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2
votes
2answers
180 views

Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function \begin{align} f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\ &+\ldots \\ &+ p_n ...
1
vote
0answers
123 views

Complexity of Nested Linear Optimization

My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint: ...
4
votes
2answers
200 views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...
0
votes
1answer
52 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 ...
6
votes
1answer
279 views

Has this generalization of a determinant (assigning multiplicities to the rows) been studied?

I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant: Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
-3
votes
1answer
462 views

Closed form solution for Least Squares Problem [closed]

I'm looking for closed form solution for the following equation: $argmin_{X,y}(\sum_i{\parallel{a_i-Xb_i-y}\parallel^2})$, where $X \in\mathbb R_{m\times n}$ is a matrix and $y\in\mathbb R_{m\times ...
3
votes
1answer
148 views

Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function: $$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
1
vote
1answer
377 views

Minimizing sum of absolute deviations

Suppose we want to find coefficients $b$ in $\underset{b}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n | y_{i}-b_{1}x_{i}-b_{0}\mid$. If we rewrite this problem in terms of linear ...
3
votes
0answers
124 views

Variant of orthogonal Procrustes problem

The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
1
vote
1answer
142 views

Eigenvalue problem with quadratic constraints

$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$ where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = ...
2
votes
2answers
164 views

Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method

If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. ...
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
435 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 ...
0
votes
1answer
66 views

About the suboptimality of linear estimators

Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$. ...
5
votes
0answers
167 views

A linear optimization problem on a graph

Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
2
votes
1answer
172 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
96 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 ...
4
votes
2answers
222 views

point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
0
votes
1answer
572 views

eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
2
votes
1answer
406 views

Finding integer points inside of a parallelogram

Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...
1
vote
0answers
153 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 ...
3
votes
1answer
161 views

Explicit formula for an LMI solution

Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil): $$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$ (The notation $X \succeq Y$ means that ...
2
votes
0answers
27 views

In what paper was the shrinkage parameter introduced to the nelder-mead simplex direct search algorithm?

I have read lots of papers referencing a 4th shrinkage parameter when talking about the Nelder Mead Simplex method. However, I cannot see any shrinkage parameter in the flow chart of the original ...
2
votes
0answers
132 views

existence of lattice point in polytope

This question was probably asked before but here goes. I have a convex polytope given by $Ax\leq b$ for a specific integer matrix $A$ and integer vector $b$. I need a simple method/result on how to ...
2
votes
1answer
266 views

Why does the LP Formulation of the MST Problem need Topology Constraints?

I am looking for an example that demonstrates the necessity of either subtour-elimination or of connectivity constraints in the LP formulation of the MST In the internet I only could find the LP ...
0
votes
1answer
361 views

Nonconvex optimization problem

I have a nonconvex optimization problem. It is actually optimizing a linear objective function over a set of linear constraints and a set of nonlinear, non convex constraints. Is this problem ...
0
votes
2answers
128 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 ...
1
vote
2answers
173 views

Name of operations on two vectors

Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$. I could define the mapping $$ T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m} $$ as follows $$ T(x,y) = ( ...
0
votes
2answers
177 views

positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix

Given that $A$ is a positive semidefinite matrix, $x$ is a vector, $\lambda_0 \in [0, +\infty) $ is a real non-negative number. I want to know the answer to the following optimization problem. $$ ...
2
votes
2answers
96 views

LP constraint enconding

I have an objective function to be maximized $obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$ With multiple constraints of the form: $\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i ...
1
vote
1answer
115 views

Deducing Linear Inequalities

Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form $X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$ with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...
1
vote
0answers
76 views

Arrangements of graphs of linear functions

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Assume $X$ is unbounded. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form $$ f(\bar{x})=a_1x_1+\ldots+a_nx_n+b $$ for some ...
2
votes
1answer
158 views

Arrangements of hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form $$ f(\bar{x})=a_1x_1+\ldots+a_nx_n+b $$ for some $a_i,b\in\mathbb{R}$. Suppose we ...
0
votes
1answer
688 views

What does “Vertex Solution” mean?

Hello! I come across the word "vertex solution" in the context " We can also assume that x and y are vertex solutions,so that the sequence {x,y} remains in a finite set." Could anybody know any ...
0
votes
0answers
70 views

Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...
6
votes
1answer
696 views

Inverse of a totally unimodular matrix

A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$. A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
2
votes
1answer
437 views

Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k . I am wondering if anyone can give me more information about a ...
2
votes
0answers
44 views

Put positive polynomial in finite intersection of half-spaces

This is a cross-posting of a MSE question (which did not attract any attention there so far). Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, ...
2
votes
1answer
93 views

Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?

In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
1
vote
1answer
361 views

Reference Request for Integer factorization with LP/ILP

Have there been attempts to factor integers with Linear Programming? Searching the internet suggests that for Integer Factorization only Number Theoretic algorithms, like sieves, are taken into ...
1
vote
1answer
332 views

Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized. ...
0
votes
2answers
532 views

Efficient algorithm finding 'a' solution of system of linear inequalities

I'm working on rational number field $\mathbb{Q}$. Is there an efficient algorithm finding a solution of system of linear inequalities? In many computer algebra systems like Sage or Maple, there ...
1
vote
2answers
118 views

complexity of finding optimal matchings of given fixed size

It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...
1
vote
0answers
659 views

Robust optimization in matlab using fmincon [closed]

I am trying to implement the following optimization (from this paper) in Matlab using fmincon: $\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$ where $\Sigma$ is a positive definite ...
0
votes
1answer
429 views

Find edge weights that fit given node weights

Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes. Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbitrary given node ...
1
vote
1answer
900 views

Schur complement and negative definite matrices

Hello, My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc} A & B\\\ B^T & C \end{array}\right) $. According to the lemma $M\geq0$ iff $C>0$ ...
10
votes
2answers
804 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 ...
1
vote
0answers
100 views

Matrix Minimax problem

I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The ...
3
votes
1answer
175 views

How to implement linear constraints that include several absolute values

Dear all, I am trying to implement a linear constraint that includes several absolute values in the form: Abs(A) + Abs(B) + Abs(C) + Abs(D) + ... = 1 Since the minimization problem includes quite a ...
0
votes
1answer
690 views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...