Questions tagged [linear-programming]

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

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0answers
149 views

Feasibility checking

I have a question regarding feasibility checking. I need to check whether the system $\{ x: Ax=b , x \geq 0 \}$ has a feasible solution. What is the best worst case running time for this decision ...
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4answers
4k views

Percentages - Find Maximum value.

3 candidates A,B and C contest an election. A gets at least 40% of all the votes. B gets at least 20% of the number of votes that A gets and cannot get more than 80% of number of votes that c gets. ...
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1answer
797 views

LP unboundedness

Does there exist a way to check if a linear programming problem is unbounded without solving it directly? In other words, How the unboundedness of an LP can be realized from its structure. Assume the ...
10
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1answer
6k views

Strict inequalities in LP

How should we deal with strict inequalities in a linear programming problem? For example: inequalities such as $ax< b$;
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2answers
835 views

Solving geometric problems using Linear Programming

Is it possible to find an LP formulation to test whether $n$ points in the plane are in convex position?
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2answers
2k views

$\ell_0$ Minimization (Minimizing the support of a vector)

I have been looking into the problem $\min:\|x\|_0$ subject to:$Ax=b$. $\|x\|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time has ...
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0answers
216 views

Simplex method with zero value constraints

The usual problem is to maximize some linear function $f(x_0, x_1 ... x_n)$ subject to linear constraints $g_i(x_0, x_1 ... x_n) \leq b_i$. My question is: What happens when all (or most) of the $b_i$...
4
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0answers
870 views

Book recommendation on Applied Integer Programming/Combinatorial Optimization/OR

Having some very basic and theoretical knowledge about these topics from my study, I'm looking for a book (or other good sources) that explains the stuff from a practical point of view. On the one ...
1
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1answer
290 views

Totally Uni-modular Matrices

A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
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1answer
6k 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
643 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 ...
0
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1answer
186 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
89 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
1k 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 ...
2
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1answer
95 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 ...
0
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1answer
990 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)
2
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1answer
4k 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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0answers
391 views

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

I have to minimize the function: $$F(x) = \sum_{i=1}^{M}\left\|x_{i+1} - x_i - K\left(\frac{x_{i+1} + x_i}{2}\right)\right\|^2 + \|x_1-c_1\|^2 + \|x_N-c_2\|^2,$$ where $x$ is a vector of $N$ scalars, ...
1
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2answers
845 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 ...
0
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1answer
77 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 columns ...
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1answer
84 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
49 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?
1
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1answer
915 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
1k 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 ...
1
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1answer
57 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 ...
1
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1answer
557 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 ...
1
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1answer
272 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 ...
-1
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1answer
924 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
361 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
526 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 $x\in {\mathbb R}$, $$ \phi(...
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1answer
397 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: ...
0
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1answer
230 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
194 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 ...
0
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1answer
217 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 d$...
4
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1answer
235 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 non-...
2
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2answers
444 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 ...
1
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1answer
171 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 $$S_2 : y^{\intercal}A=0,\\ y \geq 0,\\ y^{\intercal}b < 0 $$ ...
2
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0answers
172 views

Linear optimization, homework problem. [duplicate]

Please help with the following problem: Given a $m \times n$ 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. ...
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0answers
213 views

Linear optimization problem. [closed]

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 $x$ (respectively, $y$) is a feasible ...
1
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2answers
312 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 &...
2
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1answer
4k 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 $x+z<=6$. ...
1
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1answer
497 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 $...
1
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1answer
305 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 \...
3
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1answer
3k 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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1answer
2k 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
222 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?
6
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0answers
115 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 ...
3
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1answer
124 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 argument....
4
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1answer
1k views

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

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 algorithms known are no less complex than algorithms ...
1
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2answers
308 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 $$ ...+...=1$$