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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2
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1answer
162 views

Can the Power Method be used here?

Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected ...
4
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0answers
84 views

What Derivative-free optimization method should I use when my initial guess is very good?

I am trying to minimize a function where my initial guess is quite close to the minimum. I'm trying to minimize $$f(q) = \text{angle}(qw_1q*, v_1) + \text{angle}(qw_2q*, v_2) + \text{angle}(qw_3q*, ...
5
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0answers
233 views

conjugate gradient for Newton's method with non positive definite Hessian matrix

I want to minimize a non-linear function $f(x)$ using Newton's method. At each optimization step, I compute a descent direction $d$ to update $x$ using a second-order approximation of $f(x)$: $$ \...
3
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0answers
87 views

Inconsistency in optimize.minimize

I am trying to fit a time-dependent curve at each time step. I do so in minimizing along $x_c$ the quadratic error between the curve and a reference solution $ 1/(1 + \exp\left(\sqrt{S}(x-x_c)\right) $...
3
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1answer
56 views

Software for finding a minimum vertex cover for a hypergraph

A hypergraph $H = (V,E)$ consists of a finite set of vertices, say $V=\{1, \dots, n\}$ and a set of hyperedges $E \subseteq \mathcal{P}(V)$. We call $H$ a $k$-hypergraph if all $|e| = k$ for all $e\in ...
5
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1answer
136 views

Generate approximately semi-orthogonal tall matrix approximately satisfying constraints

I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where: Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative. Each $A_j\in\mathbb{R}^...
6
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1answer
501 views

Solvers for Quadratically Constrained Quadratic Programs (QCQP) with complex variables

I'd like to know whether there are any publicly available tools for solving QCQP with complex variables (and constraints therefore expressed through Hermitian matrices). What I have found so far is ...
4
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1answer
216 views

How to solve the following Frobenius norm-minimization problem?

Background We know how to solve the following minimization problem $$ \min_{X} \lVert AX - B \rVert_F^2 $$ But what about the extended version? $$ \min_{X} \lVert A \begin{bmatrix} X & X^2 \...
3
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1answer
196 views

How to solve the inverse problem of least-squares?

Focusing on following least squares problem: $$\min\limits_{V} \lVert Z - WV \rVert _{_F}^2$$ $$Z∈{R}^{m\times n},\quad W∈{R}^{m\times k},\quad V∈{R}^{k\times n},\quad k\lt m\lt n $$ This problem ...
1
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1answer
208 views

How to use CSDP to express a semidefinite program?

I am trying to use CSDP and am struggling with it. Consider, for example, the following semidefinite program $$\begin{array}{ll} \text{minimize} & 0\\ \text{subject to} & Q - A' Q A - \...
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1answer
682 views

FMINCON Step Size Tolerance

I get following error after implementing the attached code. Error Message "fmincon stopped because the size of the current step is less than the default value of the step size tolerance but ...
2
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0answers
207 views

How to prevent BFGS from getting stuck on astronomically large gradient?

I have implemented BFGS myself from scratch in order to solve minimization problems. Part of BFGS, as I understand it, is that the approximation to the Hessian is supposed to be positive definite, ...
1
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0answers
33 views

Best Possible Convex bounds for optimization problems

Suppose we have a primal problem $$ p^{*}=\min_x f(x), \\\text{s.t.}~~ h_i(x) \leq 0, $$ where $f(.)$ and $h_i(.)$ are possibly non-convex. Then its Lagrangian is $$\mathcal{L}(x,z_i)= f(x) + \...
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0answers
69 views

Nonlinear least square optimization

Problem description Given data at many time instance $t$, $$\min _{\alpha, \Lambda, \beta} \lVert y(t) - \alpha e^{\Lambda t} \beta \rVert_F$$ with $$ \lVert \alpha \rVert_2^F = 1 $$ where $y(t) \...
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0answers
45 views

Space covering optimization

I have the following problem: In the space $E=\{1, 2, \dots, N_x\} \times \{1, 2, \dots, N_y\}$, I want to define $N_R$ rectangles $R_k=\{x_k^0, \dots, x_k^1\}\times\{y_k^0, \dots, y_k^1\}$ which ...
1
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1answer
116 views

Some proof that linear translations and rotations of a bound-constrained function are equivalent

For example, I have a function to optimize: $$f_1(x,y) = x^2+y^2, \quad x_{lb}\le x\le x_{ub},\quad y_{lb}\le y\le y_{ub}$$ Then I apply rotation by $\theta$ plus translation by $x_0$ and $y_0$: $$f_2(...
0
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1answer
120 views

ill-conditioning

I am struggling with the following exercise from the book of Nocedal, Numerical optimization, chapter 2, exercise 2.12: Suppose that a function $f$ of two variables is poorly scaled at the solution $...
3
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2answers
194 views

How do I check if a loss function can achieve its minimum?

For example, the convex function $f(t)=e^{-t}$ doesn't achieve its minimum 0 on the real line. In a linear regression with $p$ predictors $X$, the loss function $f(\beta)=||Y-X\beta||^2$ achieves its ...
0
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1answer
175 views

Can I convert CUDA core to CPU core and use it as cpu core while running any program?

I was using Metatrader5 and have designed a strategy for trading using MQL5 programming language. While I was running a Strategy Optimization process, I saw the it will need 10,00= iterations or ...
0
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1answer
49 views

Conflicting definition of limit point

This question was raised at a different place without sufficient answers. Definition 1: We say that a vector $x \in R^n$ is a limit point of a sequence $\{x_k\}$ in $R^n$ if there exists a ...
6
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1answer
315 views

What is required of the objective function in order to use Gauss Newton method?

From what I understand, the Gauss-Newton method is used to find a search direction, then the step size, etc., can be determined by some other method. In addition to that, are the following ...
1
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1answer
43 views

How to decide on what techniques to adopt in Genetic Algorithm optimization

Does it really matter which techniques we use in the process of GA optimization? For instance, if I use the Roulette Wheel Technique instead of Tournament Method for selection, two-point crossover ...
0
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1answer
374 views

I've developed a derivative-free optimization method, looking for comments

Here is the URL: https://github.com/avaneev/biteopt I've tested it on numerous global optimization benchmarking functions (included), and on real-world hyperparameter optimization problems I have. ...
3
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1answer
283 views

How to numerically minimize a functional?

How to numerically minimize a functional, for example, $$J[y]=\int_{x_1}^{x_2}L(x,y(x),y'(x))dx$$ An equivalent problem is to solve the Euler equation for this functional as a differential equation. ...
1
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1answer
42 views

Distribute sources among destinations

There are $n$ sources with the following positive volumes: $p_1, ..., p_n$ and there are $m$ destinations with the following positive volumes: $q_1, ..., q_m$. It is known that $p_1+ ...+ p_n=q_1+ ...+...
7
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3answers
971 views

Finding the first N roots of transcendental equation

I need to find the first $n$ roots of the transcendental equation \begin{equation} F(k) = J_m'(kr)Y_m'(k)-J'_m(k)Y'_m(kr) \end{equation} for integer values of $m$ and any $r \in [0,1)$ where $J'$ ...
0
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1answer
672 views

Defining a soft constraint in cvxpy

I am using cvxpy to do a simple portfolio optimization. I implemented the following dummy code ...
4
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1answer
121 views

Nonlinear least-squares solvers vs. generic minimization

A nonlinear least-squares problem with $F:\mathbb{R}^m\to\mathbb{R}^n$, $$ F(x) \to \min_x \quad (\text{in the least-squares sense}) $$ really means minimizing $$ \frac{1}{2} \|F(x)\|^2 \to \min_x. $$ ...
2
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1answer
161 views

What is the fastest way to solve Ax=b (subject to constraints and an absolute term)

I am trying to solve/optimize $Ax=b$ in the least squares sense subject to box constraints; a few (less than 5) equality/inequality constraints; and an absolute function penalty (or some other ...
1
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0answers
21 views

How does the MADS algorithm work in practice

Mesh Adaptive Direct Search (MASH) is an algorithm for black box optimization I want to understand an implement this method to solve some 2D multivariate blackbox function $f(x,y)$, but am having ...
6
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2answers
1k views

What is the most appropriate derivative free optimization algorithm

We can use random optimization/ derivative free/ direct search to find the minimum of some black box function $f$. If I have some 2D black box function, $f(x,y)$ - which I know to be convex - what ...
3
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1answer
383 views

Linear constraints for L-BFGS-B

I know L-BFGS-B only supports simple box constrains of the form: $l_i \leq x_i \leq u_i$, where $l_i$ and $u_i$ are constants. For my specific optimization problem, I need to specify some simple ...
1
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1answer
1k views

How to define the derivative for Scipy.Optimize.Minimize

I am trying to use scipy.optimize.minimize to minimise a quadratic objective function: $f(x) =x^\top Q x$. As a start, I have successfully implemented this using the built-in Nelder-Mead Simplex ...
0
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1answer
191 views

CPU and GPU influence on task parallel execution performance

This question is mainly about hardware, but also about software. In my current work I have approximately 68 millions of combinations that I am iterating through, in parallel. For each of those ...
1
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0answers
42 views

Genetic Algorithm: Need some clarification on selection and what to do when crossover doesn't happen

I'm writing a genetic algorithm to minimize a function. I have two questions, one in regards to selection and the other with regards to crossover and what to do when it doesn't happen. Here's an ...
1
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0answers
211 views

Levenberg-Marquardt for root-finding: just square the function?

This question might be so obvious and trivial that I'm having a hard time googling it. I have a multivariate root finding problem that I'm trying to solve in C# and the library that I'm trying to use ...
0
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1answer
57 views

reduced system: primal-dual interior point method for nonconvex constrained problem

When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method? For ...
1
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0answers
16 views

Procedure to identify characteristic properties of unknown functions in a DAE model

I have a system of 1st order odes given by $$ \dot{x_1}(t) = \alpha_1 f_1(x_1,t) + \beta_1 u(t) \\ \dot{x_2}(t) = \alpha_2 f_2(x_2,t) + \beta_2 u(t) $$ They are constrained by an algebraic equation ...
1
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0answers
65 views

overlapping additive schwarz [closed]

I am solving 1D laplace problem discretized with finite differences (3-point stencil). I would like to use additive Schwarz method in classical form: $U_{k+1}=U_{k}+M^{−1} r_k,$ where $r_k=F−A U_k$...
3
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1answer
141 views

Obtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints

I would like to obtain a feasible solution for an under-determined system of linear equations, $$Ax=b$$ where, $A \in \mathbb{R}^{7\times9}, \, x \in \mathbb{R}^{9\times1}\text{and } b\in\mathbb{R}^...
4
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1answer
506 views

Optimization of a blackbox function with an equality constraint?

I believe this would be an interesting problem. I have a blackbox function which can take 2-60 input variables $(X_1,X_2,...X_n)$ which are to be optimized. I'm calling this objective function as a ...
0
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1answer
65 views

Adaptive gradient descent

I want to minimize some multivariable function $\Delta(\alpha, \beta)$. I know that this function has a zero point, $\Delta(5, 5) = 0$. Starting from some $(\alpha, \beta)$ close to $(5,5)$ (e.g. (4....
6
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1answer
746 views

Ways to speed up solving an LP with Google's ortools

I'm having an issue solving an LP of the form: $$\min z = c^Tx$$ $$\text{s.t.}$$ $$Ax \geq b$$ $$x\geq p$$ $1 \leq a_{ij} \ll b_i$, $p \leq 0$, and $c \geq 0$ The specific problems I'm running into ...
3
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0answers
102 views

Are the No Free Lunch Theorems Useful for Anything?

I have been thinking about the No Free Lunch (NFL) theorems lately, and I have a question which probably every one who has ever thought of the NFL theorems has also had. I am asking this question here,...
1
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3answers
180 views

Optimization of a blackbox function

Let's say that we have an objective function $f(\mathbf x,\mathbf y)$ which has the parameters $\mathbf x=[x_1\ldots x_n]$ and $\mathbf y=[y_1\ldots y_n]$. Here, $\mathbf y$ is a blackbox variable ...
1
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0answers
38 views

Meaning of “points where the first- and second-order optimality conditions fail”

I am reading a paper that summarizes a set of simulations. Basically, the authors are trying to minimize some function using different optimization algorithms. They conclude: "Our findings point to ...
4
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1answer
75 views

Large-scale almost-linear optimization

I'm currently trying to minimize the following function: $$\min_{\mathbf x,\mathbf y,s} \|F\mathbf x-\mathbf a\|_2^2 + \|SF\mathbf y-\mathbf b\|_2^2 + \lambda \|\max(\mathbf x, \mathbf y)\|_1,$$ $$\...
1
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0answers
92 views

Sequential Quadratic Programming for Quadratically Constrained Quadratic Programs

A standard Quadratically Constrained Quadratic Program (QCQP) is of the form: $$ \underset{x}{minimize} \frac{1}{2}x^TP_{0}x + q_{0}^{T}x $$ $$ subject \; to \quad \frac{1}{2}x^TP_{i}x + q_{i}^{...
0
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1answer
95 views

Free access to Xeon Phi clusters

Apologies if this is not the right forum to ask, but there has been a somewhat related question here. I am working on a piece of software (nonlinear constrained optimisation, coded in C++11 and ...
2
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2answers
80 views

Find combinations of variables with variable bounds to get summed up to a required value (using Python)

Let's say I have n variables. Each variable has lower and upper bounds. I want to find all suitable combinations of these variables to sum up to a required value. An example with two variables: $$ ...

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