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

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1answer
2k views

Simplex: outgoing variable cannot re-enter the basis next iteration

How can I prove that in the simplex method, a variable that has just left the basis cannot re-enter the basis on the very next iteration? The pivoting rule is Dantzig's.
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1answer
429 views

Linear programming: the optimum of the shortest path problem is attained by $x \in [0, 1]^m$

Let $G=(V,E)$ be a graph, where $|E|=m$, and suppose we formulate the shortest path problem on $G$ as follows: minimize ${}^t(1,\dots,1)x$ such that $Bx={}^t(1,-1,0,\dots,0), x\in \{0,1\}^m$, where $B ...
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1answer
78 views

Solution for assigning independent tasks to independent individuals

I have $n$ tasks that I wish to delegate to $m$ independent individuals, where $m$ is a factor or divisor of $n$. Each of the tasks $T_{1} ... T_{n}$ is independent. From the following two extremes, ...
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0answers
75 views

A programming problem requiring mathematical optimization

This is a problem statement in one of the online Judges for programming. I am looking for an algorithm that gives optimized solution, not the best solution. I'm given a set of triplet of balls, each ...
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2answers
640 views

Minimizing Sum of Product

I'm given 3 multisets $A$, $B$, and $C$ each with $n$ elements. Now I'm to form $n$ (say $D_1$ to $D_n$) multisets of 3 elements each from $A$, $B$, and $C$, such that each of these $n$ multisets ...
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1answer
79 views

How can I fairly distribute identical goods bought at different prices amongst customers so that they all pay the same price?

I'm trying to allocate a product bought at different prices to different clients in a fair way. Initially, each of the $n$ client asked for a specific quantity of the product $a_1\ldots a_n$ The ...
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1answer
368 views

Totally Unimodular Matrices and Identity Matrices

I know that if a matrix $A$ is Totally Unimodular (TU), then the matrix $(A\; I)$ is unimodular. Can I then say that the matrix $(A\; -I)$ is also TU? ($I$ is the identity block matrix)
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1answer
836 views

Why does the “auxiliary problem” method work to find a feasible dictionary?

To quote my Linear Programming textbook, One way of getting around [the obstacles that arise when an LPP has an infeasible origin] uses a so-called auxiliary problem, $\min x_0$ subject to ...
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343 views

Can this non-linear optimisation problem be converted to a linear?

I have to minimize the function: $F(x)$ $F(x) = \sum_{i=1}^{M}||x_{i+1} - x_i - K(\frac{x_{i+1} + x_i}{2})||^2 + ||x_1-c_1||^2 + ||x_N-c_2||^2$ , where $x$ is a vector of $N$ scalars, $c$ are ...
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2answers
611 views

Sign restriction on the Lagrange multiplier? Why?

Say we are given a linear program where the goal is to minimize $c^Tx$ with the constraints $Ax\ge b$. Why is there a sign restriction on the Lagrange multiplier associated with the active constraints ...
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1answer
63 views

primal simplex procedure

Minimize $-2x_1-x_2+2x_3$ subject to $x_1 +x_3 = 4$, $-2x_1 +x_2 = 8$ s.t. $x_1,x_2,x_3\geq 0$. In my book, the augmented matrix is defined as $[A : 0 : b; -c^T : 1 :0]$ (where : separates ...
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1answer
70 views

Chaotic solutions to mixed integer linear problems

Is there a way to get the branch and bound algorithm to converge to a solution "close" to an initial value? One way I can think of, is to adding a "distance from initial value" term to the cost ...
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1answer
46 views

How do I solve a LP problem when constrains have different inequalties?

How can I solve this LP problem: Maximize p=x subject to : x+y <=30 x-2y <= 0 2x+y >=30 x>=0 , y>=0 using simplex method?
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1answer
416 views

Convex hull of sets defined by (in)equalities

If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
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2answers
736 views

Gradient solver

my question is about gradient algorithms. Lets have function f like: $f(x) = \|Ax-b\|^2$ and i want to find its minimum (according to x). So i can use some gradient method, for instance gradient ...
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1answer
49 views

Assume $x\in \operatorname{cone}\{a_1, a_2,\ldots, a_m\}$. Is there a systematic way to find out the coefficients of $x$ with respect to $a_i$'s?

Assume $x\in \operatorname{cone}\{a_1, a_2,\ldots, a_m\}$. Is there a systematic way to find out the coefficients of $x$ with respect to $a_i$'s? When $a_i$'s are independent, it should easy. What ...
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1answer
148 views

Techniques for (upper-)bounding LP maximization

I have a huge maximization linear program (variables grow as a factorial of a parameter). I would like to bound the objective function from above. I know that looking at the dual bounds the objective ...
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1answer
161 views

trying to read quadratic programming problem in cplex, get error

I am trying to load a CPLEX LP file in to CPLEX using the "read" command. I believe that in this problem, I have a set of constraints that are quadratic. But, from what I understand CPLEX will still ...
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1answer
299 views

How to express $y = x\ \mathrm{mod}\ 2$ as an ILP?

Using the signed modulo operation: $(x\ \mathrm{mod}\ 2) = \begin{cases} 0\ \mathrm{if}\ x\ \mathrm{is\ even} \\ 1\ \mathrm{if}\ x > 0\ \mathrm{and}\ x\ \mathrm{is\ odd} \\ -1\ \mathrm{if}\ x ...
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1answer
163 views

Continuity of a Parametric Linear Program

Consider the convex optimization problem $$ \min_{x \in X, \ y \in Y } x $$ $$ \text{sub. to } \ x A + B y + C = 0 $$ where $X = [0,1] \subset \mathbb{R}$, $Y \subset \mathbb{R}^M $ are compact ...
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1answer
284 views

Does the triangle inequality suffice to prove all minimum results on sums of absolute values of affine functions?

The title says it all ... more formally : let $n \geq 1$, and let $a_1, a_2 , \ldots ,a_n$ be positive numbers, let $b_1, b_2 , \ldots ,b_n$ be real numbers. Consider for $x\in {\mathbb R}$, $$ ...
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1answer
185 views

creating a set in ZIMPL (which creates .LP for SoPlex & CPLEX)

I am looking for some help creating a set dynamically in ZIMPL. I have a parameter table: ...
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1answer
188 views

Solving linear programming problem with global opt method

why not solve a linear programming problem with a global opt method, or a local search method as SQP or Newton methods? I am writting a solver facing linear and non linear problems, and I wonder ...
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1answer
133 views

Optimization for large scale linear problem with equality constraint

Given the wide range of optimization methods, which is the appropriate method to use? I am thinking of using either linear programming (interior-point methods) or augmented Lagrangian methods. Which ...
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1answer
168 views

Linear Programming Duality (Basic optimization)

Suppose that $A$ is an $m\times n$ matrix, $D$ is a $p\times n$ matrix, $b$ is an $m$-vector, and $d$ is a $p$-vector. Prove that there does not exist $n$-vector $x$ satisfying $$Ax \geq b, Dx \leq ...
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1answer
179 views

Efficiently solving a special integer linear programming with simple structure and known feasible solution

Consider an ILP of the following form: Minimize $\sum_{k=1}^N s_i$ where $\sum_{k=i}^j s_i \ge c_1 (j-i) + c_2 - \sum_{k=i}^j a_i$ for given constants $c_1, c_2 > 0$ and a given sequence of ...
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2answers
311 views

Need help with a Linear Programming homework.

Please help with the problem: A polyhedron P in $R^n$ is given by the system of m linear inequalities (of n variables). Furthermore, let P have k vertices (that is, k vectors satisfying all m ...
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1answer
86 views

Casting a linear (in)equality into a linear program problem

Suppose I have the systems $$S_1: Ax \leq b$$ where $A \in \mathbb{R}^{n\times m}$, $x \in \mathbb{R}^m$ and $b \in \mathbb{R}^n$, and $$S_2 : y^{\intercal}A=0,\\ y \geq 0,\\ y^{\intercal}b < 0 $$ ...
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0answers
127 views

Linear optimization, homework problem.

Please help with the following problem: Given a $m$ x $n$ matrix $A$, $m$-vectors $b$ and $y$, and $n$-vectors $c$ and $x$. Write the dual $LP$ problems $P$ and $P^d$ in the standard form. Whether ...
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0answers
168 views

Linear optimization problem.

I have copied the entire problem from the book. It has 7 parts. Please show me how to do any 1-2 of the parts. I mostly understand the problem, but need to see a fully woked out problem. Given a $m$ ...
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2answers
145 views

Solving LP from tableau

$$\begin{array}{cccccc} & x1 & x2 & x3 & x4& x5 \\ -4& 2 & 0& -2 & 0& 3\\ 3 & 1 & 0 & -1& 1 & 3\\ 2 ...
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1answer
1k views

What are the algorithms for integer programming in which constraints are dependent?

What are some ways to deal with dependent constraints in integer programming? For example, suppose I want to maximize $x+3y+2z$ subject to (i) $x+y<=3$ and (ii) if $y+z>=2$ then $x+z<=6$. ...
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1answer
176 views

Reverse Linear Programming Formulation

My question is about having an LP in the standard form $Ax \leq b$ and the set of basic feasible solutions. For each basic feasible solution (bfs) does there exist an appropriate objective function ...
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1answer
177 views

Linear Program feasibility

Let $A$ be an $m \times n $ matrix, $b \in \mathbb{R}^n$, and consider the linear program $$\max\{ 0^Tx: Ax = b, x \ge 0\},$$ and its dual $$\operatorname{min}\{y^Tb : y^TA \ge 0 \}.$$ Here $x \in ...
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1answer
266 views

What is the complexity of computing the minimum distance between two convex polyhedra that both have $n$ faces?

EDIT: (in response to what deinst said) sometimes using a sledgehammer for some menial task is rather convenient - especially if it also has the complexity $O(n)$ (which is what my question is about) ...
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1answer
2k views

How to show a primal program is unbounded by using weak duality?

In weak duality theorem, we assume $x_i$ and $y_i$ are feasible. But how could we show a primal program is unbounded by this theorem? Suppose we have a primal program: $\max \mathbf c^\top \mathbf x, ...
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526 views

non-degeneracy in linear programming

How to do the sum? Consider the standard form polyhedron $P = \{\mathbf{x} | A\mathbf{x}=\mathbf{b}, \mathbf{x} \geq 0\}$. Suppose the matrix $A$, of dimensions $m \times n$,(m<=n) has linearly ...
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1answer
705 views

Verifying optimality for given primal and dual solutions to a linear program

Consider the following linear program: maximize $\sum\limits_{j = 1}^n {{p_j}{x_j}}$ subject to $\sum\limits_{j = 1}^n {{q_j}{x_j}} \le \beta$ $\begin{array}{*{20}{c}} {{x_j} \le 1}&{j = 1,2, ...
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0answers
152 views

All optimal solutions of a linear program

Is there a software package that can output all optimal solutions of a linear program if there are multiple such solutions?
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91 views

Does the Hirsch conjecture hold for $n < 2d$?

The Hirsch conjecture states that the graph (i.e. $1$-skeleton) of a $d$-dimensional polytope with $n$ facets has diameter at most $n - d$. It was known for a long time that it sufficed to prove it ...
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1answer
79 views

The Hirsch conjecture in $3$-dimensions

What I am wondering is if the Hirsch conjecture has a simple proof (just a few lines) in $3$ dimensions, perhaps by using Steinitz's theorem or Kuratowski's theorem and some kind of induction ...
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1answer
500 views

How to determine whether a system of linear inequalities has a POSITIVE solution or not?

The question I'd like to ask is as in the title: How to determine whether a system of linear inequalities has a POSITIVE solution or not? Is there any poly-time algorithm to do this? Or the best ...
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2answers
231 views

Dual LP from Primal LP, how?

$$\min x +y + z$$ so that $$ x + y =2$$ and $$y + z = 3$$ where, $x,y,z >0$. How to create the dual? [Something like this?] $$ \max 2s + 3t$$ so that $$ ...+... =1$$ and $$ ...+...=1$$ ...
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1answer
1k views

Simple Minimization Problem: A few questions regarding the mechanics of solving

Consider the following elementary minimization problem: Minimize: $\phi = 2700x + 2400y + 2100z$, subject to: $\text{Constraint 1}: 55x + 45y + 35z \geq 41000$ $\text{Constraint 2}: 30x + 35y + 50z ...
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1answer
355 views

How to apply the simplex method to prove that the following problem is unbounded?

$\max 6t_1 + 4t_2$ $-t_1 + t_2 \leq 6$ $t_1 - t_2 \leq 1$ $t_1 - 2t_2 \leq 8$ $t_1, t_2 \geq 0$ Anyone?
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1answer
42k views

shadow price in linear programming

I am quite confused about the meaning of shadow price from explanations on the internet. It can be understood as the value of a change in revenue if the constraint is relaxed, or how much you would ...
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1answer
384 views

Multicommodity flow in polynomial size

The original linear program for multicommodity flow has exponentially many variables. How to find equivalent linear program that has polynomial size? Linear program of multicommodity flow $maximize ...
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1answer
313 views

Underlying assumption in a Primal/Dual table

I just read in one of the questions answered by @MikeSpivey that the following table is provided in Sierksma's Linear and Integer Programming: Theory and Practice, Volume 1, page 144. ...
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1answer
244 views

Multiple solutions for both primal and dual

If matrix $A$ in an LP (or $A^T$ in its dual) has full row (column- in dual) rank, is it possible that both primal and dual have multiple solutions?
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1answer
73 views

Does the sparsest cut always have a solution?

How do I prove that the sparsest cut always has an optimal solution which is the cut for some vertex-subset? It looks like it should be a kind of fundamental theorem for sparsest cut. But I didn't ...