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1
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35 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 ...
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0answers
30 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
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1answer
144 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
110 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 ...
0
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0answers
28 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
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1answer
27 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
42 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
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0answers
17 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
44 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
83 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
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0answers
35 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
117 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
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0answers
35 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
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0answers
29 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
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0answers
67 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
55 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
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0answers
42 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
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0answers
51 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 ...
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0answers
11 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
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0answers
65 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
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0answers
41 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
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0answers
44 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
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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
127 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
188 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
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2answers
118 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
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0answers
232 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
23 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
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0answers
30 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
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0answers
46 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
34 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
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2answers
326 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 ...
0
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1answer
40 views

Directive on Dimensionality Reduction

I have a data set (24 data records) which is in $\mathbb{R}^{13}$ and I need to project it to a lower dimension (at least to $\mathbb{R}^{3}$). My objective of the dimensionality reduction is to ...
0
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0answers
18 views

Difference between two model fitting schemes

We have some experimental data, $x \mapsto \hat{f}$ and we're trying to fit a known model of the form $$f(x\ \left|\right.\ a_1, a_2, a_3, b_1, b_2, b_3) = a_1 F(b_1, x) + a_2 F(b_2, x) + a_3 F(b_3, ...
1
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1answer
62 views

A minimization problem

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \left\|\frac{w}{x}\right\|,~w,u\in \Bbb{R}^n$$ where $$\frac{w}{x}=\left(\frac{w_1}{x_1},\ldots, \frac{w_n}{x_n}\right)$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ ...
1
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1answer
45 views

A minimization problem [duplicate]

Define $$L(w,u)=\frac{1}{2}\|w-u\|^2+\beta \|\frac{w}{x}\|,~w,u\in R^n$$ where $$\frac{w}{x}=(\frac{w_1}{x_1},\dots, \frac{w_n}{x_n})$$ $$\|x\|=\sqrt{x_1^2+\cdots+x_n^2}$$ Given $u$, $x$ and $\beta$, ...
0
votes
1answer
464 views

Optimization of a function of two variables.

Suppose we have to minimize a function $f(\mathbf x,\mathbf y)$ where $\mathbf x$, $\mathbf y$ are vectors in Euclidean space. The function is convex in $\mathbf y$ when $\mathbf x$ is kept constant ...
2
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0answers
66 views

Steepest Descent/Newton

Suppose these over-determined system of equations: $$ |\mathbf{x}^T\mathbf{v_n}| = A, \qquad n = 1,2,\cdots,N-1 $$ $$ \mathbf{v_n}= [1 \quad w^n \quad w^{2n} \quad \cdots \quad w^{(N-1)n}]^T , ...
0
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0answers
156 views

An example of the Sequential Quadratic Programming (SQP)

Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to ...
1
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1answer
98 views

Minimize a nonlinear sum subject to a quadratic constraint

Currently I am solving an optimization problem that could be written as follows: $$\min J= \sum_{i=1}^N {(q_i^H\Lambda q_i)}^{\frac{1}{3}} $$ subject to $\{q_i\}_{i\in [1..N]}$ forming an ...
2
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1answer
94 views

minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$

Suppose there are $n$ points $(x_i, y_i)$ for $i = 1,\ldots,n$. Please find another point $(x, y)$ to minimize function: $$\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$$
3
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1answer
275 views

Solving a system of non-linear (trig) equations:

I am having trouble trying to solve the following equations: $\sin(\alpha)+\sin(\beta)=\dfrac {1000} A$ $\sin(\alpha)+\sin(\gamma)=\dfrac {800} A$ $\dfrac {20(1+\cos(\alpha-\beta))} {\cos(\beta)} ...
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0answers
123 views

Optimization: Minimizing Quadratic Vector Valued Functions

I have a vector valued function $F: \mathbf{R}^{n} \rightarrow \mathbf{R}^{m}$, which consists of quadratic taylor approximations. So one could say that function F consists of stacked approximations ...
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0answers
78 views

Create theoretical model for scheduling and allocation problem

I want to create a theoretical model for my scheduling and allocation problem instance, but I'm having some difficulties with a clear definition, since I have slightly different circumstances than the ...
0
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0answers
86 views

How to optimize a function with several variables

I need to develop code to optimize a set or variables based on the following conditions. I don't have the source of function. The function gets a point (x,y) and generate a mapped point (x',y') ...
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0answers
35 views

Formulating square packing as a form of optimization

I was looking at square packing problem which is defined as: Given a number N... Find the smallest square that can pack N unit squares Each square can be associated with a 3 dimensional point ...
1
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2answers
31 views

how do i find $\max\{x+z\}$ and $\max\{1+y^2\}$ where ?$x\ge0 $,$ y\ge0$,$ z\ge0$ and$xy+xz+yz=1 $

how compute $\max\{x+z\}$ and $\max\{1+y^2\}$? such that $x$,$y$,$z$ satisfied $$\begin{cases} xy+xz+yz=1 & \\ x\ge0 \\ y\ge0\\ z\ge0\\ \end{cases} $$ i face with this problem when i try ...
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0answers
91 views

Gradient descent/ nonlinear optimization intuition needed

all. I'm taking an introductory AI class, and we're using the gradient descent algorithm to find the optimized/ lowest cost of a set of thetas (variable coefficients) to best fit a regression line. In ...
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0answers
36 views

A non-linear optimization question

Suppose we want to solve the following optimization problem minimize $f(x_1,...x_n,y_1,...y_n)$, subject to $g(x_1,...,x_n,z_1,...,z_n) \leq b$, ...