All Questions
685 questions
2
votes
0
answers
47
views
Why not use global optimization algorithms like PSO to solve decentralized control problems?
I do not see many works that use global optimization algorithms to solve decentralized control problems. Here the decentralized control problem means some entries of the feedback matrix are ...
- 21
0
votes
0
answers
165
views
Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
0
votes
0
answers
137
views
Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
0
votes
1
answer
131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
1
vote
0
answers
54
views
Nested, successive minimization solved by asympotic minimization?
I am curious about the general relation between nested, successive minimization (M1) and asymptotic minimization (M2) as defined in the following. What one wants is to implicitly minimize a sequence ...
- 390
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781
0
votes
1
answer
213
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 143
1
vote
0
answers
162
views
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 $x$:
$$
\mathrm{arg}\max_X \,\mathrm{trace}(X^...
- 377
2
votes
0
answers
76
views
Polyhedron coordinate bound
Given a polyhedron
$$Ax\leq b$$
where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
- 13.9k
1
vote
0
answers
43
views
Detecting non-negativity of a single constraint by polyhedral constraints - $II$
Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
- 13.9k
0
votes
1
answer
110
views
Detecting non-negativity of a single constraint by polyhedral constraints - $I$
We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
- 13.9k
0
votes
1
answer
116
views
Iterations of Dantzig-Wolfe Decomposition for a Simple Linear Programming problem
This arises from an engineering problem I am working on. Let $\mathbf{c}_i,\mathbf{a}_i,\mathbf{b}_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the ...
- 1,421
0
votes
0
answers
93
views
Number of vertices in a polyhedron
Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
- 13.9k
1
vote
1
answer
115
views
$\mathrm{LP}$ formulation for $\mathrm{k}$-$\operatorname{opt}$ moves
$\mathrm{k}$-$\operatorname{opt}$ moves are an idea to improve non-optimal Hamilton cycles in weighted symmetric graphs by exchanging $\mathrm{k}$ tour-edges with $\mathrm{k}$ edges that do not belong ...
- 13.2k
1
vote
0
answers
172
views
continuity of linear programming
I have the following conjecture:
Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane ...
- 37
1
vote
1
answer
98
views
Optimality gap between a joint linear program and decoupled sub programs
Let $\mathbf{c}_i,\mathbf{s}_i$ be given entry-wise positive $n\times 1$ vectors for $i\in[1,\dots,d]$. Let $\tau, \alpha_1,\dots, \alpha_d$ be given positive constants.
Consider the linear ...
- 1,421
1
vote
0
answers
81
views
Algorithm for deciding feasibility of linear programs [closed]
Suppose I have the simple linear program
$$Ax \geq 0, \quad x \geq 0$$
We know that this system has a solution (for example, $x=0$). But, what if we made this rule for this system?
$$Ax \geq 0, \quad ...
4
votes
1
answer
90
views
Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?
Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
- 167
1
vote
0
answers
920
views
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
0
votes
1
answer
146
views
What to call a function that is negative on a set
Let $Y$ be a nonempty region in $\mathbb{R}^n$. I am designing an algorithm which given a point $x_0$ outside $Y$ in a finite number of steps lead to a point $x_n∈ Y$. The way I do it is that I have a ...
- 679
5
votes
1
answer
242
views
Minimizing the largest eigenvalue of matrix product
Let $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times k}$, $C\in\mathbb{C}^{k\times m}$ be given complex matrices. The objective of the optimization problem is
\begin{equation}
\mathop {\arg \min ...
- 377
16
votes
1
answer
508
views
Can solutions to Thomson's problem have pentagons?
Thomson's problem asks for the minimum-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/...
- 1,203
1
vote
0
answers
322
views
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
$$ \...
- 111
0
votes
1
answer
76
views
A question on graph partitioning
Given a connected un-directed simple graph $G=(V,E)$, is there a polynomial time algorithm to find the smallest subset $S$ of $V$ such that each node in $V \setminus S$ has at least 50% of its ...
- 1,216
0
votes
1
answer
329
views
Gradient-descent "type" Methods for non-convex and non-smooth functions
Most (stochastic) "gradient descent" type algorithms (such as Nesterov-accelerated gradient-descent or ADAM) seem to be well-defined only for functions which are either:
lower semi-...
- 5,405
0
votes
1
answer
159
views
Prove zero slope point is global maximum for constrained function with binomials. Without restriction, objective function is non-concave
How to prove the zero slope point is a global maximum in this non-concave program for a function with binomials?
I need to find the (global) maximum of the following constrained problem:
$$\max_{CAP} \...
- 49
1
vote
1
answer
1k
views
Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball [closed]
Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the ...
- 11
0
votes
1
answer
445
views
Properties of $l_q$-balls
For a given $q\in (0,1]$, define the $l_q$-ball as
$$\mathbb{B}_q(R_q)\mathrel{:=}\left\{\theta\in\mathbb{R}^d\,\middle\vert\,\sum_{j=1}^d \lvert\theta_j\rvert^q\leq R_q \right\}. $$
For a given ...
- 100
3
votes
0
answers
87
views
Additional symmetries of the Traveling Salesman Polytope
Given the complete graph $K_n=(V,E)$, the Traveling Salesman Polytope is a convex polytope in $\Bbb R^E$ obtained as the convex hull of the indicator vectors of (edge-sets of) Hamiltonian cycles in $...
- 13.6k
1
vote
1
answer
157
views
Constructing representations of probability revision functions
Let $P$ be a probability distribution over a finite Boolean algebra $\mathfrak{B}$, and fix a parameter $t_{P} \in (\frac{2}{3}, 1)$. Define the `revision function of $P$', $R_{P}: \mathfrak{B}\...
- 631
1
vote
1
answer
628
views
Allowing an "OR" option between equations in a linear program
I am looking for a way to express an "or" option in a system of linear inequalities for a linear program I am working on.
I will explain what I mean precisely: Lets say I have a set of ...
- 141
2
votes
0
answers
49
views
A question about strong slopes (nonsmooth analysis)
Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
- 6,853
8
votes
2
answers
1k
views
Minesweeper as a linear algebra problem
I've written a computer program to generate and solve minesweeper games. Once I've eliminated the obvious mines and safe squares I look at each remaining connected setsin turn and formulate a linear ...
- 361
0
votes
0
answers
68
views
Convex optimization under asymmetric loss in infinite dimensional space
The following problem is common in financial economics
$$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$
That is, given a random variable $y(\theta)$ ($\...
- 45
4
votes
0
answers
236
views
Fréchet subdifferentiation on riemannian manifolds
Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.
Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \...
- 6,853
2
votes
1
answer
139
views
linear programming with $n$ choose $r$ variables
Given parameters $r < n$, define $m = {n \choose r}$ and let $A$ be the $n\times m$ matrix whose columns are all the vectors with $r$ $1$'s and $n-r$ $0$'s. Let $b$ be a positive $n$-vector. Is ...
- 51
2
votes
1
answer
763
views
Integer solution of optimal transport
Let us consider two vectors $\mathbf{a}=(a_1,...,a_n)$ and $\mathbf{b}=(b_1,...,b_m)$ so that each quantity is an integer $a_i,b_j \in \mathbb{N}$. It represents for example supply and demand. Let $\...
- 407
0
votes
0
answers
36
views
Optimizing upper and lower bounds
Let $L_i:X\rightarrow [0,\infty)$ be continuous (objective) functions defined on a metric space $X$ and suppose that
$$
L_1(x)\leq L_2(x)\leq L_3(x)\qquad (\forall x \in X).
$$
Here, I imagine that $...
- 5,405
1
vote
1
answer
3k
views
How to minimize l1-norm constrained by "infinity norm"
Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m $. I have the following two problems:
P.1.
\begin{equation}
\underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|_1 \\
\text{s.t. } \| x \...
- 13
0
votes
0
answers
242
views
Exponential map and optimization
Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential ...
- 93
0
votes
1
answer
126
views
An otherwise linear matrix equation with the presence of a signum function : reference request
Consider the equation $$\pmb{c}+\text{sign}(G\pmb{c}) = L$$
$\pmb{c}$ is a $n\times1$ matrix.
$G$ is a $n\times n$ matrix which is also positive definite.
matrices $G$ and $c$ are real.
$L$ is a $n\...
- 698
3
votes
1
answer
252
views
Measurable selection for argmin to distance
Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with ...
- 5,405
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
0
votes
0
answers
108
views
Solutions to matrix equations in the non-negative integers
For an integer matrix $S$, and an integer vector $y$, I'm looking for solutions to $xS = y$ where the entries in $x$ are in the non-negative integers.
I've been doing this with Sage's mixed integer ...
- 101
1
vote
0
answers
68
views
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
1
vote
0
answers
147
views
Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
- 53
1
vote
1
answer
149
views
K-means clustering benchmarks [closed]
What benchmarks do you use for evaluating clustering algorithms, especially for evaluating the performance of K-means vs. another algorithm?
I am especially interested in looking at the correctness of ...
- 21
-1
votes
1
answer
103
views
How to solve MILP problem on several linear subspaces
I have a set of close mixed-integer programming problems. More exactly, all the problems share the same set of (binary and continuous) variables, the same set of linear inequality constraints, and the ...
- 39
3
votes
0
answers
136
views
What is the convergence rate of the iterative optimization method?
For the following optimization problem:
$$
\mathrm{min}_{A,B} \|I-A^{T}XB\|^2 + \lambda\|B\|^2,
$$
where $A$ and $B$ are the two variables ($\|A\|^2 \le \rho$ where $\rho$ is a constant, e.g. 1), the ...
- 31
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