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4
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0answers
27 views

Nonlinear optimisation of Expectation

I am preparing for my exams and I can't get my head around the following question. I know there exists a general method for solving these problems but I don't know where to start. I would greatly ...
2
votes
2answers
58 views

$\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$

Solve $\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$, such that $x_0 = 0, x_{20} = 5$ and $x_{k+1} - x_k = u_k$. I think I know how to solve problems like these recursively, but I ...
0
votes
0answers
29 views

BFGS method with weights

Let me put the following question relating to the non-linear least squares $$\min_{x\in\mathbb{R}^{m}}\frac{1}{2}f^{T}(x)f(x).$$ Consider $J(m,n)$m, $m>n$, to be the Jacobian matrix and $W(m,m)$ ...
4
votes
2answers
82 views

Maximize $\prod\limits_{i=1}^n m_i$

Someone visits a market where one hundred different types of fruit are sold. All types cost $1$ euro per pound. The utility the buyer receives from buying $m_1$ pounds of the first type of fruit he ...
0
votes
0answers
50 views

KKT minimization problem

Solve $x^2 - 2y \rightarrow \min$ subject to $\max\{3x^2, e^y + 2\} + \sqrt{x^2 + y^2 - 2x + 1} \leq 6x + \sqrt{5}$ and $ \sqrt{x^2 + y^2 - 4x - 4y +8} -2x+2y \leq 0$ I tried computing the ...
0
votes
1answer
35 views

What is inexact steepest descent method

Is there anybody knowing what is the inexact steepest descent method for solving non-linear optimization problems? Any reference or formal definition available online? I was asked by someone, but ...
0
votes
1answer
100 views

KKT maximization problem

$x^2y \rightarrow$ max, such that $x^2 + 4xy \leq 1, x \geq 0$ and $y \geq 0$. I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone ...
0
votes
0answers
50 views

High dimensional constrained non linear optimisation

Short question: Does anyone know of any good optimisation routines for very large dimensional problems with equality and inequality constraints. In particular effective ways to deal with the ...
0
votes
0answers
60 views

Simulated annealing with constraints

Say we have an objective function with many local optima, and we wish to find the global optimum. Further, assume the method of choice is simulated annealing. Now, suppose there is a constraint on the ...
1
vote
1answer
356 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): ...
0
votes
0answers
30 views

How fast Interior Point method can be when solving Quadratic Programming problems?

Given the following Quadratic Programming problem: $\;\;\;\;\; \min x^TQx+c^Tx $ s.t. $Ax=b$ $\;\;\;\;\; x\ge0$ where $x\in \mathbb{R}^n$, $Q \in \mathbb{R}^{n \times n}$ is a positive ...
1
vote
1answer
106 views

What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
0
votes
2answers
186 views

Why nonlinear programming problem (NLO) called “nonlinear”? What does “nonlinearity” actually mean? Is it “not linear” or something different?

My teacher in the course Mat-2.3139 presented the same definition as in Wikipedia for the nonlinear programming problem here but he did not specify what the nonlinearity actually means or what it ...
0
votes
1answer
64 views

(pseudo-/quasi-)convexitiy of ratio between quadratic and affine function

Let $X\subseteq\mathbb{R}^n$. I have the following function $f:X\rightarrow\mathbb{R}$: $$ f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i +\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}\enspace.$$ All ...
0
votes
0answers
58 views

This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
1
vote
1answer
51 views

Help to understand the setting up of this Lagrangian

So..I understand up to step 4..but then there are these things I dont get, to start with , it says on (5) that the utility function depended only on the ratios p1/w p2/w ?? why does it say that? ...
1
vote
0answers
25 views

Nonlinear optimization using parallel input/output

I have a system that accepts a vector and returns a function value. The goal is to change the elements of the vector such that the function value is minimized using a derivative-free solver, eg. using ...
1
vote
0answers
37 views

Regression/compressive sensing with non-linear constrains where the coefficients are assumed to be integer or binary {0,1}

The following regression problem $$ \mathbf{y} = \mathbf{A}\mathbf{x} $$ where $\mathbf{y}$ is a $N\times 1$ column real vector, $\mathbf{A}$ is a $N\times M$ real matrix where each column ...
0
votes
0answers
31 views

How to find optimal function that solves the following program

Can I solve this problem ananlytically or how to show a optimal solution $g(\cdot)$ exists?Thanks. $\max_{\{g(\cdot),\theta^*\}} \int_0^{\theta^*}x\phi(1-g(x))dx$ s.t. $0\leq \theta^*\leq 1$ ...
7
votes
1answer
166 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
2
votes
1answer
129 views

Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
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 ...
1
vote
1answer
29 views

Non linear programming please HELP

Hey guys, I never do that but just found out that I have an assignment due in a few hours..thought it was for later, any help/solutions for this one?
0
votes
0answers
46 views

Solution feasibility of bilevel optimization problem

I have a sequential (Stackelberg) game formulated as a bilevel optimization problem. I try to understand whether there must be a feasible solution to this problem. My questions are: Does a solution ...
0
votes
0answers
21 views

Iterative optimizer with two functions and monotone constraint.

I have some issues with a problem I have to solve for university: I have a energy function which should be minimized: $\Omega = \sum_{i=1}^N \sum_{j=1}^P [g(Z_{i,j}) - F_{i} - b_j]^2] + \lambda ...
2
votes
0answers
46 views

the objective function $\|F\|_F^2$ is quasiconvex in the optimization?why?

I have read a paper, but I can not understand one optimization thoroughly.Generally, Frobenius norm of one matrix, $\|F\|_F^2$, as the objective function is convex, so we can resolve it not using the ...
0
votes
1answer
103 views

How to get the closed form solution of a non-convex optimization problem?

I want to know if there is a closed form expression for the optimal objective function? How can I get it, if it does exist? Condition: $h,f\in \mathbb{C}^{N\times1}, \epsilon > 0 $. $\max \ \ ...
1
vote
0answers
37 views

how to minimize this convex function?

$x_i$ and $y_j$ are variables. I intend to minimize this function and obtain the optimal value of $x$ and $y$: $\begin{align} ...
0
votes
1answer
145 views

maximizing a function of a positive semi-definite matrix with bounded trace

I need to maximize a function $f(A)$ where $A$. With the constraints that $A$ is positive definite and has a trace $tr(A) \leq K$. $tr(A)=K$ will work for my problem too. I can differentiate towards ...
1
vote
0answers
36 views

On solving non-linear programming problem and the relevant software

I have a non-linear programming problem, in which all the inequality is linear and only the optimization goal is in a non-linear form. The problem is as following. $x_j$ is the variables and $a_{k,j}$ ...
0
votes
0answers
31 views

Can I put a constraint in the optimization problem that my solution is a low pass filter?

My problem consists in finding a vector B: $$ min ||AB||_1 + ||A-AB||_2 $$ $$\text{subject to}$$ $$\text{B is a low pass filter}$$ $$\text{number of non zero elements of B is smaller than some number ...
1
vote
0answers
75 views

Distinction between linear and nonlinear model

[I have already asked this question on CrossValidated but until now received no answer] I have read some explanations about the properties of linear vs nonlinear models, but still I am sometimes not ...
0
votes
1answer
62 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
0
votes
0answers
46 views

Solution of nonlinear matrix optimization problem

I have the optimization problem ($L,A$ are regular matrices and $A$ is Hurwitz stable): min $||L||_2$ subject to $LAL^{-1} + L^{-T}A^{T}L^{T}<0$ Can the nonlinear problem be formulated as a LMI? ...
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 ...
0
votes
0answers
12 views

Variation of Optimal Solution with other Parameters

I have the following kind of optimization problem. $$\min_{f_1,f_2,\cdots\ ,f_L}\sum_{i=1}^L \mu_{i}D_i(\lambda_i,f_i,\gamma)$$ sub. to $$\sum_{i=1}^Lf_i=1-\delta\\ f_i\ge 0\quad i=1,2,\cdots \ ...
1
vote
0answers
71 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
0
votes
1answer
47 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
1
vote
0answers
49 views

Linear Complementarity Problem - multiple solutions, which one will it find?

If I have a inequality constrained system: w = Mz + q <= 0, z<=0, z^T w = 0 that for some given properties M and ...
0
votes
0answers
47 views

Store equilibrium points of a 3D system in a (x,y,z) array

I have a complicated 3d nonlinear differential system which I solve algebraically using maple's solve command. I aim to find the equilibrium points of the system using MAPLE and then plot these ...
1
vote
0answers
145 views

Dimension analysis and explaining the $\varepsilon$

Reference of this post (page no 6 from equation 39) The time averaged total energy, $\bar E$, has the following $\varepsilon$ expansion in $D$ dimension: \begin{equation} ...
0
votes
1answer
159 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
2
votes
1answer
221 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
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} ...
1
vote
0answers
243 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
0
votes
0answers
24 views

Optimization: How can I limit parameters variability?

I'm optimizing the normal of a 3D plane minimizing the re-projection error (with the difference of the intensity of some pixels in two images that corresponds of some points that lies on that plane). ...
0
votes
0answers
31 views

Optimization on restricted domain

I have the following function: $y(x)= 1+ K \times x^{\beta-1} + \frac{z(x)}{x}$ for $x>0$. Here $K>0$ is a constant. $\beta>1$ is also constant. $z(x)$ is a function of $x$ with properties ...
0
votes
0answers
49 views

Optimisation methods for boundary-constrained non-linear problem, local minima desired

I have a non-linear multivariate function that I want to minimize, subject to boundary constraints. I can also evaluate the Jacobian cheaply, so I'm not looking into derivative-free methods. I'm ...
0
votes
1answer
41 views

Approaches to fitting noisy oscillatory data?

I have observations $\hat{f}$ from data at points $\mathbf{x}=\{x_1,\ldots,x_N\}$, that is modeled as a known oscillatory form $f(k\ x)$ (for example, the sinc function), where $k$ controls the ...
2
votes
2answers
552 views

Which optimization algorithm converges faster?

everyone. I'm having a large scale unconstrained optimization problem. If I treat the unconstrained problem as a constrained problem with infinity constraints, I should be able to use both the ...