# Tagged Questions

A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for ...

40 views

### Find a solution of optimal problem with an inequality constraint

Let $a,b,x$ be vectors in $R^n$, A be a matrix, $c,d \in R, c<d$. Solve the following problem: $$\begin{cases} \text{minimize} \quad (b-Ax)^T(b-Ax)\\ (a^Tx-c).(a^Tx-d) \leq 0 \end{cases}$$ Assume ...
26 views

64 views

### Nature of the Hessian of the dual function?

I originally posted this over at MathOverflow but it did not receive much (...any) attention. I'm hoping someone can point me in the right direction over here. Consider a nonlinear optimization ...
27 views

### Nonlinear integer programming problem

I am trying to maximize the following function $$f(m,n) = \frac{m \log 3 + n \log 2}{\sqrt{m^2+n^2}}$$ where $n$ and $m$ are integers, not both $= 0$, although one could be $0$. This is ...
20 views

### trust region - choice of scaling matrix

According to many resources, TR algorithms often suffer from bad scaling. The simplest remedy is to use scaling matrix D in following way \begin{align} \min_d \ f + g'd + \frac{1}{2}*d'Bd \\ ...
15 views

### Can the low-rank approximation problem be formulated as the following convex model?

Given a three-order tensor $\mathcal{Y}$, our aim is to find a tensor $\mathcal{X}$ to approximate it and $\mathcal{X}$ should satisfy the following property: $\mathcal{X}$ can be well approximated ...
7 views

15 views

### NLP, Recatungular or Polar? Sine or No Sine?

Let's say we have the following complex number, $V$: $V=V_i+jV_j=V_m\angle{V_a}$ Which type of representation is better in an NLP (polar or rectangular)? The polar form leads to one nasty ...
28 views

### Can we solve this system of inequalities analytically?

Let $A$ be positive real number and $k$ a positive integer. How to find the analytical solution of this system? Find the $a_i$ \begin{align} \begin{cases} ...
59 views

### Why does $f(x)=\frac{x^T Ax}{x^T x}$ always have a minimum value?

$f$ is defined for all $x\in\mathbb{R^n}-\{0\}$ nd $A$ is a symmetric matrix $n \times n$. I have to proof that $f$ has a minimum $f(x^*)$ and write a formula for $x^*$ using the spectral ...
30 views

### A supremum problem

Let $a=\underset{\|u\|_c\leq 1, \|v\|_r\leq 1}{\sup}v^TY^Tu$. If $\lambda<a$, $\underset{u_m, v_m}{\sup}v_m^TY^Tu_m-\lambda\|u_m\|_c\|v_m\|_r=+\infty$. While if $\lambda > a$, then ...
43 views

### Intersection of a power function with a line: how to compute?

How to compute $x$ from $$q x^p = 1 - x$$ where $x$ and $q$ are positive, while $p$ is a real number? When $p > 0$: it's two monotonic functions, one increasing and one decreasing, and having ...
133 views

### Nonlinear optimization: Optimizing a matrix to make its square is close to a given matrix.

I'm trying to solve a minimization problem whose purpose is to optimize a matrix whose square is close to another given matrix. But I can't find an effective tool to solve it. Here is my problem: ...
18 views

### How to regress certain non-linear data

How can I perform a regression onto data of that follows this shape: $$U(x):=\sum_{i=1}^N\, a_ix^ie^{-b_ix}$$ where the $a_i\in \mathbb{R}$ and the $b_i \in (0,\infty)$ ...
18 views

### Proving inequality from convexity of function

I am having trouble proving the following inequality for all $x,y>0$ from "The Mathematics of Nonlinear Programming" by Pressini, Uhl. The book states that it follows from the convexity of an ...
12 views

### Condition of two related matrices

I have a data matrix $\text{X} \in R^{n \times m}$ (n - number of variables; m - number of experiements) and two parameter vectors $\beta_{p} \in R^{p \times 1}$ and $\beta_{l} \in R^{l \times 1}$. ...
32 views

### What algorithms are applicable to solve a inequality constraint Quadratic Optimization?

Suppose that we have a quadratic optimization problem $$(QP) \qquad \min \lbrace\frac{1}{2}x^TQX+ q^TX\rbrace$$ s.t. $$AX=a;$$ $$BX\le b;$$ $$X \ge 0;$$ where $Q \in \mathbb{R}^{n \times n}$ ...
7 views

### Mapping from variable space to ccriterion space in Multiobjective Linear Fractional Programming

I would like to ask about the properties of the criterion-objective space of a Multiobjective Linear Fractional Program with two linear fractional objectives for maximization and linear constraints. I ...