0
votes
1answer
42 views

Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...
0
votes
0answers
25 views

Hessian of a non-linear Matrix function

Apologies if this is a silly question, but I am really confused. I am trying to find the Hessian of a non-linear function $f$. I understand that the Hessian of $f$ with respect to $A$ is the Jacobian ...
0
votes
1answer
27 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
1
vote
1answer
38 views

Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
3
votes
1answer
68 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
0
votes
0answers
46 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
0
votes
0answers
52 views

Newton step for functions which takes matrix arguments

I want to minimize a function $f(X)$ which takes a matrix $X$ as an argument, i.e. $\min_X f(X)$. Using a descent method I start at step $k$ with feasible matrix $X^k$ and get to the next $X^{k+1}$ by ...
3
votes
2answers
134 views

Optimization for constrained problem

I'm reading about Lagrange multipliers from a Pattern recognition book appendix and on one point the following is stated: $\begin{align} ...
0
votes
1answer
42 views

Non-Linear Programming: Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$

I am required to work on the following optimization problem: Max. $(a-\alpha)Q -(b-\beta)Q^2$ subject to $Q\geq 0$ How do I do so? My FOC is $(a-\alpha) -2(b-\beta)Q-\lambda(-Q)=0$ and ...
0
votes
0answers
60 views

Minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$. Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy: ...
0
votes
1answer
52 views

non linear equations solving methods?

I need to find $l_{2}$ and $\theta$ numerically by solving below equations. How could I do that? At least do i have some iterative way of finding those two unknowns. All others parameters are ...
5
votes
1answer
1k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
2
votes
1answer
957 views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
4
votes
4answers
415 views

Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
2
votes
1answer
105 views

Nonlinear optimization question

For (x,y) in $\mathbb R^2$, consider f(x,y) = $x^2 -2xy + \frac{4}{3}y^2 - 4y$ Find the local minimum of f. Is it a strict local minimum? Compute the $\lim\limits_{|(x,y)|\to \infty}$ f(x,y) to decide ...
-1
votes
0answers
109 views

Jacobian of Bilinear Cost Function

Could anyone help me with the Jacobian of: Given the following Matrices: ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ F \left( {C}^{2}, {E}^{2} \right) = {\left \| {C}^{2}{E}^{1} - {A}^{21} ...
1
vote
2answers
357 views

How to calculate the Hessian of the Lagrangian at x and lambda

I'm working on a project that needs to solve a constraint optimization function. Currently, I'm using Knitro solver and it needs to calculate the the hessian of the lagrangian at x and lambda. I don't ...
5
votes
1answer
323 views

How to optimize a rational function

Just a calculus problem: As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $$\begin{cases} 1 \leq a \leq c \\ 1 \leq b ...
3
votes
2answers
428 views

A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure

Motivation: Example. To solve a problem on evaluating the maximum of a product of $n$ real variables subject to an equality constraint on its sum $S$ ($=100$), I used the Lagrange multipliers method ...