Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [nonlinear-programming]

Questions about the theory and numerical algorithms for optimizing (minimizing or maximizing) nonlinear functions, possibly subject to equality and/or inequality constraints.

0
votes
1answer
41 views

Slightly change two vectors to satisfy a constraint

$\vec{a}\cdot\vec{b} \approx c$ $\vec{\alpha} \cdot \vec{\beta} = c$ $\vec{\alpha}$ is close to $\vec{a}$ and $\vec{\beta}$ is close to $\vec{b}$ Given $\vec{a}$, $\vec{b}$ and c, how to find $\vec{\...
1
vote
1answer
69 views

Why does Newton's method with Linear Equality Constraints use KKT condition?

Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint. The problem only has equality constraint. Why ...
1
vote
1answer
81 views

Piecewise-Linear Quadratic Optimization for an “Almost Convex” Problem

I have a 7-14 dimensional piecewise linear cost function I'd like to minimize with two quadratic terms of the form: $$ f(X) = X^tCX + d \sum_i |x_i-x^*_i|^2 + \sum_i P_i(x_i-x^0_i) $$ $$ \sum_i x_i ...
2
votes
0answers
154 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
4
votes
1answer
184 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 \...
0
votes
0answers
27 views

Artificial Neuron, impossible for convergence?

I'm trying to learn AI and my idea was to build an artificial neuron network to construct the nonlinear generator. I initially tried to let each vertex have 5 dendrites to the next vertex, each ...
1
vote
0answers
61 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) \...
2
votes
1answer
25 views

Constraints 'exactly/at most one non-zero element' without binary variables

In a much larger MINLP problem, I have set of variables $\{a_{ij}\}_{m,n}$, such that $0 \leq a_{ij} \leq 1 $ for all $i$, $j$, which I could think of as a matrix, for which I have two requirements: ...
0
votes
1answer
54 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 ...
4
votes
1answer
80 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}^...
3
votes
0answers
46 views

Nonlinear Sylvester-Like Equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
1
vote
0answers
63 views

Nonlinear global optimization algorithm that can use dynamic programming

I've asked this question on stackoverflow 2 weeks ago, but, judging by zero response, that probably was the wrong forum. Therefore copying it here: Let F0,...,Fn ...
0
votes
1answer
56 views

Why can't we discretize continuous domains in distributed non-convex constraint optimization problems?

Consider a non-convex distributed optimization problem. We have $X$ = a set of $n$ decision variables: $x_i$ where $i=1..n$ and $x_i \in R$, the set of Reals. We have $F$ = a set of $m$ constraint ...
1
vote
1answer
107 views

Correct way to model an embedded reinforcement (non linear FEM)?

I need to add to an existing FEM solver some embedded reinforcement element. This would give me the possibility to model/solve concrete structure (reinforced with steel rebar) taking into account the ...
3
votes
1answer
158 views

Why do active set methods or the simplex method pivot only one variable at a time?

Why do active set methods or the simplex method pivot only one variable at a time? Ostensibly, we could add multiple columns to the basis during pivoting, but the standard presentation of the methods ...
1
vote
2answers
85 views

How can i solve this non-convex multi-variable optimization problem?

I want to solve the following optimization problem: $$\min_{A,B,X} \|Y-AX\|_F^2 + \lambda_1 \|Z-BX\|_F^2+ \lambda_2 \|B\|_F^2$$ $$s.t ~~x_{ij}~ \geq 0$$ in which, $Y$ and $Z$ are data matrices and ...
6
votes
1answer
81 views

$L_2$ projection with integer constraints and prescribed sum

Suppose I am given a vector $v^0\in\mathbb{R}^n$ and integers $k,\ell\in\mathbb{Z}$. Assuming $k$ is close to zero (e.g. $0\leq k\leq5$), is there an algorithm for solving the following integer ...
1
vote
0answers
47 views

Object-oriented non-linear solving in python

I'd like to build a system in Python, consisting of (broadly speaking) objects which are internally described with (not necessarily just linear) equations, that I can connect with each other - similar ...
1
vote
0answers
61 views

Minimizing the products of variables

My problem Maximize $$\min_{i} \{\ c_i \cdot \prod_{j \in A(i)} {x_{j}} \prod_{j \in B(i)} {y_{j}} \} $$ Subject to \begin{align} &\sum_{j \in C(k)} x_{j} = 1,\ \forall k \\ &l \leq x_{j}...
2
votes
1answer
101 views

Optimization of multiple functions

I have 3 functions which consist of 6 variables $p_1,p_2,p_3,p_4,p_5,p_6$. The value of each function is equal to $x$ (say): \begin{align} f_1 &= \operatorname{sign}(2-p_1) \sqrt{|2-p_1|} + \...
0
votes
0answers
140 views

Solving optimal control problems using third-party NLP solvers

I've been doing a bit of work on the so-called pseudospectral optimal control for trajectory optimization of a launch vehicle into orbit. However, comparing my results to the literature, I seem to be ...
1
vote
0answers
159 views

Constrained optimization: Stationary point vs. Nash point

1s question: definition of stationary point for constrained optimization As far as I know, a stationary point of a constrained optimization problem is a stationary point of the Lagrangian (that has ...
1
vote
0answers
44 views

How is the Gastner-Newman equation implemented to create value-by-area cartograms?

There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
0
votes
1answer
201 views

Example Problem to Demonstrate BiCGStab

So our team has been able to code up a BiCGStab implementation for a class project, and we'd like a potential example problem to try it out on. So far, we've talked about a 1D Laplacian with Neumann ...
6
votes
1answer
121 views

Largest hypercuboid inside a polyhedron

Given a polyhedron $\mathbf{Ax} \leq \mathbf{b}$, how to find the largest hypercuboid, with unknown center $\mathbf{x_{0}}$ and side lengths $2\epsilon_{i}$, which are aligned along the co-ordinate ...
0
votes
1answer
3k views

Backtracking-Armijo Line Search Algorithm

EDIT based on comments below: I add the mathematical formulation of my problem below. I am trying to solve an equation of the form $$ \partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv ...
1
vote
2answers
38 views

Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available

I have a computationally expensive objective function, whose analytical form is not available. The only input argument to the objective function is an integer variable. The goal is to compute the ...
4
votes
2answers
63 views

Gradient of function after renormalization of variables

I have to minimize a function $f(\mathbf{x})$, where the vector $\mathbf{x}\in\mathbb{R}^n$ satisfies $|\mathbf{x}|=1$. So I tweaked the code of $f$ so that it renormalizes $\mathbf{x}$ as the first ...
0
votes
1answer
62 views

Perturbation in bounds given the perturbation to constraints

Given a feasibility problem with both inequality and equality constraints, I'm interested in the sensitivity of the bounds of the region to changes in the constraints. To help with answering the ...
1
vote
0answers
2k views

Split step method applied on nonlinear Schrodinger equation does not result in self focusing

I'm trying to simulate self focusing in the case of anomalous dispersion and positive Kerr nonlinearity in the nonlinear Schrödinger equation $\frac{\partial a}{\partial t} - i\frac{\partial^2 a}{\...
2
votes
0answers
60 views

Global optimization with known distributions of some variables

I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
1
vote
0answers
84 views

2D Multilateration with constraints as distance information

In my current problem, I am looking for an algorithm to reconstruct the position of multiple points in the 2D euclidean plane with incomplete distance information, $d_{ij}=||x_j-x_i||\in\text{dom}_a$. ...
1
vote
1answer
98 views

SOCP: Recovering primal from dual

Consider the following second-order cone program (SOCP): $$ \begin{array}{rl} \min_x & c^\top x\\ \mathrm{s.t.} & \|A_ix+b_i\|_2 \leq c_i^\top x+d_i \ \forall i \end{array} $$ Suppose I solve ...
3
votes
1answer
168 views

Minimizing linear objective on intersection of convex sets

Suppose I wish to solve the following optimization problem: $$ \begin{array}{rl} \min_{\mu\in\mathbb{R}^n} &\mu^\top c\\ \textrm{subject to} & \mu\in C_1\cap C_2\cap\cdots\cap C_k, \end{array} ...
2
votes
0answers
185 views

Using Line Search Method for Constrained Optimization

Suppose we have a $f(x)$ to be minimized (we only know that $f(x)$ is three-differentiable), and a feasible, convex set of $S$ such that all $x$ belong to $S$. Using line search method, how we can "...
1
vote
0answers
140 views

Numerically solving a system of stiff nonlinear PDEs

I am attempting to numerically solve the following: \begin{align} \frac {\partial y_1}{\partial t} &= i(y_2y_3 - y_2^*y_3^*) - y_1 \tag{1}\\ \frac {\partial y_2}{\partial t} &= y_1^*y_3 - y_2 ...
1
vote
0answers
106 views

Solving a large system of nonlinear equations, where timeseries are the unknown

I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations: $$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
1
vote
0answers
117 views

Minimization of the sum of convex function and non-linear non convex function

I'm trying to minimize the unconstrained scalar sum of a quadratic convex function (to which a convex optimizer is readily applied) and a non-linear and non-convex function which is differentiable. ...
1
vote
0answers
111 views

Scaling a vector-valued non-linear function for numerical optimization/minimization

I am trying to minimize a non-linear vector-valued function in MATLAB. As a test case for my code, I try to minimize a function whose solution I know apriori. The problem is that one of the ...
1
vote
0answers
38 views

Alternative to two “for” loops in finding best neighborhoods for TSP?

I am trying to solve Travelling Salesman Problems using tabu search. I have been able to successfully find "near enough" optimal solutions (as well as one optimal, yay!). For the moment I am using ...
1
vote
2answers
301 views

How to solve a constrained optimization problem using minFunc or minConf

I am trying to solve the following optimization problem: \begin{align} &\min\limits_{s} \rm{tr}\left(S^T S\right) + \mathrm{tr}\left(\left(S^T S\right)^{-2}\right)\\ &\text{subject to }\rm\...
1
vote
1answer
131 views

How to efficently solve: min $\sum_{ij}(a_{ij}x_{ij}^2 + b_{ij}x_{ij})$ s.t

I am trying to solve the following problem, where $a_{ij} \ge 0 \ \forall i,j$: \begin{align} \mbox{minimize}\quad & \sum_{i=1}^m\sum_{j=1}^n (a_{ij}x_{ij}^2 + b_{ij}x_{ij})\\ \mbox{subject to}\...
1
vote
0answers
120 views

Software Implementing Jiles-Atherton

I have been doing much research on the internet to see if there was a FEM simulator that natively implemented the Jiles-Artherton model whether it be commercial or open source So far, my conclusion ...
4
votes
1answer
216 views

Sum of Inverse of Variables in an Optimization Problem

I have the following optimization problem: $$ \begin{array}{ll} \text{Minimize} & \frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{d_n} \\ \text{Subject to} & A x \leq b \end{array} $$ where ...
8
votes
3answers
2k views

Is there a constrained nonlinear optimization library like IPOPT that runs on GPUs?

Somebody on my team wants to paralelize IPOPT. (at least some of functions of it). I have not been able to find a GPU implementation of it or a similar package. Nor I've found anything on their docs. ...
3
votes
0answers
134 views

Find constrained vectors maximizing angles between them - methods?

This is related to a question I had asked earlier, with the distinction that earlier I did not have a non-linear objective functional to minimize. The problem is reproduced below with added ...
3
votes
1answer
209 views

Symmetric nonnegative matrix factorization

Suppose $A\in\mathbb{R}_+^{n\times n}$ is symmetric. I would like to factorize $A\approx UU^\top$ by solving $$ \begin{array}{rl} \min_U & \sum_{ij} \left(A_{ij}\ln\frac{A_{ij}}{[UU^\top]_{ij}}+[...
1
vote
0answers
144 views

Oscillating convergence in my Resilient BackPropagation (RPROP) implementation

I have implemented in matlab a neural network that uses rprop's algorithm to update its weights. Strangely the error on the training set does not converge to a local minimum, but oscillates. Here is ...
2
votes
0answers
88 views

non convex, non linear optimization involving matrix differential equation solution

I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure ...