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

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0
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35 views

Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space. I am interested in informative examples and applications of such systems. I know ...
0
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0answers
25 views

Connection between maximal regularity and optimal control [on hold]

I would like to know the connection between maximal regularity and optimal control of some equations. I do not find any related topic on internet. Anyone can help me for this question ?
-2
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0answers
31 views

Correct definition of submodularity [on hold]

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...
1
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0answers
12 views

Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions. I've a univariate nonlinear function y=f(x). where f(x) ...
2
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0answers
45 views

Is there a name for this variant of the MST and the TSP?

Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...
0
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0answers
18 views

Bounding the error of the optimal solution from an approximated objective

Let $f$ be a smooth convex function defined in a bounded region $X$ and its Hessian is bounded $m I \le |\frac{\partial^2 f(x)}{\partial x^2}| \le M I$ for some $M>m>0$. Let $x^*=argmin_{x\in X} ...
0
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0answers
63 views

Hessian of composite function

Given a (globally, but not necessarily locally) surjective smooth map: $V:\mathbb{R}^n \rightarrow SU(4)$ (with $n >> 4$) and the function $J_{G}: SU(4) \rightarrow \mathbb{R}$ defined by: ...
2
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0answers
44 views

Are singular critical points isolated for control systems on comapct semi-simple Lie groups

Given a control system on $SU(n)$ (or any other compact, semi-simple Lie group I suspect) of the form: $\frac{d U_t}{dt} = (A + w(t)B)U_t$ where $A,B \in \mathfrak{su}(n)$ generate the algebra (to ...
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0answers
23 views

solution of an infinite horizon optimization problem

Give the following formulation: $\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$ $s.t. ...
0
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1answer
52 views

Solving a nonlinear optimisation problem

I have the following nonlinear optimisation problem arising in my model. $$\min \sum_{k=0}^{N-1} (\tau-t_k)^+\quad \text{ s.t. } {\mathbf{x}^\top\mathbf{w}\le W,\ \mathbf{x}\ge0}, t_k=t_{k-1}+x_k ...
1
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2answers
168 views

The term for problems “like” Brachistocrone?

Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be ...
1
vote
2answers
75 views

Orthogonal Procrustes problem with Absolute Values

Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising $$ \left\lVert \, |O| - T \right\lVert_F,$$ where $\lVert \cdot \lVert_F$ denotes the Frobenius norm, and $| \cdot |$ ...
2
votes
0answers
85 views

Solve non-linear Optimization Problem [closed]

I have to find $x$ that minimizes: $$ \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ...
1
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0answers
158 views

A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...
0
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0answers
21 views

Maximizing modular function subject to supermodular constaint

I'm trying to solve a constrained optimization problem with submodular functions and get some nice properties of the solution. Unfortunately, I think I am in a setting where Topkis' theorem does not ...
3
votes
1answer
117 views

Singular curves of affine distributions on a Lie group

Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group? Specifically the case of a right invariant affine distribution: $D_{U} = ...
2
votes
0answers
65 views

Learning rule for recurrent neural network with flexible time steps

Summary: I want to train a recurrent network to output some answers, but the recurrent network is allowed to re-iterate through itself a flexible number of times for each input-output pair. Why this ...
-1
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0answers
24 views

Duality for Generalization of standard Convex

in accordance to the previous question about KKT condition for generalization to standard convex, here I look for the dual problem to the generalized convex problem. the clear questions are : is it ...
2
votes
0answers
46 views

Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation: $\frac{d x(t)}{dt} = F(x,u)$ one can consider the ...
1
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0answers
71 views

A (non-convex) minimization quadratic programming problem with d constraints

Minimize $0<\omega_{dd}<2$ subject to $$\sum_{j=1}^{d}(\omega_{dj} - \omega_{ij})^{2} \geq 4, i=0,1,...,d-1,$$ where $-2<\omega_{ij}<2$ is known for $0 \leq i \leq d-1$ and $1 \leq j \leq ...
1
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0answers
92 views

For of a special case of $Ax>=b$, are there always integer solutions?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$. The output is a matrix $A(n\times n)$ ...
4
votes
2answers
89 views

Questions concerning convergence rate of Iterated Projections

Assume for simplicity $C_1,C_2,...,C_n\subset \mathbb{R}^m$ to be closed and convex subsets with $\underset{i=1}{\overset{n}{\bigcap{}}}C_i\neq\emptyset$. Let $x^0\in\mathbb{R}^n$ and define the ...
2
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0answers
51 views

Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
1
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0answers
66 views

Fréchet differentiability of functional defined by a integral [closed]

I want to prove that if the functional $I: \mathcal{C}^1[t_0,t_f] \rightarrow \mathbb{R}$ defined by $$ I(x) = \int_{t_0}^{t_f} F(x, \dot{x},t)\,dt $$ is Fréchet differentiable if $F$ is ...
2
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0answers
70 views

About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...
5
votes
2answers
122 views

The use of modules in control theory

So far I have seen the use of vector spaces in control theory and other notions from linear algebra; So I wonder if there's a use of this abstraction of modules over rings in control theory? any ...
0
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0answers
55 views

Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function: $$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$ such that ...
3
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1answer
41 views

Parameter uncertainties in LQR

I have a question about LQR. I apply optimal controller by solving Ricatti equation based on normal plant. Suppose that I have one or few parameter variations in the plant that changes some values in ...
3
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2answers
121 views

Level sets on $SU(4)$

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$? Can they be written only in terms of abstract linear maps, not ...
2
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1answer
204 views

Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves: $\frac{d U_s}{ds} = (a + w(s)b)U_s$ for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...
0
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0answers
39 views

Continuous time dynamic programming: Quadratic guess for value function

In a control problem like so: $$J = min \int_0^{t_f} Qx^2 + Ru^2 dt $$ $$\dot{x} = Ax + Bu$$ $$x(0) = x_0$$ The regular Linear Quadratic Regulator is attained by asssuming that the optimal value ...
10
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2answers
235 views

A generalization of Chebyshev polynomials

What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$. Now suppose ...
0
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0answers
47 views

Nonlinear optimization problem with inequality constraints

Consider a real valued function $g(x_i)=\frac{1}{a_1+ \frac{a_2}{x_i}}, \forall i=\{1,2,3,...,n\}$. The objective function $H$ is $H=\sum_{i=1}^{n}\frac {1}{g(x_i)-a_3x_i}$ The optimization ...
1
vote
1answer
31 views

Finding an unfrustrated set of local linear constraints with given minimal value

Let $ \{F_{i}\} $ be a finite set of linear functionals on a convex compact subset $B\subset\mathbb{R}^{n}$ such that each $F_i$ is $k$-local (acts on $k\ll n$ variables only). Assume ...
2
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1answer
75 views

Do the values of the differential of a function on a Lie group with a single maximum span the Lie algebra?

Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a ...
0
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0answers
42 views

Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix ...
5
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0answers
160 views

Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...
0
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1answer
90 views

Constrained optimal control problem

I am trying to solve an optimization problem which is probably reminiscent of optimal control theory but all of this is not exactly my field of expertize and I am a little bit lost in translation. If ...
0
votes
1answer
108 views

Gradient on $SU(n)$

I'm trying to calculate the gradient (wrt to the bi-invariant metric) of the following functions $F_1, F_2 : SU(n) \rightarrow \mathbb{R}$ defined by $F_1(U) = | Tr (G^{\dagger} U) |^2$, $F_2(U) = \Re ...
0
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0answers
47 views

Finding a movement taking out of a convex set

There is a convex set $S$ as the hull of M points in an D-dimensional Euclidean space and a point $\vec P$ in the set. Then, there is a set of vectors $\vec W$ taking the form $\vec W=\sum_{i=1}^N ...
2
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1answer
75 views

Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...
4
votes
1answer
103 views

Basic result in semi-infinite linear programming

Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
0
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0answers
40 views

Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...
4
votes
2answers
94 views

Convex optimization with full subdifferential information

Can anyone direct me to any algorithms or theorems that describe the difficulty of solving a non-smooth convex optimization problem for the special case where the full subdifferential is available? ...
0
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0answers
97 views

Duality results for quadratic equality constrained optimization problem

Consider the optimization problem $$ \begin{align} \min_{x\in\mathbb{R}^n}&\quad f(x) = x^TA_0x+b_0^Tx+c_0\tag{P1}\\ \nonumber \text{subject to } \quad & g_i(x) = x^TA_ix+b_i^Tx+c_i = ...
13
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4answers
431 views

Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant $$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$ over all $a_0,\dots,a_{n-1}$ such ...
6
votes
2answers
225 views

Do computational geometers use Lagrange multipliers?

Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual ...
2
votes
1answer
203 views

Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like $$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$ Here $f(k,x)$ is actually a probability coming from a ...
0
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0answers
99 views

Maximizing Expected Utility

I am currently trying to solve a maximization problem given by $\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$. Or in other words, I have a utility ...
0
votes
1answer
173 views

Prove that the following two optimization problems are equivalent

I am trying to solve the following optimization problem for the vector $ y $, where $ A_i $ are some given matrix (maybe low rank) and $ x_i $ are unconstrained $$ \min_{y, x_i} \sum_{i=1}^J || y - ...