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

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6
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59 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
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143 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: ...
3
votes
0answers
70 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 ...
2
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0answers
81 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
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0answers
187 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
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0answers
167 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
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0answers
141 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
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0answers
30 views

Changing a nonlinear equality constraint into some conic inequality plus rank constraint

If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
1
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64 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
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133 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
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125 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 ...
1
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379 views

Solving a system of complex non-linear equations

I have a set of five equations which can be described as follows: $m_{i}=\frac{k_{1}}{(x+a)^{i}} + \frac{k_{2}}{(b+d)^{i}}+ \frac{k_{3}}{c^{i}}$ for i=1 to 5 where $$\eqalign{ ...
1
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223 views

Multiobjective semidefinite programming

Let $C$ be size $n \times n^{2}$. Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$. There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$. $B$ is ...
0
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35 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}} ...
0
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0answers
28 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 ...
0
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24 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
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33 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: ...
0
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0answers
61 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
0answers
37 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
0answers
20 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 ...
0
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48 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
0
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0answers
51 views

Solution of a nonlinear system of two equations

Given the matrix $A_{M,N}$ with $N\gt M$, the vector $y$, I have to find the vectors $x$ and $u$, satisfying the following equations: $$D(x)x=A^Tu$$ $$y=Ax$$ where: $$D(x) = \left| \begin{array}{ccc} ...
0
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37 views

A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as \begin{align} ...
0
votes
0answers
77 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
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99 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
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0answers
51 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
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0answers
24 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
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102 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, ...
0
votes
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67 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
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55 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
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0answers
54 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 ...
0
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0answers
117 views

Question on non-linear parametric mixed integer program

I am trying to solve a mixed integer minimization problem, where there are a number of parameters, and there are products of parameters with variables appearing in the objective function. I assume ...
-1
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29 views

Optimal time control for the system of two non-linear ODE

I have the following system of two non-linear ODE with one control variable (modified model of Lotka-Volterra): $$\left\{\begin{matrix} \frac{dx}{dt}=(\alpha-\beta y)x \\ \frac{dy}{dt}=(-\gamma + ...
-2
votes
0answers
28 views

Solving nonlinear system of ODEs

I have the following system of differential equations: $$ \begin{cases} \frac{dx}{dt} = (1 - y) x - 0.4 xu \\ \frac{dy}{dt} = (x - 1)y - 0.2yu \\ \psi_1' = - \frac{dH}{dx} = (-1 + 0.4u)\psi_1 + y ...