The nonlinear-optimization tag has no wiki summary.
9
votes
2answers
385 views
A numerical optimization problem with a convolution in the constraint
I have a problem of the following form:
minimize $\|Dx\|_2$
subject to $\|x*x\|_2 = 1$
where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, ...
7
votes
3answers
357 views
Optimization problem for a parity-check code
I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
6
votes
5answers
1k views
Operations research book to start with
for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
5
votes
2answers
126 views
How to solve mixed integer nonlinear programs?
I'm not a math expert so sorry for possible trivial questions.
I have written this mixed integer nonlinear program (MINLP):
$$
\begin{align}
\min & \sum_{i \in ...
5
votes
1answer
191 views
Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$
Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
5
votes
1answer
228 views
How to optimize a rational function
Just a calculus problem:
As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $$\begin{cases} 1 \leq a \leq c \\ 1 \leq b ...
4
votes
4answers
252 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 ...
4
votes
1answer
159 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 ...
4
votes
2answers
170 views
discontinuous optimization
I'm solving the following problem:
$$ \max_\rho \;\; \rho \; \min\left[\left(
\frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$
where all constants and variables are defined ...
4
votes
2answers
896 views
Solving an overdetermined system of nonlinear equations
I'm wondering what the "best" way to approach solving a system of the following form would be:
$A_1X + Be^{CY} = A_2$
$A_3X + Be^{CY} = A_4$
$A_5X + Be^{CY} = A_6$
etc.
EDIT: Coefficients $A_i, ...
4
votes
2answers
106 views
Finding an explicit expression for a minimizer
Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional
$F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$
which is well defined for continuously ...
4
votes
1answer
112 views
Kuhn-Tucker condition is not satisfied
Show that the solution to finding minimum of
$f(x)=-x_{1}$
With conditions
$-\sin(x_{1})+x_{2} \leq 0$
$x_{1}-x_{2} \leq 0$
is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
4
votes
1answer
285 views
Solving a set of 3 nonlinear equations with constraints
Problem statement:
I am given 3 sets of equations that govern the force $P$, and also the neutral axis, defined by two variables, the radius from the center $r$ and also the rotation degree in ...
4
votes
2answers
129 views
Finding good approximation for $x^{1/2.4}$
I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
4
votes
2answers
149 views
Analog of Simplex Method for Quadratic Programming
It happened so I need to work with an algorithm for solving QP problems. The main issue here is that I can't find any references to this algorithms (at least no references in sources in english). ...
4
votes
0answers
158 views
I need help about some compactness arguments
I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
4
votes
0answers
64 views
Why Compactness is Necessary at Minimax Theorem
According to Von Neumann's minimax theorem, I have
$$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$
for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
4
votes
0answers
284 views
(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …
edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty.
Edit 3: Can somebody answer?
Edit 4:I ...
3
votes
2answers
390 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 ...
3
votes
2answers
102 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,
...
3
votes
1answer
29 views
existence of solution of $Ax= \max(b-x,0) $
How do you prove the existence of a solution to the linear system:
\begin{equation}
Ax= \max(b-x,0)
\end{equation}
A is an $n\times n$ matrix and $b$ is a vector in $\mathbb{R}^n$. $x$ is the ...
3
votes
2answers
104 views
Upper bound for Maximization problem
I have an optimization problem of the form
Max $x_1+x_2+x_3+\cdots+x_n$
subject to $x_0^2+x_1^2+x_2^2+\cdots+x_n^2+x_{12}^2+x_{13}^2+x_{14}^2+ \cdots+x_{1n}^2+x_{23}^2 + \cdots +x_{2n}^2+ \cdots ...
3
votes
1answer
70 views
$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)
This problem arose in my stereo vision project.
$$
P_{1c} = A*P
$$
$$
P_{2c} = B*P
$$
where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
3
votes
1answer
154 views
Optimization with constraint on solution of a linear system
I'm facing this optimization problem:
$$\text{minimize} \quad a^T x$$
$$\text{s.t. the solution of $A(x) z + B(x) = 0$ belongs to a convex set $S$}$$
Here $A(x)$ is a linear matrix function of $x$ ...
3
votes
3answers
204 views
simple-looking non-convex optimization problem
I want to solve the following problem:
Maximize $\sum_{i=1}^n\log(1+\lambda_i^2)$ subject to $\lambda_i >0$ and $\sum_{i=1}^n\lambda_i = M$. I was wondering how I could cast it as a convex problem. ...
3
votes
1answer
96 views
solving linear program with rank constraint?
I have a linear program where the variables are n vectors. Now I'd like to impose an extra constraint that k (k<=n) of the n vectors are linearly independent, or the matrix with the n vectors as ...
3
votes
0answers
173 views
Optimizing non linear programs of two variables
The scenario is;
We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
3
votes
1answer
143 views
Optimization problem with ratio objective
I need to solve the following optimization problem
$$
\text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad ||x||_1=1 \quad (\text{or alternatively} \quad c^T|x|=1),
$$
...
2
votes
2answers
344 views
Are the Karush-Kuhn-Tucker conditions applicable when one or more of the constraints are nonlinear?
I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to ...
2
votes
1answer
47 views
Maximalization of a cubic puzzle
What is the maximal volume of a post package of length $L$, width $W$ and height $H$,
subject to the following restrictions:
$L+W+H \leq 90 $
$L \leq 60$, $W \leq 60$, $H \leq 60$
Intuitively I ...
2
votes
1answer
45 views
Linear stability analysis on a constrained three-dimensional system of ODE
Let $\begin{cases} \dot x = f({\bf u}) \\ \dot y = g({\bf u}) \\ \dot z = h({\bf u})\end{cases}$ be a well-defined nonlinear system with ${\bf u} = (x,y,z)$ and restricted to domain $x,y,z \geq 0$. ...
2
votes
1answer
102 views
non linear optimization
How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in ...
2
votes
1answer
237 views
Prove every local minimum is a global minimum
Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$.
Minimize
$$f(x)= ...
2
votes
1answer
130 views
Nonlinear optimization with rotation matrix constraint
I'm trying to optimize the equation
|| R - W || = minimum
where W is a predetermined 3x3 matrix and R is the 3x3 matrix that I'm trying to optimize, with the ...
2
votes
1answer
271 views
Lasso with linear constraints
I want to efficiently solve the following optimization problem:
\begin{align}
\min &\quad \left\|\mathbf{x}-\mathbf{x}_0\right\|_2^2 + \lambda\left\|\mathbf{x}\right\|_1\\
\text{Subject to}& ...
2
votes
2answers
417 views
Modified Cholesky factorization and retrieving the usual LT matrix
I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 ...
2
votes
1answer
235 views
Condition for existence of Lagrange-multiplier
Using the implicit function theorem one can prove the following:
Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If ...
2
votes
1answer
37 views
Formulate the Langrangian function of a non-linear optimization problem and solve it for $y\geq0$
Consider the (non-linear) optimization problem ($P$)
$$max \quad3x_1 + 4x_2$$
$$s.t. \quad x_1^2 + x_2^2 \leq 25$$
$$ \quad x_1,x_2 \geq 0$$
Formulate the Lagrangian function ...
2
votes
1answer
86 views
Lagrange Multipliers for Function Spaces
For some constant $A > 1$ I am trying to solve the constrained minimization problem
minimize $F(u)$ in $C$
subject to $H(u) = 0$.
Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
2
votes
2answers
90 views
optimality of quadratic programming problems
Suppose we have a general quadratic programming problem:
\begin{align}
\min_{x}\,\,&c^Tx+\frac{1}{2}x^TQx,\\
\mbox{s.t.}\,\,& Ax=b,\\
&x\geq0,
\end{align}
where $Q$ is positive ...
2
votes
1answer
72 views
Control on Conformal map
Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
2
votes
1answer
161 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 ...
2
votes
1answer
75 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 ...
2
votes
1answer
86 views
Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)
Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian.
...
2
votes
1answer
212 views
Lagrangian Multipliers
I have a fundamental question about Lagrange multipliers. Here it is:
I have a function to maximize with respect to a parameter say $\theta$, subject to two constraints. Lets assume that the first ...
2
votes
1answer
208 views
Nonlinear Optimization/Programming: A good counter text
I am currently taking a nonlinear optimization course and the text is Bertsekas' "Nonlinear Programming 2e".
I think the book does a decent job but I am a much more "hands-on" and visual learner so ...
2
votes
1answer
183 views
When $\min \max = \max \min$?
Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets.
Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$.
Say under which condition we have
$$ \min_{x \in ...
2
votes
1answer
113 views
Multilinear optimization
Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms
...
2
votes
2answers
582 views
Algorithm for GRG2 method of solving non-linear least square
I have been looking for quite a while for an algorithm for the GRG2 method either in a .net assembly or an algorithm i could program myself but I cant find a decent representation of the algorithm to ...
2
votes
1answer
179 views
Finding a function that satisfies constraints numerically
I have the following system of equations for function $p(y)$ and I need help debugging my solution:
$$\begin{align}
0&=\log(p(y))+1-\lambda-\gamma y^2-\eta ...
