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
3
votes
0answers
66 views
Large-scale optimization of nonlinear equations
I'm looking to find a computationally efficient solution to a large system of nonlinear equations. I'm trying to maximize the following function:
$$ f(\vec{x}) = \sum_i^N C_i (x_i-A_i)x_i^{\epsilon_{...
0
votes
0answers
30 views
Method for implementing QP solver with matrix terms?
I am trying to implement (in code) a QP solver for the following equation:
$$\min_{u} u^{T} Wu$$
$$s.t. \; \beta u = \tau_{ref}$$
$$ Au \leq b $$
See this document, section 5.1 (Page 35)
$u$ is a ...
3
votes
2answers
110 views
MINLP with GEKKO - Modeling discrete variables
I'm trying to define a MINLP optimization problem with GEKKO in Python, and I want to use some variables with fixed values.
For my first variable, x1, I need to define the following values (as would ...
1
vote
1answer
78 views
Preconditionning for solving a non-linear system of equations with least squares
I am trying to solve a large system of non-linear equations (about a few hundred equations and variable but with less variable than equations). Given that the system is really sparse and large I am ...
3
votes
0answers
27 views
How to set up and solve acceleration-limited trajectory optimization problems?
I've been trying to learn how to solve simple acceleration-limited trajectory planning problems. I'm working in C++ and I've been using the Eigen library to do linear systems solving. I'm doing the ...
3
votes
1answer
74 views
Geometric Programming - symbolic version
I am interested in finding minimizers of functionals of the type $\sum x^ay^bz^c$ where the exponents are 1, 0 or -1. I have codes to find such minimizers when they exist up to machine precision, ...
1
vote
0answers
45 views
Finite dimensional optimization problem over dynamical system
I am interested in solving numerically the following mathematical problem
Consider an ode of the form
$$
\dot q(t) = f(q(t),t_1,\ldots, t_N),\qquad t\in [0,T],
$$
where $q\in \mathbb{R}^n$ is the ...
2
votes
1answer
76 views
Nonlinear conjugate gradient with orthogonality constraint
I have to solve a set of nonlinear optimization problems in the subspace defined as the orthogonal space to a given vector.
More precisely,
$$
\arg\min f(\vec x) \qquad \text{with} \qquad \vec x \...
1
vote
0answers
97 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
49 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
95 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
206 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
216 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:
$$
f(X) = X^tCX + d \sum_i |x_i-x^*_i|^2 + \sum_i P_i(x_i-x^0_i)
$$
$$ \sum_i x_i ...
2
votes
0answers
269 views
Convergence of a very large non-linear least squares optimization
(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it)
I'm trying to solve the following problem: I have a lot (~80000) surface ...
4
votes
1answer
191 views
How to solve the following Frobenius norm-minimization problem?
Background
We know how to solve the following minimization problem
$$
\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
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) \...
2
votes
1answer
31 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
124 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
52 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
72 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
63 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
120 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
229 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
89 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
214 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
206 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
47 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
274 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
134 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
$$
\partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t) \equiv ...
1
vote
2answers
42 views
Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available
I have a computationally expensive objective function, whose analytical form is not available. The only input argument to the objective function is an integer variable. The goal is to compute the ...
3
votes
2answers
65 views
Gradient of function after renormalization of variables
I have to minimize a function $f(\mathbf{x})$, where the vector $\mathbf{x}\in\mathbb{R}^n$ satisfies $|\mathbf{x}|=1$. So I tweaked the code of $f$ so that it renormalizes $\mathbf{x}$ as the first ...
0
votes
1answer
86 views
Perturbation in bounds given the perturbation to constraints
Given a feasibility problem with both inequality and equality constraints, I'm interested in the sensitivity of the bounds of the region to changes in the constraints. To help with answering the ...
2
votes
0answers
64 views
Global optimization with known distributions of some variables
I'm solving simple single-objective multidimensional global optimization problem using various stochastic algorithms like Monte-Carlo, GA and other evolutionary approaches. The task is formulated as ...
1
vote
0answers
93 views
2D Multilateration with constraints as distance information
In my current problem, I am looking for an algorithm to reconstruct the position of multiple points in the 2D euclidean plane with incomplete distance information, $d_{ij}=||x_j-x_i||\in\text{dom}_a$.
...
1
vote
1answer
110 views
SOCP: Recovering primal from dual
Consider the following second-order cone program (SOCP):
$$
\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}
$$
Suppose I solve ...
2
votes
1answer
209 views
Minimizing linear objective on intersection of convex sets
Suppose I wish to solve the following optimization problem:
$$
\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
206 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
166 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
130 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
128 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
44 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
341 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
135 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}\...
