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

**5**

votes

**0**answers

47 views

### Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...

**4**

votes

**0**answers

115 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
...

**2**

votes

**0**answers

31 views

### QR-Decomposition of matrix valued function

Suppose I have a matrix valued function
$$
F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T
$$
where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, ...

**2**

votes

**0**answers

77 views

### Techniques for proving that a set of constraints over the integers are inconsistent

I have a problem which boils down to showing that a set of constraints has no solutions. A simplified version of this constraint system would be the following system:
$$
\left\{
\begin{array}{l}
...

**2**

votes

**0**answers

130 views

### Subtour Elimination in Travelling Salesman Problem using MTZ

I am trying to formulation a problem similar to a Traveling Salesman with Time Window constraints.
To eliminate subtours, I need to use some constraint similar to a generalization of MTZ constraints ...

**2**

votes

**0**answers

113 views

### An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking.
These subgradients are (assume $x \in$ ...

**2**

votes

**0**answers

121 views

### Quadratic optimization with parameter in constraint

Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum.
Question: Given a function $q: \mathbb R^{N\times N}\mapsto ...

**1**

vote

**0**answers

57 views

### Estimation of part of parameters from an ODE

Suppose, we have an ODE
$$ \frac{dy}{dt}= f(t,y;p',a)$$
or alternatively
$$ \frac{dy}{dt}= f(t,y;p)$$
where the set of all parameters $p = (p',a)$. We only need to estimate part of parameter set ...

**1**

vote

**0**answers

114 views

### Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X ...

**1**

vote

**0**answers

99 views

### Modifying a QP to incorporate more constraints

Consider the following problem:
$$\min \sum_{i=1}^n (Y_i - Z^{(i)})^2 \\
\text{subjected to}~ \epsilon_k^{\top}(X_j-X_k) \leq Z^{(j)}-Z^{(k)} ~ \forall k,j = 1 \ldots n. $$
where $\epsilon_1, ...

**1**

vote

**0**answers

39 views

### What is the sufficient condition for a “strict local optimal point” to be “isolated local optimal point”(or strong local optimal point)?

I encountered a case that seems obvious that the local optimal point are isolated, yet I can only prove the local optimal points are strict rather than isolated.
I know under certain peculiar ...

**1**

vote

**0**answers

115 views

### Recovering a partition from spectral properties of the graph Laplacian

Let $G$ be a weighted graph with vertices $V$. Let $W$ be its real-valued, non-negative, $|V|\times|V|$ adjacency/affinity matrix. Let $L = \mathrm{diag}(W\mathbf1)-W$ be the (unnormalized) graph ...

**0**

votes

**0**answers

74 views

### Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...

**0**

votes

**0**answers

46 views

### No Strong Duality In Spite of Slater's Condition

I was reading some course notes here.
On Page 8, it says:
Note that strong duality holds here (Slater's condition), but the
optimal value of the last problem is not necessarily the optimal
...

**0**

votes

**0**answers

49 views

### What is the best way to optimize this matrix equation

What is the best way to optimize this Procrustes like formulation:
$\min\quad\|AX-B\|^2_{\rm F} + \|X^Tc\|^2,$
s.t. $X^TX = I$
Here A and B are $n \times p$ matrices and $c$ is a $p \times 1$ ...

**0**

votes

**0**answers

18 views

### weakly Complementary slackness

Suppose we have a general pair of primal-dual semidefinite programming and strong duality holds.
What does mean the term "weakly Complementary slackness condition" in optimality?

**0**

votes

**0**answers

125 views

### k-means type clustering of binary data, under capacity constraints per cluster. Proof of NP-hardness?

Suppose you are given a set of $I$ binary vectors in ${\mathbb R}^N$, a number of clusters $k$, and positive integers $\{ c_i \}_{i=1}^k$ where $\sum_{i=1}^k c_i = I$.
I am interested in finding a ...

**0**

votes

**0**answers

64 views

### Optimization with differential inequality constraint

Consider the closed set $[t_1,t_2]⊂R_{>0}$ and $V(t):[t1,t2]→R_{>0}$ being a continuous and piecewise continuously differentiable function. We want to find a continuously differentiable function ...

**0**

votes

**0**answers

39 views

### Minimum of a real-valued function of multiple variables

Suppose we are given two functions $$\phi(x) = \frac{\sigma^2}{2\mu^2} \left(1 - e^{-(2\mu/\sigma^2)x}\right) - \frac{x}{\mu}，x\in \mathbb R$$ and $$g_0(x) = \frac{\sigma^6}{4\mu^4} ...

**0**

votes

**0**answers

59 views

### Big eigenvalues of a special stochastic matrix

Given a matrix $M$ of size $n\times n,$ we write its different eigenvalues by $x_1,x_2,\ldots,x_m$ with $m\leq n$ such that $|x_1|>|x_2|>|x_3|>\cdots|x_m|,$ and call $x_2\doteq ...

**0**

votes

**0**answers

47 views

### functional maximization

Define a functional space of functions of the form $F(t)=p_1 exp^{-\mu_1(\delta-t)}+p'_1 (1-exp^{-\mu_1(\delta-t)}))$. $p_1,p'_1,\delta,\mu$ are parameters in [0,1] and trivially, variation of these ...

**0**

votes

**0**answers

52 views

### Minimizing inside a spherical uncertainty region

I am trying to figure out how to solve:
$\min_U r_{p}$
where $r_{p}=\alpha^\intercal\omega$ and $U$ is a sphere centered at $\alpha$ with radius equal to $\chi|\alpha|$ . ( $\omega$ is a vector or ...