3
votes
2answers
34 views

The quadratic case in nonlinear programming

I'm reading about nonlinear programming and I stumbled into the following statement where I started to wonder a bit: Consider the function $$f(\textbf{x}) = ...
0
votes
1answer
25 views

Is there any theoretical upper bound on the second derivative of a twice-differentiable function?

Lets assume that f(x) is a twice-differentiable and nonlinear function, where x is bounded by the interval l ≤ x ≤ u, and the function itself is bounded by L ≤ f ≤ U. We know the values of l, u, L and ...
1
vote
1answer
78 views

First-order necessary condition for relative minimum point

I'm studying linear and nonlinear programming and I came across with the following proposition : given $\rm x\in\Omega$ we are motivated to say that a vector $\mathbf d$ is a feasible direction at ...
1
vote
2answers
23 views

Show that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution

How can I show (preferably using the Bolzano-Weierstrass theorem), that $x_1^2+x_2^2+(x_1-x_2)^3 \rightarrow \min$ has no solution? I can see that it is true, but how can I show it?
2
votes
1answer
146 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
0
votes
5answers
127 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
1
vote
1answer
90 views

Minimizing a linear function on a strictly convex set.

All the theorems that I know considering the uniqueness of a solution to a minimization/maximization problem requires the strict convexity/strict concavity of the objective function. But consider the ...
0
votes
1answer
26 views

Help in Linear summation of n(any given number)

The original formula is this. We're computing the complexity of an insertion sort. How did the first formula turned into the second formula?
1
vote
1answer
354 views

Taylor's theorem for vector valued functions

I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it): ...
2
votes
0answers
35 views

Calculating second derivative of $g(\alpha) = f(\textbf{y}(\alpha))$

I'm having problems with the second derivative of the function $g(\alpha) = f(\textbf{y}(\alpha))$ (which I will define more precisely below). I tried calculating it myself, could anyone just simply ...
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 ...
1
vote
1answer
515 views

A sufficient condition for a unique maximum of the product of two concave functions

Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that $h(x)=f(x)g(x)$ have a unique maximizer on an interval $[a,b]$ for $a<b$?
3
votes
2answers
154 views

Maximum of a product of a polynomial with positive coefficients and a finite sum of exponentials with negative coefficients on $[0,+\infty)$

Prove or disprove that $$ f(x)=\left(\sum_i a_i x^i\right)\left(\sum_j b_j e^{-\lambda_j x}\right) $$ where $\forall i, a_i>0$, $\forall j, b_j>0,\lambda_j>0$, and both sums are finite, ...
7
votes
1answer
270 views

Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
2
votes
1answer
294 views

Condition for existence of Lagrange-multiplier

Using the implicit function theorem one can prove the following: Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If ...
3
votes
2answers
429 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 ...