**0**

votes

**1**answer

215 views

### A problem about a matrix norm on $\mathfrak{su}(4)$

Given a fixed $B \in \mathfrak{su}(4)$ is it possible to solve for $F$:
$\sigma^{\text{max}}\left(\frac{A}{F(A)} + B \right) = 1$, $\forall A \in \mathfrak{su}(4)$. A theorem in ...

**2**

votes

**1**answer

36 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 ...

**34**

votes

**1**answer

2k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**0**

votes

**1**answer

285 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

**12**

votes

**4**answers

347 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 ...

**1**

vote

**0**answers

43 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**

votes

**0**answers

31 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 ...

**0**

votes

**0**answers

51 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 = ...

**3**

votes

**1**answer

55 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? ...

**6**

votes

**1**answer

324 views

### An S-lemma for polynomials of degree 4 in three variables

Might the following be true:
Let $p,q\in\mathbb{R}[x,y,z]$ be homogeneous polynomials, with $\deg(p)\leq 4$ and $\deg(q)= 2$. Suppose $q(x,y,z)>0$ for some $x,y,z\in\mathbb{R}$. Then the ...

**2**

votes

**1**answer

126 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 ...

**5**

votes

**2**answers

186 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 ...

**0**

votes

**0**answers

21 views

### Methods for RCPSP

I have an Resource Constrained Project Scheduling Problem (RCPSP) with and additional strict precedence graph $H$, where $(j, j') \in H$ means $j'$ should stay closely after $j$.
Can you advise any ...

**0**

votes

**0**answers

42 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

**1**answer

135 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 - ...

**2**

votes

**0**answers

113 views

### Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem

Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...

**10**

votes

**2**answers

413 views

### Optimal inspection path on a sphere

Suppose you would like to "inspect" every point of a unit-radius
sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$
on $S$, but you can only see a distance $d$ from where you stand.
...

**9**

votes

**1**answer

251 views

### Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...

**15**

votes

**1**answer

2k views

### A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...

**2**

votes

**2**answers

157 views

### constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...

**1**

vote

**0**answers

376 views

### Partial feedback linearization (Control theory)

I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, ...

**1**

vote

**0**answers

47 views

### Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...

**2**

votes

**1**answer

127 views

### Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time ...

**3**

votes

**1**answer

52 views

### Kálmán filter with delayed (outdated) information

We have a linear system with observation as follows:
$x(t+1)=Ax(t)+Bu(t)+w(t)$
$y(t)=Cx(t)+z(t)$
for given information $y(0\sim t)$, we can construct the dynamics of $\hat{x}(t)$ using the Kálmán ...

**2**

votes

**1**answer

111 views

### Largest subgroup of $SU(n)$ for which the adjoint action preserves specific inner product on $\mathfrak{su}(N)$

Given $A,B \in \mathfrak{su}(n)$ such that $K(A, B)=0$, I am looking for the largest subgroup $H$ of $SU(n)$ for which:
$K \left(A, Ad_{U}(B) \right) = 0, \ \ \forall U \in H$ where $K$ is the ...

**2**

votes

**1**answer

84 views

### Functions that are easy to compare to a norm

Let $X$ be a subset of $\mathbb{R}^d$, let $\|\cdot \|_p$ be a norm with $1\leq p\leq\infty$, and let $f:\mathbb{R}^d\to\mathbb{R}$ be a function. I'm trying to find examples of $X$, $p$, and $f$ for ...

**2**

votes

**0**answers

43 views

### LQR solution when there are linear terms in the cost function?

I am trying to solve the following Bellman Equation:
$V(s) =\max_u \left[a'u - (u-s)'Q(u-s) + V(u)\right]$
In the equation above, $s,u,a\in \mathbb R^n$, $Q\in \mathbb R^{n\times n}$ is positive ...

**0**

votes

**0**answers

37 views

### Global optimisation of the real part of impedance

I have the following global optimisation problem:
$$
\underset{\omega}{\min}-c^{T}\left(\omega^{2}\mathbf{1}+A^{2}\right)^{-1}b
$$
where $A$ is a $n \times n$ real matrix, $c$ and $d$ are ...

**3**

votes

**0**answers

111 views

### existence of optimal control

I'm looking for an existential result in optimal control for the following class of problems:
Given $T > 0$, $\bar x, \hat x\in\mathbb R^d$, an instantaneous cost function $c:\mathbb R^d\times ...

**1**

vote

**0**answers

28 views

### Minimizing sum of functions, while keeping their values non-negative

Suppose we have data $\mathbf{x}_i$, $i\in \{1, \ldots, K\}$, and we're trying to find parameters $\hat{\mathbf{\theta}}$ such that
$$\hat{\mathbf{\theta}} = \underset{\mathbf{\theta}} ...

**10**

votes

**0**answers

135 views

### Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...

**4**

votes

**3**answers

552 views

### Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...

**5**

votes

**0**answers

58 views

### Finding the optimal mixture of two convex functions

I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where ...

**0**

votes

**0**answers

27 views

### Complexity of optimizing a bi-objective function with integer constraints

I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...

**5**

votes

**1**answer

166 views

### The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors:
$$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$
where $s_j$'s can only take values of $+1$ or $-1.$ I ...

**4**

votes

**1**answer

75 views

### Analysis of first-order methods for constrained convex optimization with approximate oracles

In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal ...

**11**

votes

**8**answers

2k views

### How do I approach Optimal Control?

Other than learning basic calculus, I don't really have an advanced background. I was curious to learn about Optimal Control (the theory that involves, bang-bang, Potryagin's Maximum Principle etc.) ...

**6**

votes

**3**answers

625 views

### Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...

**2**

votes

**0**answers

121 views

### Fixed area, largest mass — is there a name?

Let $x\in \mathbb{R}^n$ and let $s_k(x)$ denote the sum of the $k$ largest entries of $x$. The function $s_k(x)$ is well-known to be convex and is often used in optimization, such as
...

**0**

votes

**1**answer

90 views

### How to minimize this sparse quadratic function?

There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...

**1**

vote

**1**answer

85 views

### What's the most efficient way to solve this euclidean projection on non-negative affine space constraint?

I've come across this convex optimization problem in my research where I need to project a matrix $X_0$ onto a non-negative affine space constraint and box constraints. Concretely,
$X \in ...

**2**

votes

**2**answers

171 views

### Why the 'S' in S-procedure/S-lemma?

The S-procedure (also called as S-lemma) is a technique from V. A. Yakubovich that is used to relax a system of quadratic inequalities to a linear matrix inequality problem. It is used largely in ...

**1**

vote

**1**answer

48 views

### characterization of a certain closed convex cone

Consider $x_1,\cdots,x_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by
$$\mathcal{K}(\underline{x}):=\{(\varphi(x_1),\cdots,\varphi(x_n)):\varphi \textrm{ convex on ...

**0**

votes

**0**answers

59 views

### Projected Alternating Minimization

Assume that $f(x,y)$ is a non-convex function for $x,y\in \mathbb{R}$. Assume that we want to minimize this function (even locally) with respect to $x$ and $y$ such that $x \in \mathcal{X}$ and $y \in ...

**1**

vote

**0**answers

33 views

### Optimization of a multilinear function over a product of hypersimplices

Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...

**2**

votes

**1**answer

168 views

### Necessary conditions for optimality in Banach spaces

Let $X$ denote the non-negative "orthant" of the Banach space $L^2$ (or whatever you call the set of functions in $L^2$ that are non-negative), and let $C$ be a closed, convex subset of $X$. Let $f$ ...

**2**

votes

**2**answers

284 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**1**

vote

**0**answers

55 views

### Theorems on stochastic Lyapunov function

Let $X_n$ be a sequence of random variables such that
$$P(X_{n+1} \in A|X_m,x_m,m\leq n)= \int_A p(dw|X_n,x_n)$$
It is called a controlled Markov process.
Now, suppose there exist $\epsilon_0$, ...

**3**

votes

**3**answers

3k views

### Maximum flow with negative capacities?

I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...

**3**

votes

**1**answer

411 views

### Do upper-semicontinuous polyhedral multifunctions have Lipschitz continuous selections?

We are interested in the following question (definitions and references are given below):
Main Question: Given an upper-semicontinuous polyhedral multifunction $F:R^n \rightarrow R^m$, is there ...