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

**1**

vote

**0**answers

18 views

### When is a convex program continuous in its constraint vectors?

Consider $$F(z)=\min ae^{-x}+b e^{-y} s.t. x\ge 0, y\ge 0\text{ and } x+y=z$$
I checked if this function is continuous, but it is not at $z=0$. $F(z)=2\sqrt{ab}e^{-z/2}$ when $z\ne 0$, and ...

**0**

votes

**0**answers

27 views

### Binary Operator and Vector Combination Problem [closed]

Given a set of vectors, a set of binary operators, and a solution constraint. Is there a mathematical way to combine the vectors and operators to attain the solution constraint?
The vectors are all ...

**0**

votes

**0**answers

20 views

### Global and local maxima in a weighted sum of logarithms of linear functionals?

Initially posted on math.stackexchange, was recommended that this is a more relevant forum:
Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters ...

**1**

vote

**0**answers

27 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}} ...

**0**

votes

**0**answers

55 views

### Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is
positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...

**3**

votes

**0**answers

95 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 ...

**0**

votes

**0**answers

20 views

### Necessary Conditions for Saddle Value point [migrated]

This questions is from the Kuhn-Tucker paper "Nonlinear Programming" in Section 2 Lemma 1. I don't understand how those conditions are necessary for a saddle point. I always thought that a saddle ...

**0**

votes

**0**answers

16 views

### Is it possible to design an algorithm an optimization problem with non-convex and coupled constraints, using decomposition methods?

Recently, I am working on an optimization problem in wireless networks. I formulate the problem to be an optimization problem with non-convex and coupled non-linear constraints. And I want to design a ...

**2**

votes

**1**answer

111 views

### Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...

**3**

votes

**1**answer

42 views

### Constrained vs Unconstrained Optimization

I'm currently working on an optimization problem with a linear objective with linear and nonlinear constraints, i'm facing difficulties reaching a good solution, so i was advised to move the nonlinear ...

**1**

vote

**0**answers

64 views

### Minimize Product of Sums of Squared Distances

The Question
Given two sets of vectors $S_1$ and $S_2$，we want to find a unit vector $s$ such that
$$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\}
\cdot
\{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...

**0**

votes

**0**answers

18 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 ...

**0**

votes

**0**answers

90 views

### How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla ...

**4**

votes

**2**answers

99 views

### Convexity of a (non-symmetric) function of matrices

Let $f : H_{n\times n} (\mathbb{C}) \rightarrow \mathbb{R}$ be the function on Hermitian, positive semidefinite matrices $f(A) = \frac{M_i (A)}{\det(A)}$ where $M_i(A)$ is the determinant of the the ...

**3**

votes

**1**answer

61 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 ...

**2**

votes

**1**answer

102 views

### Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**6**

votes

**1**answer

460 views

### Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the ...

**1**

vote

**0**answers

35 views

### Likelihood convexification

I am doing constrained vector optimization using a non-convex non-linear likelihood function. My problem is of the following form:
$$\begin{align*}\hat Q &= \underset{\vec Q}{\arg\min} -\log ...

**2**

votes

**0**answers

62 views

### How to solve the following generalized quadratic programming problem [closed]

I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...

**0**

votes

**0**answers

58 views

### Specific optimization problem solution procedures

Is there a standard procedure to solve following two optimization problems?
$$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...

**4**

votes

**2**answers

214 views

### Convexity of a function of matrices

Let $A$ be an $n\times n$ positive-definite matrix. Let $0<\lambda _1 \leq \lambda_2 \leq \lambda _3 \ldots \leq \lambda _n$ be the eigenvalues of $A$. Let $n\geq k\geq 1$. Is the function $f(A) = ...

**1**

vote

**0**answers

40 views

### modern exposition of exact ground state of classical XY model or Ising model

What is the state of art technique in solving exact ground state of Heisenberg model, meaning minimization of the H terms (hamiltonian) provided infinite spin space?
...

**5**

votes

**1**answer

161 views

### Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty).
For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$.
Given some $1\leq s < d$, consider ...

**1**

vote

**0**answers

32 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, ...

**0**

votes

**0**answers

27 views

### non-coherent estimation problem

I have the following signals
$$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...

**2**

votes

**0**answers

110 views

### Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
...

**2**

votes

**0**answers

46 views

### Sequence transformations that are entropy invariant

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Define entropy of $\mathcal{A}=\{a_i\}_{i=1}^m$
by ...

**5**

votes

**1**answer

162 views

### Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem:
$Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...

**1**

vote

**0**answers

42 views

### Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution ...

**1**

vote

**1**answer

104 views

### Limiting Entropy of deterministic sequences - 1

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.
Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at ...

**1**

vote

**1**answer

77 views

### Entropy difference dominance of sequences

Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$.
Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution ...

**1**

vote

**1**answer

56 views

### Entropy dominance of certain restricted sequenes

Say you have positive $\{a_i\}_{i=1}^n$ and you have $p_i=\frac{a_i}{\sum_{i=1}^na_i}$, then assume you have a $C$ such that $C<2a_n\ll\sum_{i=1}^na_i$ (that is $C$ is not very large), then define ...

**2**

votes

**1**answer

67 views

### Entropy dominance

Let $0<a<b<c$ be distinct positive reals.
Define four different probability distributions:
$$\mathcal{P}_{ab}:P_{a,ab}=\frac{a}{a+b}=1-P_{b,ab}$$
...

**1**

vote

**1**answer

52 views

### Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...

**1**

vote

**1**answer

74 views

### Smooth unit vector field on a tetrahedron to interpolate vertex constraints

For a tetrahedron $T\subset \mathbb{R}^3$ with vertices $r_i\in \mathbb{R}^3$ , $i=1,\ldots,4$, and unit vectors $u_i\in \mathbb{S}^2$ at each vertex $i=1,\ldots,4$
consider the (energy) functional
...

**11**

votes

**2**answers

572 views

### Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture).
(The question was first asked at math.SE, ...

**0**

votes

**1**answer

59 views

### Finding maximum of a function with unfixed number of variables

Can anybody solve this:
For a constant positive integer $n\geq6$
find $k$ and positive integers $a_{1},a_{2},...,a_{k}$
that maximize the expression
...

**0**

votes

**0**answers

60 views

### Best s-term approximation and unit balls in weak $\ell^p$ norm

In the book "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut there is the following asymptotic estimate at page 332, equation (11.1):
$$\sup_{\mathbf{x}\in B^N_{r,\infty}} ...

**1**

vote

**1**answer

81 views

### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...

**0**

votes

**0**answers

34 views

### How can be a conservative field constraint be efficiently implemented in a continuous optimization problem?

Suppose we have the following continuous optimization problem:
$$
\underset{x}{\mathrm{minimize}}f\left(x\right)
$$
subject to
$$
\exists X:\nabla X=Jac\left(X\right)=x
$$
where $f$ is a function ...

**-3**

votes

**1**answer

55 views

### Convert constraint to do convex optimization or use Lagrange multiplier method [closed]

$w_1, w_2, w_3 ... w_n$ are the weights I need to find
I have the following constraint:
$|w_1| + |w_2| + .. |w_n| <= 5$
That is the sum of the absolute values of the weights has to be less than ...

**4**

votes

**2**answers

271 views

### Convex Sets and Nearest Neighbors

For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is uniquely determined. It is known that ...

**3**

votes

**0**answers

82 views

### Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...

**0**

votes

**0**answers

27 views

### Optimization problem involving an entrywise function

Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...

**0**

votes

**0**answers

52 views

### Convergence of Coordinate Descent / Alternating directions

My question regards this method
http://en.wikipedia.org/wiki/Coordinate_descent,
where at each step a function $f$ is minimized along one coordinate axis (or block of coordinates).
Assume that $f: ...

**1**

vote

**0**answers

71 views

### Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background
I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below:
Definition: Maximally Uniform ...

**0**

votes

**0**answers

45 views

### minimizing concave function

I am interested in using the algorithm of Harold Benson described in his 1991 paper: "A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set". In the paper, he ...

**0**

votes

**0**answers

24 views

### Recursively calculate Tikhonov regularizer in b-spline objective function

I'm trying to write a program to calculate cubic b-spline based on set of inputs. But I can't figure out how to calculate value of Tikhonov regularizer.
My b-spline function is this:
I have ...

**7**

votes

**2**answers

432 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...