All Questions
281 questions with no upvoted or accepted answers
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106
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Knot invariants in WZW CFT via Holographic Principle
In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...
- 5,230
1
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0
answers
94
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Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
1
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0
answers
178
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$BMO$ is not reflexive
It is well known that $BMO$ is the dual space of the Hardy space $H^1$, which is the dual space of $VMO$. I believe that $BMO$ is not reflexive, but I am not quite sure that the above information is ...
- 16.2k
1
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36
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Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
- 33
1
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0
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96
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On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
- 61
1
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0
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335
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Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 988
1
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0
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47
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Nash Equilibria change linearly in (some) game parameters. Already known / follows from a more general result?
EDIT: The key thing that I am wondering about is the linearity of the P2 strategy, not the constancy of P1. (The latter is straightforward.)
Question: Is the following result already known? Or is it a ...
1
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0
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64
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Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?
Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
- 199
1
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0
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59
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How do I incorporate Ito's lemma into the solution for a finite-horizon stochastic cake-eating problem?
I'm interested in finite-horizon, continuous-time cake-eating problems in which the agent has a time-horizon $W$ over which to eat the cake, and then chooses an optimal consumption path $\{h_t\}_0^W$, ...
- 11
1
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0
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51
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Application of the $\operatorname{BMO}$, $H^1$ duality
Let $f\in \operatorname{BMO}(\partial \Delta)$, then there exists a Carleson measure $\mu$ in $\Delta$ such that
$$f(\zeta)-\int_{\Delta}P_{z}(\zeta)d\mu(z)\in L^{\infty}(\partial \Delta),\ \zeta\in\...
- 111
1
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0
answers
45
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Transport-type duality for preduals of $C^{k,1}$-functions
Let $\Omega$ be a non-empty, simply connected, and open subset of $\mathbb{R}^d$ for some positive integer $d$. Let $k$ be a non-negative integer. Consider the Banach space $C^{k,1}_0(\Omega)$ ...
- 5,405
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96
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References: properties of $L_p$ spaces involving time and probability space
Questions are from the theory of PDEs\SPDEs
Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...
1
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0
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195
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Gross-Hopkins duality
$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
- 448
1
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1
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331
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Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
- 51
1
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0
answers
61
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Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
1
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0
answers
98
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Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 287
1
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0
answers
35
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How to chose the start vector for the MTZ variables
In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
- 13.2k
1
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0
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58
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Second-order envelope theorem for linear programming
Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
- 373
1
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0
answers
37
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Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
- 11
1
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138
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$L_p$ estimate in mixed boundary problem for elliptic equation
Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...
- 46
1
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0
answers
162
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Optimization problem on trace of complex matrix product
Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$:
$$
\mathrm{arg}\max_X \,\mathrm{trace}(X^...
- 377
1
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0
answers
43
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Detecting non-negativity of a single constraint by polyhedral constraints - $II$
Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
- 13.9k
1
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1
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115
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$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
- 13.2k
1
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0
answers
172
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continuity of linear programming
I have the following conjecture:
Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane ...
- 37
1
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0
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191
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Dual of union of Reproducing Kernel Hilbert Spaces
I have a union of Reproducing Kernel Hilbert Spaces $\mathcal{B}$. I am interested in finding the dual of $\mathcal{B}$. Knowing what the dual is might help to write an alternate formulation for the ...
- 171
1
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0
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920
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Maximizing a piecewise-linear convex function
Crossposted on Operations Research SE.
I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables:
...
- 111
1
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0
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322
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Decomposition of Polyhedral - An example
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
- 111
1
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1
answer
228
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Weak duality sign
Let $\left\{\begin{matrix}
\operatorname{min}_xc^Tx\\Ax\leq b
\\ x\in \mathbb{R}^+
\end{matrix}\right.$ be a LP primal problem and $\left\{\begin{matrix}
\operatorname{max}_yb^Ty\\A^Ty\geq c
\\ y\in ...
- 11
1
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0
answers
68
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Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
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0
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147
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Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
- 53
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0
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82
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What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
1
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0
answers
78
views
Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
1
vote
0
answers
67
views
What are the corners of this polytope?
Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...
- 11
1
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0
answers
231
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Dual space of mean-free Sobolev space
I am considering the space $V:=\{v \in H^1(\Omega): \int_\Omega v = 0\}$ of mean free functions. What is the dual space of this space? Is the dual space given by $D:= \{f \in (H^1(\Omega))^*: \langle ...
- 11
1
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0
answers
245
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Dual varieties and nodal sections
Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ ...
- 828
1
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0
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303
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Continuity of the Legendre transform of a Lipschitz function
Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
- 31
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0
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282
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total unimodularity of a matrix
Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
- 393
1
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0
answers
49
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Weakly reflexive algebra vs proper (residually finite-dimensional) algebra
Currently I am reading the book "Hopf Algebras. An Introduction" by S. Dascalescu, C. Nastasescu, S. Raianu. There is a Definition 1.5.20 on page 44 (boldface is mine):
An algebra $A$ is called ...
- 519
1
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0
answers
163
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Can we reduce the maximization of this integral to the maximization of the integrand?
I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
- 167
1
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0
answers
25
views
Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
- 11
1
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0
answers
43
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How do I find the Rockafellar-Fenchel dual derivation of reguarlized objective function?
I'm stuck finding the dual of this optimization problem:
$$ \max_{\tilde{u},\tilde{v}} \int \left( \frac{1}{2}\left\|x\right\|^2 + \tilde{u}(x)\right) \, ds(x) + \int \left( \frac{1}{2}\left\|y\right\|...
- 383
1
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0
answers
36
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Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
- 19
1
vote
0
answers
126
views
Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
- 1,826
1
vote
0
answers
24
views
Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
- 4,049
1
vote
0
answers
37
views
Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
1
vote
0
answers
74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
1
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0
answers
66
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On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
- 13.9k
1
vote
0
answers
62
views
LP Constraints for Bridgeless Cactus Graphs
When trying to determine the optimal bridgeless spanning cactus graph of a weighted, symmetric graph, I got stuck.
What I do not know how to capture, is
the variable number and sizes of the cycles
...
- 13.2k
1
vote
0
answers
60
views
On the defect of a flow network
This problem in graph theory was actually motivated by some problems in Theory of Fractals.
To formulate the problem I need to recall some definitions related to flow network.
A flow network is a ...
- 41.8k
1
vote
0
answers
86
views
Infinite system of equations with finitely many constraints
During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following ...
- 1,151