dominated extreme points is exponential in the input size. Are there any ...
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1answer
16 views

Model logical constraints without binary variables?

Is it possible to express "either $f(x) \leq 0$ or $g(x) \leq 0$" where $f,g$ are linear constraints by using a finite number of continuous constraints/new variables, WITHOUT breaking convexity or ...
0
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1answer
28 views

Combining multiple linear programming to minimize the sum

I have a math problem that looks like a bunch of linear programming problem combined where A matrix is shared. Here is the math definition of my problem Minimize \begin{align} & p_1 (x_{11} + ...
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2answers
26 views

Is Linear Programming a Combinatorial optimization method?

I want to know LP can be considered as a Discrete optimization or continuous. The solutions can be fractions so it should be continuous. Please suggest. thanks
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0answers
20 views

linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
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0answers
14 views

Representation of a point in terms of extreme points and extreme directions in a LP feasible region

We know that every point in a feasible region (X) of a Linear Program can be represented as a convex combination of the extreme points of X plus a non-negative combination of the extreme directions of ...
1
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1answer
34 views

Can number of constraints be less than number of variables in Linear Programming?

In standard form of LP we have $n$ variable and $m$ constraint. In simplex algorithm we set all non-basic variable to zero and at most $m$ basic variable have positive value. if $m < n$, then at ...
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0answers
69 views

Nested optimization problems solving using mixed integer linear programming

Let us have two vectors of decision variables, $\mathbf{x}$ and $\mathbf{y}$, two linear objective functions, $F \left( \mathbf{x}, \mathbf{y} \right)$ and $f \left( \mathbf{x} \right)$, and two sets ...
1
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1answer
21 views

Job scheduling to minimise squared completion times using mixed 0-1 quadratic program

I have come across an Optimization question as follows: There are $n$ jobs that have to be processed on a machine. The machine can process only one job at a time. The time taken to process job ...
1
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1answer
58 views

Problem understanding proof involving determinant

I have problems understanding Theorem 3.2, page 29 from Theory of Linear and Integer Programming. I don't understand (3): Let $M$ be a matrix in $\mathbb{Q}^{n\times n}$, and let $M_{ij} = ...
2
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1answer
27 views

Dropping Upper Bounds in a Linear Program

Can anyone explain why usually, in a Linear Program, the upperbound constraints are "redundant" and then they can be dropped? For example, consider: ...
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1answer
16 views

Equality of polyhedra

Is a minimal representation for a polyhedron unique? And if so can we use this to prove that two polyhedra are equal (or maybe the same is a better definition).
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1answer
59 views

Linear programming algorithm that minimizes number of non-zero variables?

I have real world problems I'm trying to programmatically solve in the form of $$Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ Subject to \begin{align} & a_{11} x_1 + a_{21} x_2 + \cdots + a_{n1} ...
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0answers
21 views

What is the complexity of Simplex Method's Phase 1?

What are the average and worst-case complexities of the Phase 1 of the Simplex Algorithm? Is it respectively polynomial and exponential as well? Google search did not yield any results unfortunately. ...
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1answer
51 views

Solve Karush–Kuhn–Tucker conditions

solving a constrained optimizing problem with equality constraints can be done with the lagrangian multiplier. (http://en.wikipedia.org/wiki/Lagrange_multiplier) This approach leads to a system of ...
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1answer
33 views

linear programming problem - how much additional resources should I buy?

I have the following linear optimization problem: Maximize $$\sum_{i=1}^{n}x_{i}B_{i}$$ subject to the constraints $$a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \le l_1$$ $$...$$ ...
2
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1answer
22 views

improving symbolic generation of objective function for optimization

I am currently using matlab to solve an optimization problem. I am generating the objective function using the symbolic toolbox. I planned use the symbolic toolbox to calculate the gradient and ...
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0answers
18 views

Computing the Optimal Simplex Tableau for Linear Programming

I am learning in my class about computing the optimal simplex tableau. I learned that, if you have an initial basic feasible solution, you can apply a series of formulas to compute the optimal ...
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1answer
38 views

Solving a linear optimization problem with products and work benches

I am taking a linear algebra course and I have a homework assignment of: A factory produces 5 products T1, T2, T3, T4, T5. Products are made on 3 different work benches P1, P2, P3, which can be used ...
5
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1answer
69 views

Max flow min cut from duality

I have been having some trouble deriving the max flow min cut theorem from duality, which I was told is possible. To begin with, I need to cast the problem into the form "maximize $\langle c, ...