2
votes
2answers
92 views

Boundedness of ratio of linear functions

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
1
vote
1answer
79 views

a closed form lower bound solution for linear programming

Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it? to clearly express the problem assume that $$ z(\mathbf{a,B,c})=\mathop {\inf} ...
1
vote
1answer
115 views

Name search for special Linear Integer Program

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
1
vote
0answers
43 views

Maximize minimum optimization using linear integer programming [closed]

I am trying to solve a maximize minimum optimization. I have four different items that each of them has 10 values of Rates and for each value it has a corresponding weight. Then I have a free table ...
1
vote
0answers
52 views

Are there any known bounds on the value of solutions of linear integer programming?

Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints? ...
0
votes
0answers
43 views

Multi-objective set-cover optimisation problem

I'm looking for an algorithm to solve the following multi-objective set-cover problem. We start with a 'universe' (set) of items $\mathcal{U}$, along with a partitioning $P = \{p_0,\ldots,p_m\}$ ...
1
vote
1answer
174 views

Maximizing linear objective function with absolute values

This has be asked on other forums, though couldn't find authoritative answer. I have a linear program over the reals and don't want to introduce integer or binary variables. The objective function ...
0
votes
1answer
47 views

generalization from linear programming solution [closed]

I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
20
votes
4answers
1k views

Why are optimization problems called “programming”?

Why are optimization problems often called programs? linear programming geometric programming convex programming Integer programming ...
4
votes
2answers
362 views

Simplified knapsack problem

There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
2
votes
1answer
141 views

For interior point methods of linear programming, what is the “L” in the computational complexity $\mathcal{O}(n^3 L)$?

My question is about interior point methods of linear programming. Suppose the constraint matrix $A$ has $m$ rows and $n$ columns, and $m<n$. The state-of-the-art methods, like primal dual interior ...
1
vote
0answers
78 views

Interior point optimisation using big M for L1 norm on linear system using Dikin's Affine method

I am a 4th year undergrad surveying student studying computations, specifically $L_{1}$ norm minimisation of residuals in large data sets. To start with (and probably to finish with) I'm using a set ...
1
vote
0answers
127 views

Equal maximum and minimum in a large-scale linear programming

For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...
0
votes
2answers
113 views

Rewrite optimization objective

Hi, I wanted to ask, under which conditions can one rewrite the optimization objective $\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$ as $\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$ I have particular ...
10
votes
2answers
735 views

Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
0
votes
0answers
54 views

Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?

I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise. The other variables in the linear ...
0
votes
0answers
170 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 ...
1
vote
2answers
181 views

what method can I employ to solve this optimization problem which involves \min?

The optimization problem is: maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N ...
1
vote
0answers
224 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 ...
1
vote
2answers
342 views

Sherali-Adams relaxation

I am trying to find a book or a paper, which explains, how and why the Sherali-Adams relaxation method works. The original paper (1990) is difficult for me to understand. I need a more basic ...
1
vote
0answers
582 views

How to solve simple bilinear equations under extra linear constraints

Hello, This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy: $\lambda_i^T ...
1
vote
1answer
424 views

Can one efficiently optimize over the inverse of matrix?

Hello, I have the following problem: Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
1
vote
1answer
207 views

Approximate Set Cover Problem by Rounding

Here is the simple algorithm for approximating set cover problem using rounding: Algorithm 14.1 (Set cover via LP-rounding) Find an optimal solution to the LP-relaxation. Pick all sets ...
1
vote
1answer
376 views

Need help to find an efficient algorithm for the following problem!

Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$. Given $A_{n\times n}$ is the covariance matrix of $x$. $u$ is a given n-dimensional vector of real ...
4
votes
2answers
480 views

Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
3
votes
1answer
2k views

Linear program to maximize the minimum absolute value of linear functions ?

I'd like to compute $\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$. where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$. Can this be solved with a linear ...
3
votes
0answers
478 views

Infinite Linear Programming

I'm trying to prove optimality for a continuous linear program. That is, I have a linear program with an uncountable number of variables and constraints. I'm not sure how to demonstrate feasibility ...
2
votes
4answers
352 views

Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by $f(x) = \max_i ( a_i + b_i x )$ We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...
1
vote
2answers
797 views

Inequality-constrained linear-regression, what is the covariance of the estimator?

If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$ ...
2
votes
0answers
260 views

Recovering a piecewise affine function

Lets say I have an piecewise affine convex function $f(x_1,x_2)$, on which the following operations are possible: Computing $f(x_1,x_2)$. Computing a subgradient to $f$ at $(x_1,x_2)$ Computing all ...
10
votes
1answer
1k views

Sum of difference moduli vs. sum of modulus differences

This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself. Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...