Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

Filter by
Sorted by
Tagged with
0
votes
2answers
64 views

Which is the best way to generate all $x_i$ where $\sum\limits_{i=1}^7 x_i = 1.0$?

I was wandering which is the best way to generate various combinations of $x_i$ such that $$\sum\limits_{i=1}^7 x_i = 1.0$$ where $ x_i \in \{0.0, 0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$ I can ...
4
votes
2answers
80 views

What is an approach for optimizing the values of a matrix?

My apologies if I get some terminology wrong, I don't have a formal math background; half my problem is articulating what I'm trying to do and identifying the domain of math that deals with this kind ...
4
votes
2answers
162 views

What numerical optimization method to use for this function?

In order to solve this over-determined system of equations numerically: $$ f_l(\mathbf x) = \displaystyle \left \lvert \sum_{k=1}^Kx_k^2e^{-j\frac{2\pi}Np_kl} \right \rvert , \qquad P = \{p_1,p_2,\...
3
votes
1answer
276 views

Rewrite constrained optimization objective

I wanted to ask, under which conditions can one rewrite the optimization objective $\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$ as $\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$ I have particular interest ...
2
votes
0answers
126 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
votes
1answer
141 views

Numerical Optimization methods?

What kind of functions are suitable for numerical optimization methods such as Newton, Gradient Descent, ... ? Any conditions?
0
votes
1answer
126 views

How to optimize nonlinear goal under linear constraints?

I have a "linear" equation set as follows, with nonlinear optimization goal. P(0) + P(1) = 1 P(0, 0) + P(0,1) = P(0) P(0) < 1 P(1) < 1 P(0,0) > 0 P(0,1) > 0 ...
7
votes
2answers
787 views

How can I, as a future mathematician, contribute most to Smart Grid research?

After I've finished my Master's degree in mathematics, I too want to use my powers for good. One endeavour I consider good is the pursuit of the design and implementation of a Smart Grid which will, ...
2
votes
1answer
1k views

Calculating the gradient without knowing the function

I have to develop an optimizer for a simulation. There are 7 reference values $$ r_1, r_2,\ldots,r_7 $$ (certain values which are expected to show up) and 7 corresponding actual values $$ a_1,a_2,\...
5
votes
1answer
606 views

Remez algorithm for best degree-0ne reduced polynomials with same endpoints

Given a function $f(x)$ on [-1,1], the Remez algorithm can find the degree (at most) $n$ polynomial $P_n(x)$ that minimizes the maximum error between $P_n(x)$ and $f(x)$ on that interval. It is an ...
2
votes
2answers
258 views

When $\min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y)$?

When $$ \min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y) \qquad? $$ I mean when we are minimizing a function with respect to two variables, under what conditions we are allowed to ...
1
vote
0answers
151 views

Ritz method for the buckling of a plate using calculus of variation

I need to find buckling of plate. And i have got the constant in analytical solution which can be founded from variation methods. How can i find polynomial for the same function for Ritz method? I ...
2
votes
1answer
127 views

Multiobjective optimization with two real functions over two real vector spaces

Question: Does anyone know about a book, a paper or an algorithm for the following optimization problem? What are the sufficient conditions for the existence of the joint optimum, and how to find it?: ...
1
vote
1answer
244 views

Sparse least-squares fitting of discrete probability distributions

I have an optimization problem involving $n$ discrete probability distributions and I am looking for a suitable solution for this problem that I can implement as computer program. Let the vector $\...
0
votes
0answers
123 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') using ...
6
votes
1answer
1k views

Levenberg-Marquardt - Is forcing Hessian to be positive definite OK?

I am often doing parameter estimation using Levenberg-Marquard method which involves solving the following linear system at each step: $$(H+\lambda I)\delta=r_{i}$$ where $H$ is a square Hessian ...
1
vote
1answer
4k views

How to solve transcendental equations with MATLAB?

Here's the equation: $$ - \frac{MN}{\sqrt{2\pi \left( \sigma_1^2 + \sigma_2^2 \right)}} \frac d {dy} \exp \left( -\frac {y^2} {\left( \sigma_1^2 + \sigma_2^2 \right)} \right) = k \left( x-y \right) $$...
1
vote
1answer
56 views

surface approximation using least squares

I am studying the following problem. Soppose you have two Bezièr patches with a common curve; suppose that the control points of the two patches are given by some initial guess (they are all known). ...
3
votes
0answers
244 views

Does convexity of a function guarantee tractability of finding its minimum?

Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not. ...
0
votes
1answer
221 views

Better than Runge-Kutta-Fehlberg 4(5) at high order?

I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using 128-...
2
votes
0answers
146 views

Divergence of Gradient Method

Is there any example of a continuous differentiable function out there, in which the gradient method with Armijo's stepsize-rule doesn't converge? I found it pretty hard to create one myself because ...
4
votes
0answers
175 views

Qualities of Projected Gradient Methods

Consider the following constrained minimization problem: $ min_{x \in X} \ f(x) $ where $ X \subset \Bbb{R}^{n} $ is a nonempty closed convex set and f is continuously diferentiable. I'm ...
2
votes
1answer
166 views

Formulate optimization problem

My research area has "nothing to do with mathematics" but I still find it full of optimization problems. Therefore, I would like to learn to formulate and solve such problems, even though I am not ...
2
votes
1answer
851 views

What Is The Gap Between 2 And the first IEEE number larger then 2

What is the gao between 2 and the first IEEE single format number larger than 2? What is the gap between 1024 and the first IEEE single format number larger than 1024 What is the differece between ...
1
vote
0answers
224 views

Maxima with equality constraints

I am trying to create an algorithm that finds the global maximum to a function with (in)equality constraints numerically. However, I am trying to fit that into a webpage (for example, via javascript. ...
0
votes
1answer
883 views

Finding a Fixed Point Solution

I'm trying to solve the following where $A,B >0$ in the most general case (don't assume they are equal). Wolfram Alpha cannot compute this in the allowed time, but I feel some fixed point ...
2
votes
1answer
203 views

Graphically, what is positive semidefinite-ness?

Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating: \begin{align} \mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla f(...
2
votes
0answers
73 views

Customising force-directed graph layout

I would like to implement a variant of the force-directed graph layout where some nodes are constrained to moving only along a predefined curve (e. g. circle). I looked at some implementations using ...
0
votes
1answer
190 views

Book on theoretical computational optimal control

I'm looking for a comprehensive introduction to the theoretical side of optimal control, existence of solutions and so on, including theory behind numerical solution methods. Regarding the latter I'm ...
3
votes
2answers
123 views

Numerical optimization question

I need to solve the following optimization problem. \begin{aligned} \min_{\lambda_0,\lambda_1} & \sum_{t=1}^T\sum_{n\in\{4,8,12,16\}} \left(\frac{1}{n}A_n + \frac{1}{n}B_n^\top X_t + y_t^{(n)}...
0
votes
1answer
585 views

Optimize material in a can

A can in the shape of a right circular cylinder is to be constructed to contain 1000 cm$^3$. The circular top and bottom of the can must have a radius of 0.25 cm more than the radius of the can so ...
0
votes
1answer
436 views

Solving a Minimization Problem With a Limited Set of Vector Inputs

My numerical analysis skills are a bit rusty on this, I plan to use scipy/numpy or octave to approach the solution but I need a pointer on how I should transform the problem in a way that it can be ...
3
votes
1answer
396 views

Solving ill posed linear equations

Given a set of linear equations $AX=B$, say $A$ is an ill posed matrix (has a few singular values equal or very close to zero), which numerical algorithm (conjugate gradient, least squares or steepest ...
1
vote
2answers
328 views

Compressed sensing, approximately sparse, Power law

An x in $\mathbb{R}^n$ is said to be sparse if many of it's coefficients are zeroes. x is said to be compressible(approximately sparse) if many of its coefficients are close to zero.ie Let $x=(x_1,...
3
votes
2answers
4k views

Fitting a 3D line to a 3D line point cloud

I want to fit a 3D line to a 3D line point cloud using numerical optimization. Currently, I'm using Steepest Descent and the error function is a function of $\alpha$, $\beta$, $\gamma$; that is, 3 ...
3
votes
1answer
1k views

Root Finding Algorithm for Discrete Functions

I was recently working with functions of the form $$N - \sqrt{\frac{N}{x}}\cdot\left\lfloor \frac{N}{\sqrt{N/x}}\right\rfloor + \sqrt{\frac{N}{x}} - \left\lfloor \sqrt{\frac{N}{x}}\right\rfloor$$ ...
7
votes
2answers
243 views

Non-convex optimization: $\min ||y-Ax||_p$ for very small $p$ given that $||x||_2=1$

I need to find $x$ that minimizes the cost function $\|y-Ax\|_p$ when $p$ is close to $0$, subject to the constraint $\|x\|_2=1$ where $x$ and $y$ are vectors in $\mathbb{R}^n$ and $A$ is an $n\times ...
2
votes
1answer
2k views

Optimizing integral functionals using Matlab

I am looking for some bibliography regarding solving integral optimization problems numerically (preferably using Matlab). I want to solve problems of the type $$ \min_{r \in A} \int_a^b (f(x,r(x)...
3
votes
1answer
110 views

least square problem

Let $1<p<\infty $.We define the space: $L_{V}^{p}(-1,1)=\left \{ f:(-1,1)\rightarrow \mathbb{R}:\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx<\infty \right \}$ We define the norm: $\left \| ...
1
vote
1answer
294 views

Hessian gives a worse approximation of a multivariate function

I have a real, smooth, multivariate (with 10 variables or many more) function, for which I have the exact Jacobian and Hessian. It turns out that unless the norm of the increment of the function is ...
4
votes
1answer
261 views

What does “overdetermined” mean

When we say a problem is an overdetermined system, what do we mean by that in a rigorous fashion? Thanks.
0
votes
1answer
77 views

unstable optimizer, stable objective

I am trying to minimize a convex objective numerically using gradient descent. I select the starting point randomly. I repeat the experiment multiple times. The optimal objective value I get each time ...
1
vote
0answers
133 views

Maximizing noisy unknown function

I'm interested in maximizing a function $f(\mathbf \theta)$, where $\theta \in \mathbb R^p$. The problem is that I don't know the analytic form of the function, or of its derivatives. The only thing ...
0
votes
1answer
389 views

Global optimization problem, whats a good approach? Branch and bound method?

Problem: Given a 4-dimensional parameter space. We place bounds on the parameter, say, 0 to 100, so that we have a 4-dimensional rectangle as a search space. We have a cost function that takes the ...
9
votes
2answers
560 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, ...
3
votes
3answers
2k views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
3
votes
1answer
469 views

Newton's method

I'm trying to understand how the Newton's method in optimization works. This is the algorithm: $S_0)$ Choose $x_{0}\in \mathbb{R}^{n},\rho>0,\ p>2,\ \beta\in(0,1), \displaystyle \sigma\in\...
1
vote
1answer
192 views

Optimization for large scale linear problem with equality constraint

Given the wide range of optimization methods, which is the appropriate method to use? I am thinking of using either linear programming (interior-point methods) or augmented Lagrangian methods. Which ...
1
vote
1answer
856 views

Minimization of function expressed with vectors and matrices

I need to find vector $\bf{p}$ in the following system: $$\bf{0} \approx \bf{W} \left[ \bf{C}^2 \bf{p} - \bf{d} \right]$$ $$\bf{0} \approx \varepsilon \bf{p}$$ In the above, $\bf{0}$ is a vector, $\...
1
vote
2answers
456 views

Portfolio Optimization Problem Without Correlation Info

I received this interesting problem from a friend today: Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following ...