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

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1answer
29 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
54 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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0answers
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
38 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 ...
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1answer
34 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
35 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
28 views

linear programming slater condition

I am wondering if anyone could help to come up with a such example: ...
1
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1answer
57 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 ...
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1answer
44 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
133 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 ...
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1answer
24 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 ...
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1answer
67 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
29 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
23 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
119 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
25 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
88 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
41 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 $$a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \le l_1$$ $$...$$ ...
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1answer
34 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
44 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
48 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
134 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, ...
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0answers
28 views

Degeneracy in Simplex Algorithm

According to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not ...
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0answers
120 views

Dual problem of a linear programming example with an unrestricted variable

I have been asked to find the dual of the following (P): Min $Z= x_1+ 2x_2+ 3x_3$ Such that $$-4x_1 +3x_2 +5x_3=5$$ $$x_1+ 2x_3 \geq 4$$ $$x_1,x_3\geq0$$ $$x_2 \ \text{ unrestricted}$$ I ...
2
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1answer
45 views

Description of a constraint for a mixed integer program.

Suppose we have 100 items that are labelled from the set $P = \{A, B, C, D, E\}$. My constraints are as follows: I want to choose exactly seven items. The choice should have at least one item of ...
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1answer
34 views

Removing a max function in the constraints

Can the following problem be transformed into a linear programming problem: Find $x_1,..,x_N$ which maximizes the objective function $$\sum_{i=1}^{N}x_{i}\sum_{j=1}^{n_{i}}c_{ij}$$ subject to the ...
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0answers
36 views

Generating primal solution from dual solution of a LP

How to get the primal solution from a dual solution in general? For example, let the primal problem is $$ \text {maximize } 2r_1+2r_2-2c_1-2c_2 $$ where $$ r_1-c_1\leq1\\ r_1-c_2\leq1\\ ...
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1answer
14 views

Efficient algorithm for slightly generalized attribution problem

I have what I believe is an attribution problem: Given an $m \times n$ matrix, I need to select $p = \min\{m,n\}$ elements maximizing their sum such that they do not share a row or column. More ...
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0answers
32 views

If the supremum is finite, then the value is attained.

In a linear programming proof we have the: $\sup\{c^Tx: Ax \le b\}$ This supremem can be $\infty$, or defined as $-\infty$, if there are no vectors x such that $Ax \le b$. But it is stated that ...
0
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1answer
50 views

Optimization Problem - Lowest Total Price from Multiple Suppliers

I believe this is a linear algebra problem, but if not please let me know: Say you have 4 suppliers. You want to order 4 different items. The 4 suppliers each have a different price for each item and ...
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2answers
36 views

Minimization problem, both terms in function positive

I have the following problem: Using the simplex method, minimize $z = 10x + 3y$ given the following conditions: $$2x + y \le 12$$ $$4x + y \ge 12$$ $$2x + y \ge 8$$ I've been told that minimizing ...
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1answer
39 views

MINLP optimization with matlab reaching different solutions every run

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 ...
0
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1answer
58 views

matlab MINLP optimization with ga

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. There are additional constraints but ...
0
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1answer
83 views

integer linear programing in matlab with the symbolic toolbox

I am writing a program to optimize a set of generators. I have hourly data and but dont want to necessarily optimize the whole time series. For a similar problem in the past I used the symbolic ...
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0answers
39 views

Modeling a lower-bound constraint on a euclidean distance in quadratic programming

I have been trying to model a certain problem into a mainly linear program, but with some quadratic constraints since I don't think that is something that can be avoided. I hope to solve it either ...
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0answers
27 views

Solution existence on $ax + by = c$

I have to produce an program which resolve the following equation: $ax + by = c$ With the following condition: $a$, $b$ and $c$ are known positive integer. $x$ and $y$ are positive ...
0
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1answer
51 views

Which algorithms are commonly used to solve this kind of Binary Integer Programming problem?

I want to solve the problem of minimizing $$\mathbf{c}^T\mathbf{x}$$ subject to the condition that $$A\mathbf{x} = \mathbf{b}\text{,}$$ where $\mathbf{b},\mathbf{c}$ are given vectors in ...
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2answers
51 views

Escaping from a point in linear programming

Is there a trick for explaining the following constraint as a set of linear (in)equalities? $$ \sum_{i=1}^n|x_i-a_i|>0, $$ where $a_i$'s are real constants.
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0answers
49 views

Algorithm to efficiently compute$ A^k$

If a symmetric matrix $A$ has SVD $A=U\Sigma U^{\top}$, then $A^k=U\Sigma^kU^{\top}$. What would be the most efficient algorithm to compute $A^k$ such that the worst case time complexity is as low as ...
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0answers
28 views

linear programming problem given initial solution

While dealing with a linear programming problem we usually try to start with the basic feasible solution corresponding to the identity Matrix in the coefficient matrix. I have no idea how to solve the ...
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0answers
242 views

Determining information in minimum trials (combinatorics problem)

A student has to pass a exam, with $k2^{k-1}$ questions to be answered by yes or no, on a subject he knows nothing about. The student is allowed to pass mock exams who have the same questions as the ...
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0answers
15 views

Polar cones' property [duplicate]

I am trying to prove: $A \subseteq B \implies B^\circ \subseteq A^\circ$ where $A^\circ$ is polar cone of $A$ ($A$ convex cone) and $B^\circ$ is polar cone of $B$ ($B$ convex cone)
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0answers
58 views

Prove $\lambda=\min_{i = 1,\ldots, n}\max_{0 \le k \le n-1}\left(\frac {p_i(n)-p_i(k)}{n-k}\right)$

Prove the minimum directed cycle mean cost satisfies: $\lambda = \min_{i = 1,\ldots, n} \max_{0 \le k \le n-1} \left(\frac {p_i(n) - p_i(k)} {n-k}\right)$ using the Bellman-Ford algorithm. Let ...
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1answer
45 views

Given an optimal solution to the LP, show how it can be used to construct a directed cycle with minimal directed cycle mean cost.

Let $\mathcal G = (\mathcal V, \mathcal A)$ be directed graph with associated edge costs $c_{i,j}$ that has at least one directed cycle. Define the directed cycle mean cost to be $\frac {\{\text {sum ...
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0answers
31 views

Calculating ONeill prices in combinatorial auctions using the dual (Linear programming)

I am trying to calculate the O’Neill prices[1] for the individual atoms in a combinatorial auction. The method is: Solve the IP to find the optimal allocation for the auction. Remove the integer ...
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0answers
29 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
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0answers
13 views

Optimizing a set of rules to better predict the outcome of events

I'm trying to better predict the top three finishers of the next 1000 800m mens freestyle swimming race. I've got a set of rules to rate the swimmers: 1) Add 5 points if the swimmer won his last ...
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0answers
77 views

Prove an artificial variable that leaves the basis will never return.

Prove an artificial variable that leaves the basis will never return. Edit: This is for the simplex method (I think). I have no idea how to start this. Anyone know any books with these kinds of ...
2
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1answer
168 views

Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate

How do I prove an optimal solution to dual is not unique if an optimal solution to the primal is degenerate? I have no idea how to start this. Anyone know any books with these kinds of questions (and ...
2
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2answers
86 views

Can calculus optimization problems be turned into linear programming problems?

I found a Linear Programming textbook somewhere, and I skimmed through the first few pages. While I am not nearly enough ready to go through it, the things it dealt with seemed very much like calculus ...