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

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Use Exact Non-linear formulation or a linear approximation?

I am writing a paper that discusses results to solve stochastic problems with recourse analytically. The problem is nonlinear. I can also write an approximate stochastic linear program to sove the ...
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1answer
315 views

Minimal set of inequalities

I have a set of $m$ linear inequalities in $R^n$, of the form $$ A x \leq b $$ These are automatically generated from the specification of my problem. Many of them could be removed because they are ...
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232 views

linear programming

Suppose you have won \$ 6000 from OK Grand challenge promotion and you want to invest is. Upon hearing the news , your two differnt friends Mukanya and Mhofu offer you each an opportunity to become a ...
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1answer
51 views

Can standard Linear Programming algorithms return all valid solutions without losing their efficiency?

I have a (generalized) Linear Programming problem to solve. I anticipate exactly two equally valid optimizations of my objective function. I would be happy if I could receive both these points; it ...
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2answers
466 views

Need Homework Help: A small corportion borrowed $500,000, some at 9%, 10% and 12%. Use a system of equations--how much was borrowed at each rate if…

A small software corporation borrowed 500,000 cash to expand its software line. The corporation borrowed some of the money at 9%, some at 10%, and some at 12%. Use a system of equations to determine ...
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1answer
215 views

Prove that an optimal solution $x^*$ of the problem 1 $\min f(x)$ s.t $x\in \mathbb{R}^n$ and..

Prove that an optimal solution $x^*$ of the problem 1 $\min f(x)$ s.t $x\in \mathbb{R}^n$ and an optimal solution $(\bar{x},\bar{z})$ of the problem 2 $\min z $ s.t $z\ge f(x)\,, x\in \mathbb{R}^n$ ...
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1answer
125 views

Maximizing a linear combination of certain integers

Consider some tuple $x = (x_1, ..., x_k) \in \mathbb{N}^k$ of $k$ non-negative integers such that $x_1 > x_1 > ... > x_k$ and suppose that $A \subset \mathbb{N}^k$ is such that there exists a ...
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3answers
322 views

Find a nonnegative basis of a matrix nullspace / kernel

I have a matrix $S$ and need to find a set of basis vectors $\{\mathbf{x_i}\}$ such that $S\mathbf{x_i}=0$ and $\mathbf{x_i} \ge \mathbf{0}$ (component-wise, i.e. $x_i^k \ge 0$). This problem comes ...
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1answer
304 views

LP: nonbasic solution made into basic solution, help me with this terminology

Related chat here, reading the Bertsimas book now on pages 50-51. By the way, I am gathering Linear-Programming -related studying material here, welcome to read a book and have coffee :) I cannot ...
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0answers
73 views

Linear Optimization Problem - Assign Objects to People

Say you have a 100x5 matrix of integers between -10 and 10, including zero. Each row represents an object; each column represents a person's ranking of the objects. Of the possible ranking values ...
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1answer
131 views

How relevant is mathematical optimization today?

That's it. That's all I'd love to know from you guys. Mathematical optimization, with the aid of today's software. Do you think it's still relevant in today's world?
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2answers
182 views

Reduced cost zero for the two-phase Simplex?

I cannot understand the line -12, -4, -5, 1, 1, -1, 0, 0, 0. Now the formula $\bf c - \bf A ^t \bf y$ when $c=0$ will result into the line. It is just many times a ...
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2answers
783 views

Optimality conditions and Directions in Simplex method

I am trying to understand the optimality conditions in Simplex -method, more in the chat here -- more precisely the terms such as "reduced cost" i.e. $\bar{c}_j=c_j-\bf{c}'_B \bf{B}^{-1} \bf{A}_j$ and ...
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1answer
67 views

Interactive Vizualizer of different Simplex -methods?

My book [1] around the pages 80-100 outlines the theories behind different simplex methods such as Naive-Simplex, Revised-SImplex, Full-tableau-Simplex, Dual Simplex, etc-simplex --. It is very dry ...
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0answers
58 views

Using duality to establish a relationship between in two-stage linear programming

I'm currently working on a problem that involves a two-stage linear program (LP). For simplicity, I refer to the LP in first stage as LP$_1$, and the LP in the second stage as LP$_2$. The relationship ...
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3answers
171 views

Integer combination

i want write a module to find the integer combination for a multi variable fomula. For example $8x + 9y \le 124$ The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, ...
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1answer
99 views

Solving an optimization problem involving reciprocals

I am trying to solve the following minimization problem, perhaps by getting it into a LP form: Let $u= [u_1, u_2, ...u_N]^T$ a column vector, and $v=[{1\over u_1}, {1 \over u_2}, ...{1 \over u_N}]^T$ ...
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1answer
689 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
261 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 ...
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1answer
441 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
141 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
87 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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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
209 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
450 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 ...
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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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136 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
833 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
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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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 ...
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1answer
743 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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125 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 ...
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2answers
485 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
337 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
84 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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253 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 ...
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1answer
135 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
374 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
73 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
484 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
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 ...
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1answer
312 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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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 ...
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1answer
570 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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274 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
545 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 ...