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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.

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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 ...
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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 ...
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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
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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
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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, ...
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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 ...
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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 \...
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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 ...
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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 ...
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1answer
94 views
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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{\...
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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 ...
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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 ...
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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 ...
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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 \...
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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) \...
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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: ...
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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 ...
3
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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}^...
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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\...
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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 ...
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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 ...
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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
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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 ...
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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
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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 ...
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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 ...
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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}...
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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|} + \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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
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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 ...
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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, $d_{ij}=||x_j-x_i||\in\text{dom}_a$. ...
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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 ...
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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} ...
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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 "...
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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 ...
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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{...
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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. ...
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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 ...
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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 ...
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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\...
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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}\...