Questions tagged [optimization]

This tag is intended for questions on methods for the (constrained or unconstrained) minimization or maximization of functions.

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6
votes
2answers
2k views

Non-differentiable global optimization problem

I am trying to solve the following test problem which is well-known in the community in different variants: Place N = 15 points in the 3-dim. unit cube such that the minimal distance between them is ...
9
votes
2answers
1k views

Safe application of iterative methods on diagonally dominant matrices

Suppose the following linear system is given $$Lx=c,\tag1$$ where $L$ is the weighted Laplacian known to be positive $semi-$definite with a one dimensional null space spanned by $1_n=(1,\dots,1)\in\...
6
votes
1answer
430 views

Reducing degeneracy in constrained (convex) optimization problem

DISCLAIMER: I've edited the question repeatedly for clarity and to target the most relevant answer. I have the following general problem $$ \min \|h_1\cdot h_2\|^2 $$ such that $$\|g_1\wedge g_2-h_1\...
7
votes
1answer
567 views

Jacobi iteration to reduce the quadratic function

Given certain function $f(X)$ which is quadratic in $X\in\mathbb{R}^{n\times d}$, $$\frac{1}{2}tr(X^TAX) - tr(Y^TBX)$$ for positive definite weighted Laplacian matrices $A, B\in\mathbb{R}^{n\times n}...
10
votes
2answers
2k views

How is geometric programming different from convex programming?

How is (generalized) geometric programming different from general convex programming? A geometric program can be transformed into a convex program, and is typically solved by an interior point method....
15
votes
6answers
18k views

Constraints involving $\max$ in a linear program?

Suppose $$\begin{align*} \min A &\mathrm{vec}(U) \\ &\text{subject to } U_{i,j} \leq \max\{U_{i,k}, U_{k,j}\}, \quad i,j,k = 1, \ldots, n \end{align*}$$ where $U$ is a symmetric $n\times ...
11
votes
1answer
565 views

Computing standard errors for linear regression problems without calculating inverse

Is there a speedier way to calculate standard errors for linear regression problems, than by inverting $X'X$? Here I assume we have regression: $$y=X\beta+\varepsilon,$$ where $X$ is $n\times k$ ...
5
votes
1answer
125 views

Using an approximation algorithm to adapt parameter values of a given algorithm

Problem: I have an incremental online clustering algorithm which need 4 parameters that should be specified by the user before execution. The algorithm will gives "good results" if "a good parameter ...
8
votes
2answers
275 views

One-sided non-linear least squares with linear constraints

I am trying to solve a one-sided non-linear least-squares problem with linear constraints, i.e the problem: $\min_{\mathbf{x}} \quad \sum^m_{i=1} \mathbf{r}_i(\mathbf{x}) \qquad \text{ s.t } \quad A\...
3
votes
1answer
197 views

Can BFGS be used to minimise several functions at once?

I have multiple objective functions which are related to several parameters. I want to minimise more than one objective functions using several parameters. Is it even possible using BFGS? When I used ...
6
votes
4answers
279 views

Approximately “solving” a linear system of equations without a feasible solution

A linear system of equations has the form $Ax = b$, where a matrix $A$ and a vector $b$ are given, and I wish to find a solution vector $x$. Suppose that the system $Ax = b$ has no feasible solution. ...
15
votes
4answers
5k views

Linear programming feasibility problem with strict positivity constraints

There is a system of linear constraints ${\bf Ax} \leq {\bf b}$ . I wish to find a strictly positive vector ${\bf x} > 0$ that satisfies these constraints. That means, $x_i > 0$ is required for ...
3
votes
3answers
899 views

What is the best way to solve Ax = b (with A large, spd, sparse, banded and poorly conditioned)?

I'm trying to solve $Ax = b$ given a vector $b$ and a large, symmetric positive definite, sparse, banded matrix $A$ that has a very poor condition number. I know several iterative methods that ...
10
votes
1answer
5k views

Are there any heuristics for optimizing the successive over-relaxation (SOR) method?

As I understand it, successive over relaxation works by choosing a parameter $0\leq\omega\leq2$ and using a linear combination of a (quasi) Gauss-Seidel iteration and the value at the previous ...
14
votes
5answers
11k views

Minimizing the Sum of Absolute Deviation ($ {L}_{1} $ Distance)

I have a data set $x_{1}, x_{2}, \ldots, x_{k}$ and want to find the parameter $m$ such that it minimizes the sum $$\sum_{i=1}^{k}\big|m-x_i\big|.$$ that is $$\min_{m}\sum_{i=1}^{k}\big|m-x_i\big|.$$...
4
votes
3answers
288 views

How to fill a 2D set over a cartesian lattice with as few rectangles as possible?

Suppose I have a black and white image (composed of binary pixel values in a 2D cartesian array) that contains an irregular, nonconvex shape. Let's further suppose that the shape is one connected ...
12
votes
1answer
4k views

Efficient solution of mixed integer linear programs

Many important problems can be expressed as a mixed integer linear program. Unfortunately computing the optimal solution to this class of problems is NP-Complete. Luckily there are approximation ...
-2
votes
1answer
290 views

Multi-objective optimization problem - Euclidean space

I am looking for some clues for an optimization problem. My problem consists on arriving to a image by optimizing multiple layers with the pixel position probability. This is an overview of the ...
21
votes
8answers
3k views

Software package for constrained optimization?

I am looking to solve a constrained optimization problem where I know the bounds on some of the variables (specifically a boxed constraint). $$ \arg \min_u f(u,x) $$ subject to $$ c(u,x) = 0 $$ $$ ...
25
votes
3answers
19k views

BFGS vs. Conjugate Gradient Method

What considerations should I be making when choosing between BFGS and conjugate gradient for optimization? The function I am trying to fit with these variables are exponential functions; however, the ...
4
votes
2answers
328 views

best way to optimize a function with linear/non-linear parameters

I am trying to fit some raw data using a function of the form $f(r) = \sum_{i=1}^{K} d_kS_k(n_k,\alpha_k,r)$ where $S_k(n_k,\alpha_k,r) = \frac{\alpha_k ^{n_k+3}}{(n_k+2)!}r^{n_k}\exp(-\alpha_kr)$ ...
77
votes
17answers
85k views

Is there a high quality nonlinear programming solver for Python?

I have several challenging non-convex global optimization problems to solve. Currently I use MATLAB's Optimization Toolbox (specifically, fmincon() with algorithm=<...