All Questions
281 questions with no upvoted or accepted answers
2
votes
0
answers
43
views
Weak relaxation of a strongly lower semi-continuous functional
Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak ...
- 5,405
2
votes
0
answers
46
views
Notion of distance between linear programs
Consider the linear programming problem
\begin{align}
\max_{x}&~c^Tx \\~s.t.~~a^Tx &\leq B~,~0\leq x_i \le1
\end{align}
where $c$ and $a$ are $n \times 1$ given non-negative vectors. $B$ is a ...
- 1,421
2
votes
0
answers
166
views
Do Poincaré duality algebras need to be defined over a field?
I asked the below question here on MSE, but after some time and a bounty offering I have not received an answer.
A graded commutative, connected $\mathbb{k}$-algebra $A$ is called a Poincaré duality ...
- 208
2
votes
0
answers
66
views
Proving the existence of a dual for an infinite linear program
I am concerned with proving the existence of the dual of an infinite linear program. In addition to the writings of Rockafellar, Luenberger, and Boyd & Vandenberghe on: subdifferentials, Legendre-...
- 121
2
votes
0
answers
47
views
A linear program where coordinate descent works pretty well
I am working with a polytope $P\subset \mathbb{R}_+^n$ with the property that there are at about $n!$ minimizers of $\sum_{i=1}^n x_i$, in the following sense:
Select any coordinate $j$ and set $...
- 21
2
votes
0
answers
66
views
Reference request: Matroid cryptomorphisms for arbitrary monomial ideals
For a matroid $M$ let $C$ be the circuit ideal of $M$, that is, the Stanley-Reisner ideal of independence complex of $M$. Then there are simple ideal-theoretic operations that take $C$ to the facet ...
- 984
2
votes
0
answers
148
views
Generalization of Farkas' Lemma to Hermitian Matrices
I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
2
votes
0
answers
48
views
Nondegenerate linear maps functorially associated to algebras
In the sequel, "$k$-algebra" means "associative unital finite dimensional $k$-algebra.
Apologies for the very long exposition.
If $A$ is a $k$-algebra and $s:A\to k$ is $k$-linear, we say that $s$ ...
- 1,766
2
votes
0
answers
122
views
A new topology on the dual of a locally convex space?
Working with the separable quotient problem for locally convex spaces we (with Saak Gabriyelyan) arrived to an interesting topology on the dual of a locally convex space and we would like to know if ...
- 41.8k
2
votes
0
answers
158
views
The dual of the space of continuous sections in a vector bundle
If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...
- 5,407
2
votes
0
answers
42
views
Dual representation of problems involving $f$-divergences
Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems.
Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance ...
- 6,853
2
votes
0
answers
283
views
Derivative with multiple summation operators
I have a defined utility function as Eq.(1), and I am seeking the minimized utility subjects to some constraints. The notation used is as following:
\linebreak
$V$ is the set of nodes, $v_i\in V$; $O$...
- 21
2
votes
0
answers
43
views
Partitioning $n$-space based on linear combinations
I'm trying to figure out the approximate number of areas the positive $n$-space will be divided into if we partition it as follows: we have $k$ linear functions $F_1$, $F_2$, ..., $F_k$ on $n$ ...
- 21
2
votes
0
answers
2k
views
How to find a positive solution to an under-determined linear system (if such a solution exists)?
Like the title says, if an under-determined system of linear equations does have at least one positive solution, how to find it efficiently?
Suppose we have an under-determined system:
$$Ax = b$$
...
- 121
2
votes
0
answers
80
views
Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector
I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
- 123
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
2
votes
0
answers
344
views
Linear programming with an infinite matrix
I would like to solve the following infinite linear system subject to $x_i \ge 0$ that minimizes $x_3$.
The third column contains no additional nonzero values beyond what is shown. Though the first ...
- 283
2
votes
0
answers
126
views
Unveiling hidden structures
One way to unveil a hidden structure of a undirected graph - given as an adjacency matrix - is to permute the rows and columns until a pattern with a maximal geometrical symmetry is found. (The ...
- 9,720
2
votes
0
answers
299
views
Practical application of envelope theorem for linear programs
Assume that we have solved a (standard) linear program
$$
\text{minimize}_{x\in {\mathbb R^n}}\,\, c_0^Tx, \,\,\,\,\, \text{s.t. } A_0x \leq b_0,
$$
and would like to know how sensitive is the optimal ...
- 7,182
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 909
2
votes
0
answers
71
views
Existence of probability distribution satisfying upper/lower bounds on events
Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
- 21
2
votes
0
answers
167
views
Conditions under which the dual function is self-concordant
Consider the following optimization problem
\begin{align}
\min_{x}&\quad f(x)\\
\nonumber \text{subject to } \quad&h_i(x) = 0,\,i=1,\ldots,m\\
\nonumber \quad&x\in X\subseteq\...
- 179
2
votes
0
answers
167
views
Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$
Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
- 1,981
2
votes
0
answers
177
views
Formulating shortest path as submodular minimization
I'm curious about the general question of whether any combinatorial optimization problem with polynomial time solution can necessarily be reformulated as minimizing a submodular function.
The answer ...
- 21
2
votes
0
answers
154
views
Listing all Lattice Points in a Box
Let $B := [-1,1]^n$ be an $n$-dimensional box. Moreover, let $v_1,\ldots,v_n \in \mathbb{R}^n$ form a basis of $\mathbb{R}^n$, where the entries of the $v_i$ are explicitly irrational. We can assume ...
- 173
2
votes
0
answers
210
views
Finding optimal linear transformation for intersection of convex polytopes
I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) ...
- 695
2
votes
0
answers
93
views
Do copairings provide dualities in derived categories?
Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
- 54.6k
2
votes
0
answers
148
views
Derivation of gradient of SSE in Geodesic Regression
On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
- 5,405
2
votes
0
answers
128
views
Local duality for abelian varieties
Let $A$ be an abelian variety over a p-adic field $K$. Let $I$ be the inertia group of $K$. There is a Yoneda pairing $$H^n(\hat{\mathbb{Z}},A^I) \times Ext^{2-n}_{\hat{\mathbb{Z}}}(A^I,\mathbb{Z}) \...
- 213
2
votes
0
answers
71
views
Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M
this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :)
Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...
- 21
2
votes
0
answers
185
views
Is the isomorphism between $BMO/\mathbb{R}$ and $(H^1(\mathbb{R}^n))^{\star}$ isometric?
Let $BMO$ the space of bounded mean oscillation functions on $\mathbb{R}^n$ equipped with the Lebesgue measure. If $Q\subset \mathbb{R}^n$ a cube, let $m_Q f$ the average of a function $f\in L^1_{loc}(...
- 1,616
2
votes
0
answers
91
views
Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
2
votes
0
answers
356
views
Reference Request: Algebraic Serre's Duality Theorem for Curves
Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne (...
- 440
2
votes
0
answers
90
views
Poincaré Duality of a quasi-free algebra
I'm completely stumped on this one (yet I feel it is obviously true or obviously false)
If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality?
All i need to find is a protective ...
2
votes
0
answers
120
views
integrality of a linear program -- binary equality constaints
Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...
- 127
2
votes
0
answers
39
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 ...
- 21
2
votes
0
answers
163
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 ...
- 501
2
votes
0
answers
252
views
compact embedding in duals of weighted Sobolev spaces
On the whole space $\mathbb{R}^d$ consider the weight $\omega(x)=\sqrt{1+|x|}$. Under which conditions on $k,q$ is the embedding
$$
L^p(\mathbb{R}^d,\omega(x)dx)\subset\subset (W^{k,q}(\mathbb{R}^d,\...
- 5,380
2
votes
0
answers
63
views
Put positive polynomial in finite intersection of half-spaces
This is a cross-posting of a MSE question (which did not attract any attention there so far).
Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, ...
- 621
2
votes
0
answers
230
views
Consistency of a system of linear equations
I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...
- 221
2
votes
0
answers
917
views
Guessing game with guess cost
This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
- 4,952
2
votes
0
answers
319
views
Generalized fourier transform and convolution?
Let $a(t)$ and $b(t)$ be two equal length sequences indexed by time index $t$.
We know that $a(t) * b(t)$ corresponds to $A(\omega) \odot B(\omega)$ in the frequency domain where $A(\omega)$ and $B(\...
- 13.9k
2
votes
0
answers
215
views
Number of breakpoints in parametric maximum flow problems
The parametric maximum flow problem can be formulated as
$$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right),
$$
where all $c_{ij}<0$ (so that ...
- 567
2
votes
0
answers
281
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 ...
- 567
2
votes
0
answers
5k
views
A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
1
vote
0
answers
28
views
Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
- 21
1
vote
0
answers
58
views
duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 988
1
vote
0
answers
126
views
Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
1
vote
0
answers
123
views
Duality in Spc with ∧ and [-,-]
I am thinking about two duality theorems for H-spaces and their actions.
By H-space is meant a commutative monoid in the derived (homotopy) category of based connected CW-complexes. We can consider ...
user30211