Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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Lagrangian multiplier vectorial form

When i have this problem: $min f(x)$ $h{_i}(x)=0, i=1,\dots,m$ $g{_j}(x)<=0, j=1,\dots,p$ I can use the Lagrangian multiplier to write function in: $L(x,\lambda,\mu)=f(x)+\sum_{i=1}^{m} \...
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Optimum point of $f(s) = \int_0^{\pi} \frac{ \exp(-s) y \cos(ky)}{s^2+y^2} \,dy $

Is it possible to find optimum point for the following function f(s) (i.e. $df/ds=0$): $$ f(s) = s e^{-s} \int_0^{a} \frac{ y \cos(\frac{\pi}{a} y)}{s^2+y^2} \,dy $$ or $$ f(s) = s e^{-s} \int_0^{a/...
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Numeric solution for a non-linear system

It has been a while I have not practiced mathematics but I should have enough background to get your answers if well detailed. I have an n-by-n matrix, let's call it D, where dij represents the ...
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44 views

Finding the basis functions given the boundary values and values of derivatives at the boundary

Given an interval $I=[a,b]$ we define $$P_3(I):=\{v:I\rightarrow\mathbb{R}\mid v \text{ is a polynomial of degree} \leq 3 \text{ i.e } \\v=a_3x^3+a_2x^2+a_1x+a_0 \text{ for } a_i\in\mathbb{R}\}.$$ How ...
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53 views

How to choose a proper numerical optimisation method

Given a problem in numerical analysis in finance/econometrics, how to decide whther to choose Monte Carlo, Newton Raphson , Finite Difference , Gradient descent? I had this silly misconception that ...
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171 views

What Does Coordinate Descent Actually Do? Any Intuition Behind Coordinate Descent?

We've done a bunch of theoretical stuff in my optimization class, but basically no time for the actual implementation details. I'm trying to get an understanding of coordinate descent, which if I'm ...
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1answer
310 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
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49 views

How can be a conservative field constraint be efficiently implemented in a continuous optimization problem?

Suppose we have the following continuous optimization problem: $$ \underset{x}{\mathrm{minimize}}f\left(x\right) $$ subject to $$ \exists X:\nabla X=Jac\left(X\right)=x $$ where $f$ is a function $f:...
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116 views

Newton Raphson Method Overestimating Parameters

I have implemented an almost plain vanilla algorithm to find the MLE estimates of 3 parameters in a log-likelihood function (in R.) When I test my algorithm with some simulated data it does pretty ...
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1answer
110 views

Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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182 views

Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
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340 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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158 views

Writing down the KKT optimality conditions

Consider the problem Minimize $(1/2)\times{x}^{T}\times Q\times x+{P}^{T}\times x$ Subject to $(1/2)\times {x}^{T}\times P\times x+{d}^{T}\times x≤r$ Where Q and P are n×n matrices, P is ...
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140 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve $c(u)$ of degree $4$ at the points $-1$, $0$, $1$ as: $c(-1) = 4$; $c'(-1) = 4$; $c(0) = 6$; $c(1) = -4$; $c'(1) = -6$; ...
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1answer
44 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
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82 views

Can a 6-arm star be convex

Please help me with the following question. Suppose that the constant level contours of some function $V:\mathbb{R}^{2} \rightarrow \mathbb{R}$ have the shape of a symmetric 6-arm star. Can such a ...
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280 views

What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot \...
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260 views

Dijkstra and grid cells?

I have a problem with Dijkstra's algorithm. I'm trying to calculate the shortest transport time between all cells in a grid used by a general circulation model (think oceanography or meteorology). the ...
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1answer
134 views

Solving an equation with multiple unknowns from different sets of natural numbers

Is it possible to solve an equation: a*x+b*y+....+c*z-n = 0 where {a, b,..,c, n} are natural numbers and {X, Y,...,Z} are different sets of natural numbers? Is it possible to find minima if there is ...
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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 ...
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1answer
303 views

Adaptive step size Runge Kutta: getting a specific value

A system of differential equations can be solved using a numerical integrator with adaptive stepsize $ h$, e.g. Runge Kutta Fehlberg Now suppose I want to integrate a set of differential equations $\{...
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19 views

Analysis of optimization procedure for two equivalent optimization problems

I come up with this question. Assume the optimization problem is: $$argmin_w\|\hat{y}(w)-y\|^2$$ Now suppose there is a overcomplete set for the representation of $\hat{y}$ and $y$, this set has lots ...
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1answer
348 views

Optimization - Optimal velocity for fuel consumption

I am working through a series of questions for my 1st year uni maths course and I am really stuck here so any help would be appreciated. I know how to optimize 3D objects and areas but just can't work ...
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1answer
28 views

Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares

Adjust the data up curve φ(x) = α1e^(α2x) by the method of least squares: Here's what I've done so far but I think it is wrong(and sorry for the bad english) --x | 0    | 1    | 2   | 3   | 4   ...