1
vote
1answer
43 views

Matrix Maximization

I would like to solve the following optimization problem for a matrix $X$ which is symmetric and positive-semidefinite: $$ \mathrm{maximize} \, \, \, f(X) = \log \mathrm{det} X - k_1 \log(k_2 + a^T X ...
0
votes
0answers
23 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 ...
0
votes
2answers
53 views

Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
0
votes
0answers
12 views

Monotonic transformation in numerical optimization

Taking the logarithm of the Cobb-Douglass utility function ($u = x_1^a * x_2^b$) yields a utility function whose argmin is somewhat easier to derive. Since the logarithm is monotonic for $u>0$, we ...
0
votes
1answer
46 views

Optimization problem with a minimization sub-problem as a constraint

I have a problem, for predefined $x_0,z\in\mathbb{R}$, which looks like $$\min_{\alpha,x} \sum_{i=1}^n \alpha_i f_i(x_i,z) $$ subject to \begin{align} \sum_{i=1}^n \alpha_i &= 1 \\ \sum_{i=1}^n ...
0
votes
0answers
19 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
1
vote
1answer
49 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
0
votes
0answers
13 views

Linear Quadratic Bilevel Programming Problem

How to solve this type of linear-quadratic bilevel programming problem ? Please help.
1
vote
0answers
60 views

using lsqcurvefit to fit piece-wise linear

I would like to use this function to fit piece-wise linearly to a set of data. Namely, I want to fit them with several linear segments. Including other requirements, I would not want the segments ...
1
vote
0answers
30 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
2
votes
1answer
54 views

Initialization of Limited-memory BFGS (using libLBFGS)

I am using the package libLBFGS in order to minimize an objective function, for which the first derivative (with respect to the optimization variable) is known and computable. I use the default ...
3
votes
3answers
60 views

Minimization of $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$?

I am trying to find the minimal value of the expression: $log_{a}(bc)+log_{b}(ac)+log_{c}(ab)$ I think experience gives that the variables should be equal, if so then the minimal value is 6, but ...
0
votes
0answers
49 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 ...
2
votes
0answers
70 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
0answers
92 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') ...
6
votes
3answers
474 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 ...
2
votes
0answers
173 views

Optimization over function spaces

There are many methods to solve optimization over standard spaces such as real and complex numbers or vector spaces of these. In my case I have a vector space of functions of a real value to find. ...
2
votes
3answers
597 views

how to compute the gradient of a function at an extremal point

I am writing a computer program that searches for the minimum of a multivariate function $f: \mathbb{R}^n \to \mathbb{R}$. This function is in fact the sum of many functions: $$f(x) = \sum_{i=1}^m ...
2
votes
3answers
640 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 ...
0
votes
0answers
270 views

Can this non-linear optimisation problem be converted to a linear?

I have to minimize the function: $F(x)$ $F(x) = \sum_{i=1}^{M}||x_{i+1} - x_i - K(\frac{x_{i+1} + x_i}{2})||^2 + ||x_1-c_1||^2 + ||x_N-c_2||^2$ , where $x$ is a vector of $N$ scalars, $c$ are ...
0
votes
1answer
105 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 ...
9
votes
2answers
419 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, ...