Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

**0**

votes

**0**answers

23 views

### How can I solve a constrained optimization problem with a random number of decision variables?

Here is my problem. Let $A_t$ be a random variable with Poisson-Binomial distribution with set of success-probabilities $\{q_1^{(t)},\ldots,\,q_n^{(t)}\}$, with $t\in\{1,\,2,\,3,\ldots\}$, $n\in\...

**1**

vote

**0**answers

65 views

### Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?

Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...

**0**

votes

**0**answers

23 views

### Closed-form solution of quadratically constrained quadratic program in 2 unknowns

I am interested in a closed-form solution of the following problem in two unknowns:
\begin{equation}
(\bar x, \bar y) = \text{arg}\min_{(x, y) \in \mathbb R^2}
\left\{
(x, y)^\top
\begin{bmatrix}
a ...

**1**

vote

**1**answer

59 views

### Takahashi minimization theorem for lower pseudo-continuous functions on complete metric spaces

Takahashi minimization theorem says : Let $(X,d)$ is a complete metric space, $f:X\to \mathbb{R}\cup\{+\infty\}$ is a proper(not constantly +$\infty$) lower semi continuous function, which is bounded ...

**1**

vote

**1**answer

72 views

### Closed Form Solution for Optimization Problem over the Space of Rigid Transforms

Is there a closed form solution to this constrained optimization problem:
\begin{equation}
\min_{R \in SO(3),\, \mathbf t \in \mathbb R^3} = \sum_{i = 1}^N \| M_i(R \mathbf p_i + \mathbf t) \|^2,
\...

**2**

votes

**1**answer

41 views

### Are the intersection of proximinal sets in a Hilbert Space proximinal?

Let $X$ be a real Hilbert Space and $C \subseteq X$. Let $d_C$ be the infimal distance function to $C$ and $P_C(x) = C \cap S[x; d_C(x)]$ be the metric projection. We say $C$ is proximinal if $P_C(x) \...

**6**

votes

**1**answer

137 views

### Adding constraints as penalty with $\| \cdot \|_0$ norm

In the paper Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries (page 2), the authors rewrite the minimization problem
\begin{align}
\min_{\alpha \in \mathbb R^k} \| \...

**6**

votes

**0**answers

80 views

### another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$
In 1984 S.D.Berman, a Soviet mathematician, ...

**0**

votes

**0**answers

28 views

### how to get this deterministic equivalent formulation of its original probabilistic counterpart by knapsack constraint?

I'm reading this article with title "a probabilistic model applied to emergency service vehicle location". https://www.sciencedirect.com/science/article/pii/S0377221708002336
This is a very good ...

**3**

votes

**1**answer

143 views

### it's convex sequence inequality

A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...

**0**

votes

**0**answers

34 views

### a game-theoretical / optimization problem

Does anyone know how to solve (or a good reference) for the following problem?
Let n and m be two positive integers. Let z \mapsto G^{i,j}(z) be a real polynomial, for each i=1,...,n and j=1,...,m. ...

**2**

votes

**1**answer

91 views

### Convergence of a stochastic sequence?

I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...

**0**

votes

**1**answer

48 views

### Quasiconvexity property of quasinorms

Schatten $p$ norm is convex when $p\geq1$ holds and if $p\in(0,1)$ it is quasinorm.
If $p\in(0,1)$ then is Schatten $p$ norm quasi convex? I am interested in definition of quasi convexity here https:...

**2**

votes

**0**answers

49 views

### How to solve such integer program problem?

Consider a $3$-tuple $(a,b,s)$ with $a,b\in\mathbb{Z}_+,s\in\mathbb{Q}_+$. Denote $ab-s$ by $\Delta$. Let $A$ be a positive number. What are the values of $A$ such that for any $(a,b,s)$ with $\Delta\...

**0**

votes

**0**answers

40 views

### Sensitivity of Lagrangian solution: implicit constraint

just a question about a literature reference. I am writing a paper for engineers.
Usually for the Lagrange multiplier problem ∇f(x)+λ∇g(x)=0
the sensitivity result that the multiplier λ gives the ...

**0**

votes

**0**answers

14 views

### Efficient numerical optimization of an “almost separable” function

I have come across an optimization problem with the following objective function:
$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))...

**0**

votes

**0**answers

49 views

### Using mollifiers (or some other idea) to solve constrained minimax problem

Sorry in advance if this sounds like a more SE question.
Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...

**1**

vote

**0**answers

34 views

### Minimum Preserving Transformations [closed]

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...

**3**

votes

**0**answers

50 views

### Dependency of the Wasserstein distance on the parameter: a differential perspective

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below:
$$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...

**1**

vote

**2**answers

152 views

### Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...

**8**

votes

**1**answer

367 views

### Higher dimensional scutoids?

The recent discovery of scutoids in biological structures is fascinating. Two scutoids are depicted below (from Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells), each ...

**0**

votes

**1**answer

73 views

### Subset optimization with composite aggregate functions

I have a finite set $P = \{1, 5, 3, 6, 4, ..., p_n\}$ of size $N$ and average $A$.
I want to find the most efficient way to maximize the following function:
$$
f(x, y) = \frac{1}{(1+e^{-6(x-2)})(1+e^...

**0**

votes

**0**answers

75 views

### Minimizing Frobenius norm with sparsity constraints

I am trying to solve the following minimization problem
\begin{equation*}
\begin{aligned}
& \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}}
& & \| X Y^\top A - B \|_{\text F}^2 ...

**2**

votes

**1**answer

114 views

### Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...

**1**

vote

**0**answers

52 views

### Distance between quadric surface and point or Intersection of sphere and quadric surface

I asked a similar question on math.stackexchange, but the answer wasn't quite ideal for my application. Apparently analytic solutions are surprisingly rare for general quadric distances.
Given a ...

**2**

votes

**1**answer

110 views

### Non-negativity condition for special quartic

I know that a necessary and sufficient condition for the positivity of a quartic polynomial of many variables is in general difficult. I have a somewhat special case, maybe here more can be said. Let $...

**3**

votes

**1**answer

92 views

### Is the level set of a product of affine linear functions comprised of convex curves?

Internet searches haven't helped. Can you?
Let $\, f = \prod_{i=1}^n (a_i x + b_i y + c_i).$
Is each component of $\, f^{-1}(1)$ a convex curve?
I expect so, and can prove it for $n=2,$ but I'm ...

**2**

votes

**1**answer

191 views

### Minimizing the expectation of a functional of probability distribution subject to an entropy constraint

Consider a PDF $\pi(x)$ for $x\in[0,1]$, and the following functional
$$
F(\pi) = \mathbb{E}_\pi |x-y| $$
It is minimized by any point mass, so to avoid such degeneracy I'd like to lower-bound the ...

**0**

votes

**0**answers

31 views

### Nonlinear optimization problem better efficiency

I have a highly nonlinear optimization problem that I describre in the following lines:
$A$ is a $N \times N$ known matrix, $\vec{z},\vec{M},\vec{D}$ are known vectors of length $N$ and $R,T$ are ...

**0**

votes

**2**answers

87 views

### Maximizing a convex function with a convex constraint

Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, the objective is to find the farthest point in the level set $\left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1\right\rbrace$ (Assuming that ...

**1**

vote

**1**answer

104 views

### Parametric constrained optimization

I'd like to find a way of determining if the distance from the origin of a parametric parabolic path falls below a certain value within a given range of the parameter. The parabola is expressed as:
$$...

**3**

votes

**1**answer

159 views

### A sum over a hyperplane in $\mathbb{Z}^4$

Fix $M \geq 2$. What is the smallest number $\tau = \tau(M)$ such that $$\sum_{a,b,c,d =1\\ a + b = c+ d}^M (x_a x_b x_c x_d)^{\tau/4} \leq 1,$$ for all $x , \ldots , x_M \in \mathbb{R}_{\geq 0} $ ...

**0**

votes

**0**answers

16 views

### Finding Optimal Spheric Polyhedra with Given Convex Hull Topology

I want to draw finite planar graphs in certain canonical ways.
My idea is to use a stereographic projection of the convex hull of points placed on the unit sphere in a way, that the graph induced ...

**0**

votes

**0**answers

95 views

### L1 norm constraint on product of 2 matrix

I want to solve below minimization problem
\begin{equation*}
\begin{aligned}
& \underset{A, B}{\text{minimize}}
& & ||Y-AB^T -D||_F^2 \\
& \text{subject to}
&& |A_i|_1 \leq a,...

**0**

votes

**1**answer

46 views

### Linearly constrained saddle-point optimization

Let $f(x,y)$ be a smooth (twice differentiable) saddle function (convex in $x$ and concave in $y$), where $f \colon X \times Y \rightarrow \mathbb{R}$, and $X \subset \mathbb R^n$, $Y \subset \mathbb ...

**2**

votes

**0**answers

38 views

### Clarification on FPTAS optimization in a paper

In the abstract of this paper by Hildebrand, Weismantel & Zemmer it is stated that they provide an FPTAS for $$\min x'Qx$$ over a fixed dimension polyhedron when $Q$ has at most one negative or ...

**2**

votes

**1**answer

73 views

### Linear optimization with one positive definite quadratic equality condition in P?

I have the following minimization problem in $z \in \mathbb R^n$, which contains $x_1, \dots, x_t, y \in \mathbb R$.
$$\begin{array}{ll} \text{minimize} & y\\ \text{subject to} & xQx'= y\\ &...

**0**

votes

**0**answers

57 views

### non-convex optimization with constraint

I have a special non-convex optimization problem:
$\min / \max \ f(x) + g(x) + h(x)$,
subject to $| g(x) - h(x)| < \varepsilon$,
where $f(x)$ is non-convex, but both $g(x)$ and $h(x)$ are ...

**0**

votes

**1**answer

73 views

### On an error bound for matrix constraints

Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$.
Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...

**2**

votes

**0**answers

43 views

### Optimization with bounds on the control and its derivative

I would like to understand the following optimization problem. Let $F(t,x)$ be a continuous function defined on $[0,1]\times [0,1]$, which is increasing in $t$ and convex in $x$ (I have in mind $F(t,x)...

**0**

votes

**2**answers

152 views

### Lagrange Multipliers for two constraints, degenerate case

To optimize $f(x,y,z)$ subject to $g(x,y,z)=h(x,y,z)=0$, we use the Lagrange Multiplier method and solve
\begin{equation*}
\nabla f=\lambda \nabla g+\mu\nabla h,\quad g=0,\quad h=0.
\end{equation*}
...

**0**

votes

**1**answer

87 views

### 1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation
$$
\begin{split}
\mathbb ...

**0**

votes

**0**answers

9 views

### Embedding Symmetric TSP Instances in Euclidean Spaces of Least Dimension

It is known, that the set $E_{opt}$ resembling the opitmal tour of a symmetric TSP, is invariant under the addition of so called vertex weights $w(v_i)$, i.e. to adding or subtracting the same ...

**0**

votes

**0**answers

54 views

### Suggestions to solve an optimization problem that involves quadratic forms

I am in a crucial part of my research, I have arrived at an optimization problem that I can not solve, I need to solve it to be able to perform simulations and thus complete my research, due to this ...

**0**

votes

**0**answers

26 views

### Functions of squares of gradients and the spectrum of the Hessian

This is a very ill-formed formed question but kindly indulge me!
Say one has function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and a constant $a >0$. And let $g = \nabla f$ and $i \in 1,2..,d$. ...

**6**

votes

**5**answers

259 views

### Nearest matrix orthogonally similar to a given matrix

Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...

**1**

vote

**0**answers

121 views

### Lagrange-Newton minimization and initial value for the Lagrange multipliers

Short version. How to choose the initial values for the Lagrange multipliers in the Lagrange-Newton
equality-constraint minimization method?
Introduction. The problem to solve is
\begin{equation}
\...

**4**

votes

**0**answers

204 views

### Minimization over a convex function of equal vs unequal success probabilities of Bernoulli random variables

Let $U_1,U_2,\ldots,U_n$ be $n\geq 2$ mutually independent Bernoulli random variables. There are two cases of interest:
$1.$ The random variables $U_1,U_2,\ldots,U_n$ are identically distributed;
$...

**9**

votes

**2**answers

208 views

### A (reverse)-Minkowski type inequality for symmetric sums

Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true?
\begin{align*}
\left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \...

**1**

vote

**0**answers

40 views

### Rank Optimization over semi-definite constrains

Let $X$ and $Y$ be finite dimensional Hilbert spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, $L(X)$ and $L(Y)$ be the sets of linear operators of $X$ and $Y$, $\text{Herm}(X)$ and $\text{Herm}(Y)$ be the ...