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Questions tagged [oc.optimization-and-control]

Operations research, linear programming, control theory, systems theory, optimal control, game theory

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Expressing a double Riemann Sum as a definite integral

I am reading a paper where the authors rewrite the following expression (for a continious function $\alpha$) into an integral: $$\lim_{n\to \infty} \left(\min_{1\leq j \leq n}\left(\sum_{k=1}^{j-1} \...
AspiringMat's user avatar
3 votes
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90 views

Schrödinger Bridge for other costs

Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23) $$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t ...
nico's user avatar
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Leech lattice shortest vector vs other 23 cases and E8 cases

In this paper by Viazovska, she said that: "The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at the lattice points and radius $1/\sqrt{2}$." So I think ...
zeta's user avatar
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Curve fitting with "rough" loss functions

Many real-valued classification and regression problems can be framed as minimization in the following way. Setup: Let $\Theta \in \mathbb{R}^p$ be the parameter space that we are searching over. For ...
Simon Kuang's user avatar
5 votes
1 answer
177 views

Upper bound for an inverse Laplace transform

Can anyone see how to get a tight upper bound for the function defined in terms of the inverse Laplace transform below? $$g_\text{sgd}(t)=\mathcal{L}^{-1}\left[\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{...
Yaroslav Bulatov's user avatar
1 vote
0 answers
41 views

On Designing Some Optimal Control Problems

In the context of a dynamical systems, some states may not be attainable with scalar controls from $L^1(0,T)$, but they may be reachable with feedback controls. If we know that the system is null ...
elmez's user avatar
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The existence of an optimal distributed control problem

Consider $\Omega$ as a bounded interval of $\mathbb{R}$, and let $y\in L^{\infty}(\Omega \times (0,T))$ be a mild solution of the following parabolic partial differential equation: \begin{equation}\...
elmez's user avatar
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Maximisation of a convex (quadratic) function

This post is a continuation of A variant of (discrete) optimal transport problem For $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$, $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ and $...
Fawen90's user avatar
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3 votes
1 answer
85 views

Condition for 3×3 block matrix to be stable

Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...
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How to solve $\min_{\mathbf{x}\in \{\pm 1\}^N} \lVert \operatorname{sign}(\mathbf{Wx}) - \mathbf{y} \rVert_2^2$ where $\mathbf{y}\in \{\pm 1\}^N$?

Given the matrix $\mathbf{W} \in \mathbb{R}^{N \times N}$ and the vector $\mathbf{y} \in \{\pm 1\}^N$, how to solve $$\min_{\mathbf{x}\in \{\pm 1\}^N} \left\| \operatorname{sign}(\mathbf{Wx}) - \...
user3750444's user avatar
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Relation between the left-dominant eigenvectors (eigenvector corresponding to 0 eigenvalue) of two Laplacian matrices

Let $G_1=(v,\epsilon_1)$,$G_2=(v,\epsilon_2)$ be two graphs with the same set of vertices and $\epsilon_1 \subset \epsilon_2$. $L_1$ and $L_2$ be the Laplacian matrices associated with graph $G_1$ and ...
SREEJEET MAITY's user avatar
1 vote
2 answers
179 views

How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$

Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that \begin{equation}\tag{1}\label{1} \int_a^b \...
Meowdog's user avatar
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Can a real quartic polynomial in two variables have more than 4 isolated local minima?

This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far. Finding examples of 4 ...
Jap88's user avatar
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Derivative in Sobolev space extended by zero

Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero. How to find $J'(u)$ for $$ J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;?? $$ In $L_2$ it's easy: $$ J'(u) = \left(\...
anon.for's user avatar
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1 answer
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Constrained linear optimization problem on $C^1$

I am dealing with a problem of the form ($a<b$) $$ \displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
Meowdog's user avatar
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1 answer
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Is non-convex optimisation really in NP class?

Crossposted on Mathematics SE I've seen in many optimisation papers the statement that general non-convex optimisation problem is NP-hard. If we assume that non-convex optimisation is in NP class, it ...
Dmitry Vilensky's user avatar
2 votes
0 answers
71 views

Differential inequality with convex constraint

The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me. Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\...
Denis Serre's user avatar
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1 answer
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Existence of minimal Steiner tree

I am looking for a reference proving the existence of the minimal Steiner tree in the Euclidean Steiner tree problem: Given N points in the d-dimensional Euclidean space, the goal is to connect them ...
user72829's user avatar
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3 votes
1 answer
64 views

Sensitivity of the solution of QP with respect to parameters

Given a quadratic program, $$\begin{array}{ll} \text{minimize} & \displaystyle \frac12 x^TAx + b^Tx \\ \text{subject to} & Cx \le d \end{array}$$ Suppose $A \succ 0$, so the program strongly ...
gcy's user avatar
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On a stochastic control problem

Let $(\Omega,\mathcal F, \mathbb P)$ be a probability space on which a Brownian motion $W$ is defined, and $\mathcal U$ be the set of progressively measurable (w.r.t. the Brownian filtration) ...
Fawen90's user avatar
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Traveling Salesman: Optimization over cities, not distance

In the classical traveling salesman problem, we are given a graph of cities with distances between each city and are asked to find the shortest path that traverses all of the cities. Meaning that the ...
MathManiac5772's user avatar
1 vote
0 answers
63 views

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) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
shahulhameed's user avatar
1 vote
2 answers
49 views

Monotonicity of kernel matrices with respect to hyperparameters

Let $\mathcal{X}$ be some nice space, let $\Phi$ be some ordered space, and let $K :\mathcal{X} \times \mathcal{X} \times \Phi \to \mathbf{R}$ be a positive-semidefinite kernel indexed by a ...
πr8's user avatar
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1 vote
1 answer
66 views

An inequality relating $\ell_1$ distance of input and output of a Markov krnel

Let $K$ be a Markov kernel from $\mathcal{X}$ to $\mathcal{Y}$, i.e., $K(\cdot|x)$ is a probability measure on $\mathcal{Y}$ for all $x\in \mathcal{X}$. Let $\mu$ and $\nu$ be two probability measures ...
math-Student's user avatar
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How to find a specific clique cover set?

Let $G(\mathcal{V},\mathcal{E})$ be a graph with vertex set $\mathcal{V}$ and edge set $\mathcal{E}$. Also, non-negative weights $w_i$ are assigned to each vertex $i\in\{1,\ldots,n\}$. Suppose the ...
Math_Y's user avatar
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A property of quadratic optimization solution map

Define for a given $\mathbf{b}$ the quadratic optimization solution map by $$ \mathcal{S} \ \colon \ U \mapsto \mathbf{x} \ \colon \ \mathbf{x}^\top \, U^{-1} \, \mathbf{x} = \min_{\mathbf{y} \geq \...
tsnao's user avatar
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1 vote
1 answer
146 views

Role of verification theorems in stochastic optimal control?

I am currently working on the optimal control of certain classes of stochastic processes and I have difficulties understanding the roles of verification theorems. My problem is the following: I am not ...
KBS's user avatar
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1 vote
1 answer
71 views

Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
Akira's user avatar
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0 votes
1 answer
97 views

Is there a redundant constraint in linear programming? [closed]

From wikipedia: But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice). (In order to do that, ...
Bipolo TheGod's user avatar
1 vote
1 answer
92 views

Some calculations with polynomials in the proof of the Routh-Hurwitz test

In an article on the Routh-Hurwitz test, I couldn't see why the following result is true: Let $$p(s)=a_ns^n+a_{n-1}s^{n-1}+\dots+a_{1}s+a_0$$ and $$ \eta_{*} = \frac{a_n}{a_{n-1}}$$ $$g_{\eta}(s):=p(...
Manoel Reis's user avatar
13 votes
2 answers
551 views

How to characterize the regularity of a polygon?

In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
Caio Tomás's user avatar
1 vote
0 answers
29 views

Bounds on Eigenvalues After Skew-Symmetric Perturbation

Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum: $$\mathbf{A} = \...
Leo's user avatar
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2 votes
1 answer
173 views

Nash equilibrium at another level

This is a variant of the Nash equilibrium. Let's say that there are 3 prizes: A Ferrari, a diamond watch, and a new boat. There are 6 players. 3 players with a motive while 3 players with another ...
Hass Boyouk's user avatar
6 votes
3 answers
881 views

Proof of a matrix implication

If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$. I have observed by considering many examples of $x,y,z,...
BAYMAX's user avatar
  • 51
1 vote
1 answer
119 views

Optimal transport: the existence of an optimal pair of $c$-conjugate functions

$\newcommand{\diff}{ \, \mathrm d}$ Let $X,Y$ be Polish spaces, $\mathcal C_b(X)$ the space of all real-valued bounded continuous functions on $X$, $\mathcal P(X)$ the space of Borel probability ...
Akira's user avatar
  • 713
0 votes
0 answers
19 views

Is there an approximation algo for packing interlinked 3d shapes?

Is there an approximation algorithm for packing interlinked 3d shapes? This feels like it should be possible to approximate, but I'm not sure if it has been researched, or if it has, what it would be ...
Benjamin H's user avatar
1 vote
3 answers
223 views

How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?

Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix: $$ \min_{s\in\...
Alec Jacobson's user avatar
1 vote
0 answers
133 views

How to maximize certain function of hundreds variables related to correlations between sets vectors ? (and win Kaggle :))

It might be helpful for data science/bioinformatics challenge. Consider for simplicity three rectangular matrices $Y_{true}$ , $Y_{predict0},Y_{predict1}$ of the same sizes say 70000*140. Let us ...
Alexander Chervov's user avatar
1 vote
0 answers
23 views

Convergence of a sequence of minima of a family of optimal control problems

Let $(\lambda_k)$ be an unbounded and increasing sequence of real numbers, and $f:\mathbb{R}^n \ \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be sufficiently smooth such that $\dot{x}(t) = f(x(t),u(t)...
Tadashi's user avatar
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0 votes
1 answer
60 views

How quickly can this IQP or its MILP relaxation be solved

Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem: $$\begin{align*}&&\max_{P\in\{...
alosc's user avatar
  • 71
1 vote
0 answers
57 views

A stochastic control problem with costly information

Motivation: Nate is holding an asset he knows is overpriced. At the moment there is an upward trend due to hype, but he is aware that at some point in time, the bubble will burst, leading to a crash ...
Nate River's user avatar
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1 vote
0 answers
93 views

How to solve the following optimization problem?

Let $G=(V,E)$ be a connected network with $|V|=n$. Consider the following optimization problem I'm trying to know under which conditions the following minimization problem has solution : $${\sum _{i=1}...
Goga's user avatar
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10 votes
0 answers
560 views

A simple stochastic game

An individual, henceforth called the runner starts at the center of an open two dimensional square $\Omega$ of side length $r \geq 2$. At each turn, a vector $x \in S^1$ is chosen uniformly at random, ...
Nate River's user avatar
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0 votes
0 answers
30 views

Matrix optimization to find ideal embedding

Basically, I am trying to find the embeddings so I can approximate $K \approx M(\vec{\phi})$. The embeddings are for each one of my samples $\vec{\phi}(x_i) \in \mathbb{R}^D$ so I thought it should ...
Hamza Errahmouni Barkam's user avatar
0 votes
1 answer
95 views

Method for (binary) optimization under constraints

I would like to know if there is a method to solve the Problem. Problem: Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
kris's user avatar
  • 3
1 vote
0 answers
91 views

Can I minimize a mysterious function by running a gradient decent on her neural net approximations? [closed]

A cross post from on AI StackExchange. So I have this function let call her $F:[0,1]^n \rightarrow \mathbb{R}$ and say $10 \le n \le 100$. I want to find some $x_0 \in [0,1]^n$ such that $F(x_0)$ is ...
Vladimir Zolotov's user avatar
3 votes
2 answers
209 views

$\min(\det(\mathbf{A}))$ for special matrix $\mathbf{A}$

(The construction of matrix $\mathbf{A}$ is not difficult to be understood. You can first jump to A Toy Example to take a glance. Any idea or suggestion would be appealing for me.) The Original ...
BinChen's user avatar
  • 133
1 vote
0 answers
31 views

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 $\{h_t\}_0^W$, ...
C_A_Pepe's user avatar
0 votes
0 answers
21 views

Feasibility of a basis of a linear program and a relaxed subproblem

I tend to assume that given $$\min c^\top x \\\ Mx=q$$ $$ \text{where} \\ q = \begin{pmatrix}b \in \mathbb R^m\\d \in \mathbb R^n\end{pmatrix},\\M \in \mathbb R^{(m+n)×2n},\\M=\begin{pmatrix}A\\Q\end{...
someone random's user avatar
1 vote
1 answer
57 views

Given a set of vectors how to pick $M$ such that sum of maximums of coordinates is maximized?

I asked the same on math.Stackexchange. I have $n$ (say $n = 300$) vectors $v_1,\dots,v_n$. Each of them has $K$ coordinates (say $K = 30$). For vector $v_j$ I denote it's coordinates as $v_{j1},\...
Vladimir Zolotov's user avatar

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