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

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1answer
685 views

Using max/min operators in linear programming.

I'm currently implementing a Markov Decision Process using the solver GLPK, I'm following the lecture by Vincent Conitzer, and there is a step I don't understand between the theoretical problem and ...
2
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1answer
254 views

Books on AI, programming, optimization

I'm studying math (just started) and I like programming as well (just started this too), is there a career or a branch of research including deep aspect of this two aspects? Is there someone among you ...
1
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1answer
432 views

Optimization with non-negativity and norm constraint

I am facing the following optimization problem: $$\min_x w^tx \\ s.t. ||x|| = 1, \forall i: x_i \geq 0 $$ where $w$ and $x$ are real valued vectors. How would I solve this? My background is not ...
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1answer
139 views

How low can the approval rating of a majority candidate be?

“Ostrogorski's paradox” describes a strange situation in which voters decide on candidates based on issues in platforms, but on each issue of the platform, the majority of voters disapprove of the ...
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0answers
24 views

Background of choosing standard for of a linear program as type III inequalities?

In linear programming where we seek to minimize $c^Tx \to \text{min}_{x\in P}!$ with respect to some inequality constraints, why do we choose $P$ in the form $Ax \leq b$, $x \geq 0$ as the ...
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0answers
86 views

How to get the initial ellipsoid in the ellipsoid method for solving optimization problem?

If what I assume is correct, assumption : for a maximization problem, we run a binary search over estimated values, starting with max estimated value, and narrow down to the feasible optimal value ...
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0answers
61 views

Using a corner-polyhedra-characterization to proof a connection of edges <-> base solutions

For polyhedra $P = \{x \in \mathbb{R}^n \mid Ax \leq b \}$, which I want to call type-I-polyhedra (as the inequality constraint in its definition is a type I-inequality), we have a ...
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0answers
203 views

Question simplex method application criteria?

Can we apply simplex method if one or more equation are equal to zero. tell me full criteria my question example is as follows: Maximize: $z=135x+50y$, subject to: $$\begin{align} ...
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2answers
446 views

LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
2
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0answers
110 views

Why is every nontrivial surface of a polyhedron an intersection of facets?

In the geometry of (convex) polyhedra used for linear optimization, one has the lemma: Consider the inequality $Ax \leq b$ where $A^+ x \leq b^+$ (the non-implicit inequalities of $Ax \leq b$) ...
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0answers
134 views

1s surpassing 0s in binary strings of odd length

Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in ...
2
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0answers
78 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. 1- What is the best worst case running time for this decision ...
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3answers
808 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. ...
2
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1answer
141 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 ...
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0answers
49 views

unboundedness of an LP

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 ...
1
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1answer
723 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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0answers
123 views

Checking feasibility of a system of inequalities

Consider the following system of inequalities: $$Ax=b \\ x\geq 0$$ $A$ is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How feasibility of this ...
5
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2answers
473 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
325 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
79 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 ...
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0answers
239 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
134 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
1k 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
373 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
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
72 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
472 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
votes
1answer
76 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
308 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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0answers
50 views

computing centerpoint using linear programming

I was redirected from math.stackoverflow. So I am trying to teach myself discrete geometry and have started with the centerpoint problem. Could someone please help me understand computing ...
1
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1answer
556 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
270 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 ...
1
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2answers
536 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
59 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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0answers
70 views

Solving PSD matrix in Newton's method

I have functions defined as follows: $f_1(A) = \sum\|x_i-x_j\|_A = \sum\sqrt{(x_i-x_j)^TA(x_i-x_j)}$ and $f_2(A) = \sum\|x_k-x_l\|^2_A$ where $A$ is a positive semi-definite (PSD) matrix, $x$ are ...
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1answer
69 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 ...
0
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1answer
43 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
382 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
649 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
46 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
136 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 ...
0
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1answer
121 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
197 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
143 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
236 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
158 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
175 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
129 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
156 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 ...
3
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1answer
163 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 ...