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.
122
questions
0
votes
0answers
28 views
Method for implementing QP solver with matrix terms?
I am trying to implement (in code) a QP solver for the following equation:
$$\min_{u} u^{T} Wu$$
$$s.t. \; \beta u = \tau_{ref}$$
$$ Au \leq b $$
See this document, section 5.1 (Page 35)
$u$ is a ...
3
votes
2answers
92 views
MINLP with GEKKO - Modeling discrete variables
I'm trying to define a MINLP optimization problem with GEKKO in Python, and I want to use some variables with fixed values.
For my first variable, x1, I need to define the following values (as would ...
1
vote
1answer
71 views
Preconditionning for solving a non-linear system of equations with least squares
I am trying to solve a large system of non-linear equations (about a few hundred equations and variable but with less variable than equations). Given that the system is really sparse and large I am ...
3
votes
0answers
27 views
How to set up and solve acceleration-limited trajectory optimization problems?
I've been trying to learn how to solve simple acceleration-limited trajectory planning problems. I'm working in C++ and I've been using the Eigen library to do linear systems solving. I'm doing the ...
3
votes
1answer
73 views
Geometric Programming - symbolic version
I am interested in finding minimizers of functionals of the type $\sum x^ay^bz^c$ where the exponents are 1, 0 or -1. I have codes to find such minimizers when they exist up to machine precision, ...
1
vote
0answers
44 views
Finite dimensional optimization problem over dynamical system
I am interested in solving numerically the following mathematical problem
Consider an ode of the form
$$
\dot q(t) = f(q(t),t_1,\ldots, t_N),\qquad t\in [0,T],
$$
where $q\in \mathbb{R}^n$ is the ...
2
votes
1answer
71 views
Nonlinear conjugate gradient with orthogonality constraint
I have to solve a set of nonlinear optimization problems in the subspace defined as the orthogonal space to a given vector.
More precisely,
$$
\arg\min f(\vec x) \qquad \text{with} \qquad \vec x \...
1
vote
0answers
96 views
Connection between piecewise linear basis functions and RELU activation function
ReLU activation is defined as follows
$$\sigma(x)=\max(0, x).$$
Let's assume that I have deep network of 1 hidden layer, than output from my layer has form
$$ f(x)= \sigma(Wx +b), $$
where matrix W ...
1
vote
1answer
46 views
Research articles on MultiObjective Non-Linear Programming (MONLP)
I'm looking for papers dealing with multi-objective non-linear programming which could help me implement an algorithm to solve my problem.
My problem is :
Maximize $f(x) = c \cdot x$, while ...
-3
votes
1answer
94 views
0
votes
1answer
42 views
Slightly change two vectors to satisfy a constraint
$\vec{a}\cdot\vec{b} \approx c$
$\vec{\alpha} \cdot \vec{\beta} = c$
$\vec{\alpha}$ is close to $\vec{a}$ and $\vec{\beta}$ is close to $\vec{b}$
Given $\vec{a}$, $\vec{b}$ and c, how to find $\vec{\...
1
vote
1answer
174 views
Why does Newton's method with Linear Equality Constraints use KKT condition?
Goal: Optimize convex function $f(\vec{x})$ subjected to constraint $A\vec{x} = \vec{b}$ starting at a point $\vec{x}_0$ that satisfies the constraint.
The problem only has equality constraint. Why ...
1
vote
1answer
205 views
Piecewise-Linear Quadratic Optimization for an “Almost Convex” Problem
I have a 7-14 dimensional piecewise linear cost function I'd like to minimize with two quadratic terms of the form:
$$
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
268 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
191 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?
$$
\min_{X} \lVert A
\begin{bmatrix}
X & X^2
\...
1
vote
0answers
67 views
Nonlinear least square optimization
Problem description
Given data at many time instance $t$,
$$\min _{\alpha, \Lambda, \beta} \lVert y(t) - \alpha e^{\Lambda t} \beta \rVert_F$$
with $$ \lVert \alpha \rVert_2^F = 1 $$
where $y(t) \...
2
votes
1answer
30 views
Constraints 'exactly/at most one non-zero element' without binary variables
In a much larger MINLP problem, I have set of variables $\{a_{ij}\}_{m,n}$, such that $0 \leq a_{ij} \leq 1 $ for all $i$, $j$, which I could think of as a matrix, for which I have two requirements:
...
0
votes
1answer
57 views
reduced system: primal-dual interior point method for nonconvex constrained problem
When solving a reduced KKT system of a nonlinear (and nonconvex) constrained program after eliminating slack and dual variables, how do we actually take the next step in a primal-dual method?
For ...
3
votes
1answer
118 views
Obtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints
I would like to obtain a feasible solution for an under-determined system of linear equations,
$$Ax=b$$
where, $A \in \mathbb{R}^{7\times9}, \, x \in \mathbb{R}^{9\times1}\text{and } b\in\mathbb{R}^...
2
votes
0answers
51 views
Nonlinear Sylvester-Like Equation
Maybe you can point me to some results already developed for this.
I have to solve for $X$ the following "Sylvester-like" equation:
$$ AX - XB = F(X)$$
where $A\in\mathbb{R}^{a\times n}$, $B\in\...
1
vote
0answers
71 views
Nonlinear global optimization algorithm that can use dynamic programming
I've asked this question on stackoverflow 2 weeks ago, but, judging by zero response, that probably was the wrong forum. Therefore copying it here:
Let F0,...,Fn ...
0
votes
1answer
62 views
Why can't we discretize continuous domains in distributed non-convex constraint optimization problems?
Consider a non-convex distributed optimization problem.
We have $X$ = a set of $n$ decision variables: $x_i$ where $i=1..n$ and $x_i \in R$, the set of Reals.
We have $F$ = a set of $m$ constraint ...
1
vote
1answer
116 views
Correct way to model an embedded reinforcement (non linear FEM)?
I need to add to an existing FEM solver some embedded reinforcement element. This would give me the possibility to model/solve concrete structure (reinforced with steel rebar) taking into account the ...
3
votes
1answer
217 views
Why do active set methods or the simplex method pivot only one variable at a time?
Why do active set methods or the simplex method pivot only one variable at a time? Ostensibly, we could add multiple columns to the basis during pivoting, but the standard presentation of the methods ...
1
vote
2answers
88 views
How can i solve this non-convex multi-variable optimization problem?
I want to solve the following optimization problem:
$$\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$$
in which, $Y$ and $Z$ are data matrices and ...
5
votes
1answer
89 views
$L_2$ projection with integer constraints and prescribed sum
Suppose I am given a vector $v^0\in\mathbb{R}^n$ and integers $k,\ell\in\mathbb{Z}$. Assuming $k$ is close to zero (e.g. $0\leq k\leq5$), is there an algorithm for solving the following integer ...
1
vote
0answers
55 views
Object-oriented non-linear solving in python
I'd like to build a system in Python, consisting of (broadly speaking) objects which are internally described with (not necessarily just linear) equations, that I can connect with each other - similar ...
1
vote
0answers
68 views
Minimizing the products of variables
My problem
Maximize
$$\min_{i} \{\ c_i \cdot \prod_{j \in A(i)} {x_{j}} \prod_{j \in B(i)} {y_{j}} \} $$
Subject to
\begin{align}
&\sum_{j \in C(k)} x_{j} = 1,\ \forall k \\
&l \leq x_{j}...
2
votes
1answer
182 views
Optimization of multiple functions
I have 3 functions which consist of 6 variables $p_1,p_2,p_3,p_4,p_5,p_6$. The value of each function is equal to $x$ (say):
\begin{align}
f_1 &= \operatorname{sign}(2-p_1) \sqrt{|2-p_1|} + \...
1
vote
0answers
197 views
Constrained optimization: Stationary point vs. Nash point
1s question: definition of stationary point for constrained optimization
As far as I know, a stationary point of a constrained optimization problem is a stationary point of the Lagrangian (that has ...
1
vote
0answers
46 views
How is the Gastner-Newman equation implemented to create value-by-area cartograms?
There is a paper called "Density-equalizing map projections: Diffusion-based algorithm and applications" by Michael T. Gastner and M. E. J. Newman, which explains their algorithm (which is based in ...
0
votes
1answer
264 views
Example Problem to Demonstrate BiCGStab
So our team has been able to code up a BiCGStab implementation for a class project, and we'd like a potential example problem to try it out on.
So far, we've talked about a 1D Laplacian with Neumann ...
5
votes
1answer
130 views
Largest hypercuboid inside a polyhedron
Given a polyhedron $\mathbf{Ax} \leq \mathbf{b}$, how to find the largest hypercuboid, with unknown center $\mathbf{x_{0}}$ and side lengths $2\epsilon_{i}$, which are aligned along the co-ordinate ...
0
votes
1answer
5k views
Backtracking-Armijo Line Search Algorithm
EDIT based on comments below:
I add the mathematical formulation of my problem below. I am trying to solve an equation of the form
$$
\partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv ...
1
vote
2answers
42 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 ...
3
votes
2answers
64 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
82 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 ...
0
votes
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
63 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
92 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
109 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 ...
2
votes
1answer
205 views
Minimizing linear objective on intersection of convex sets
Suppose I wish to solve the following optimization problem:
$$
\begin{array}{rl}
\min_{\mu\in\mathbb{R}^n} &\mu^\top c\\
\textrm{subject to} & \mu\in C_1\cap C_2\cap\cdots\cap C_k,
\end{array}
...
2
votes
0answers
204 views
Using Line Search Method for Constrained Optimization
Suppose we have a $f(x)$ to be minimized (we only know that $f(x)$ is three-differentiable), and a feasible, convex set of $S$ such that all $x$ belong to $S$. Using line search method, how we can "...
1
vote
0answers
161 views
Numerically solving a system of stiff nonlinear PDEs
I am attempting to numerically solve the following:
\begin{align}
\frac {\partial y_1}{\partial t} &= i(y_2y_3 - y_2^*y_3^*) - y_1 \tag{1}\\
\frac {\partial y_2}{\partial t} &= y_1^*y_3 - y_2 ...
1
vote
0answers
113 views
Solving a large system of nonlinear equations, where timeseries are the unknown
I am trying to solve a problem, which I find quite hard, like, headache-hard. I have to solve the following set of $M$ nonlinear equations:
$$F(X)=\begin{bmatrix}f_1 (X)\\f_2 (X)\\...\\f_M (X)\\ \end{...
1
vote
0answers
128 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
127 views
Scaling a vector-valued non-linear function for numerical optimization/minimization [closed]
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
43 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
338 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
133 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}\...
